TPTP Problem File: SLH0510^1.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_00191_006073__6344042_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1581 ( 499 unt; 292 typ; 0 def)
% Number of atoms : 4330 (1143 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 13888 ( 316 ~; 55 |; 265 &;11142 @)
% ( 0 <=>;2110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 1868 (1868 >; 0 *; 0 +; 0 <<)
% Number of symbols : 269 ( 266 usr; 18 con; 0-4 aty)
% Number of variables : 4565 ( 364 ^;4052 !; 149 ?;4565 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:56.750
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (266)
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thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_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__Epistemic____Logic__Ofm_Itf__a_J,type,
image_4449434806354059013c_fm_a: ( epistemic_fm_a > epistemic_fm_a ) > set_Epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_4430763730358542731c_fm_a: ( epistemic_fm_a > list_Epistemic_fm_a ) > set_Epistemic_fm_a > set_li769143395467472256c_fm_a ).
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__Epistemic____Logic__Ofm_Itf__a_J_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_2304902010338030059c_fm_a: ( epistemic_fm_a > set_li769143395467472256c_fm_a ) > set_Epistemic_fm_a > set_se4933744490392479200c_fm_a ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__a_J_001tf__a,type,
image_7228943142519265129fm_a_a: ( epistemic_fm_a > a ) > set_Epistemic_fm_a > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
image_7959425827855735423c_fm_a: ( list_Epistemic_fm_a > epistemic_fm_a ) > set_li769143395467472256c_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_7387318756023638021c_fm_a: ( list_Epistemic_fm_a > list_Epistemic_fm_a ) > set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_971165786557580383c_fm_a: ( list_Epistemic_fm_a > set_Epistemic_fm_a ) > set_li769143395467472256c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_1280919820078628965c_fm_a: ( list_Epistemic_fm_a > set_li769143395467472256c_fm_a ) > set_li769143395467472256c_fm_a > set_se4933744490392479200c_fm_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_7951968421222759263c_fm_a: ( list_l6083326122719238310c_fm_a > set_li769143395467472256c_fm_a ) > set_li1092409891198325638c_fm_a > set_se4933744490392479200c_fm_a ).
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__Epistemic____Logic__Ofm_Itf__a_J,type,
image_3376480847214465317c_fm_a: ( set_Epistemic_fm_a > epistemic_fm_a ) > set_se5208064806568342746c_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_1284921450708845483c_fm_a: ( set_Epistemic_fm_a > list_Epistemic_fm_a ) > set_se5208064806568342746c_fm_a > set_li769143395467472256c_fm_a ).
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__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_4454716909333697547c_fm_a: ( set_Epistemic_fm_a > set_li769143395467472256c_fm_a ) > set_se5208064806568342746c_fm_a > set_se4933744490392479200c_fm_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001tf__a,type,
image_4497856574827002249fm_a_a: ( set_Epistemic_fm_a > a ) > set_se5208064806568342746c_fm_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_8805014153957852165c_fm_a: ( set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a ) > set_se4933744490392479200c_fm_a > set_se4933744490392479200c_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__Epistemic____Logic__Ofm_Itf__a_J,type,
image_1898462159489238305c_fm_a: ( a > epistemic_fm_a ) > set_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_5290880835153512871c_fm_a: ( a > list_Epistemic_fm_a ) > set_a > set_li769143395467472256c_fm_a ).
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_Oimage_001tf__a_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
image_7867170089500699143c_fm_a: ( a > set_li769143395467472256c_fm_a ) > set_a > set_se4933744490392479200c_fm_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
insert6567282714061225712c_fm_a: list_Epistemic_fm_a > set_li769143395467472256c_fm_a > set_li769143395467472256c_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_a2: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
the_el2173195877760541071c_fm_a: set_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__a,type,
stalnaker_conjunct_a: list_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
accp_Epistemic_fm_a: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
accp_l7472382826029824047c_fm_a: ( list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a > $o ) > list_l6083326122719238310c_fm_a > $o ).
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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member5906877432388582473c_fm_a: list_Epistemic_fm_a > set_li769143395467472256c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member2328277852289759785c_fm_a: list_s580375451141968640c_fm_a > set_li4204741992506657632c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member2517045796615263401c_fm_a: set_li769143395467472256c_fm_a > set_se4933744490392479200c_fm_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_A,type,
a2: epistemic_fm_a > $o ).
thf(sy_v_G,type,
g: list_Epistemic_fm_a ).
thf(sy_v_i,type,
i: a ).
% Relevant facts (1275)
thf(fact_0_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_1_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_2_K__conjunction__in__mult,axiom,
! [A: epistemic_fm_a > $o,I: a,G: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( stalnaker_conjunct_a @ G ) ) @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_3_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_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_12_K__conjunct__imply,axiom,
! [A: epistemic_fm_a > $o,G: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% K_conjunct_imply
thf(fact_13_K__imply__conjunct,axiom,
! [A: epistemic_fm_a > $o,G: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) ) ) ).
% K_imply_conjunct
thf(fact_14_K__conjunction__in,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_Con_a @ P @ Q ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_conjunction_in
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_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_17_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_18_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_19_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_20_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_21_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_22_K__distrib__K__imp,axiom,
! [A: epistemic_fm_a > $o,I: a,G: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ ( epistemic_imply_a @ G @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_23_K__conj__imply__factor,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) @ R ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_24_K__conjunction__out,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 @ Q ) ) @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_25_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ! [G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ 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_26_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ 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_27_imply__conjunct,axiom,
! [G: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G @ P ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) ) ) ).
% imply_conjunct
thf(fact_28_conjunct__imply,axiom,
! [G: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) @ ( epistemic_imply_a @ G @ P ) ) ) ).
% conjunct_imply
thf(fact_29_K__thm,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_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 ) ) ) ).
% K_thm
thf(fact_30_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_31_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_32_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G: 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 ) @ G ) ) @ Q ) ) ).
% K_mp
thf(fact_33_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_34_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_35_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_36_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_37_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_38_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_39_duality__taut,axiom,
! [I: a,P: epistemic_fm_a,Q: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( 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_40_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_41_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_42_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_43_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_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_45_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_46_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47_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_48_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_49_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_50_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_51_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_52_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_53_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% K_Boole
thf(fact_54_eval_Osimps_I5_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
=> ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_55_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H4: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H4 @ ( epistemic_K_a @ I @ P ) )
= ( H4 @ ( epistemic_K_a @ I @ P ) ) ) ).
% eval.simps(6)
thf(fact_56_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_57_eval_Osimps_I4_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
& ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_58_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_59_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_60_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_61_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G: 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 @ G ) ) @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G ) ) @ R ) ) ) ).
% K_swap
thf(fact_62_conjunct_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a] :
( ( stalnaker_conjunct_a @ ( cons_Epistemic_fm_a @ P @ Ps ) )
= ( epistemic_Con_a @ P @ ( stalnaker_conjunct_a @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_63_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_64_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_65_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_66_list_Oinject,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a,Y21: epistemic_fm_a,Y22: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X21 @ X22 )
= ( cons_Epistemic_fm_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_67_list_Osimps_I9_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) )
= ( cons_l8134865115577406678c_fm_a @ ( F @ X21 ) @ ( map_li3080774272885743684c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_68_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( cons_l8134865115577406678c_fm_a @ ( F @ X21 ) @ ( map_Ep1988204514693272906c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_69_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( cons_Epistemic_fm_a @ ( F @ X21 ) @ ( map_Ep7084560364594560580c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_70_Cons__eq__map__D,axiom,
! [X3: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X3 @ Xs )
= ( map_li3080774272885743684c_fm_a @ F @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( X3
= ( F @ Z ) )
& ( Xs
= ( map_li3080774272885743684c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_71_Cons__eq__map__D,axiom,
! [X3: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X3 @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ F @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( X3
= ( F @ Z ) )
& ( Xs
= ( map_Ep1988204514693272906c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_72_Cons__eq__map__D,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( X3
= ( F @ Z ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_73_map__eq__Cons__D,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_li3080774272885743684c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_74_map__eq__Cons__D,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_Ep1988204514693272906c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_75_map__eq__Cons__D,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_76_Cons__eq__map__conv,axiom,
! [X3: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X3 @ Xs )
= ( map_li3080774272885743684c_fm_a @ F @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs
= ( map_li3080774272885743684c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_77_Cons__eq__map__conv,axiom,
! [X3: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X3 @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ F @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs
= ( map_Ep1988204514693272906c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_78_Cons__eq__map__conv,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_79_map__eq__Cons__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_li3080774272885743684c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_80_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_Ep1988204514693272906c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_81_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_82_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_83_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_84_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_85_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_86_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_87_not__Cons__self2,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_88_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_89_order__refl,axiom,
! [X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ X3 ) ).
% order_refl
thf(fact_90_order__refl,axiom,
! [X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ X3 ) ).
% order_refl
thf(fact_91_order__refl,axiom,
! [X3: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ X3 @ X3 ) ).
% order_refl
thf(fact_92_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_93_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_94_dual__order_Orefl,axiom,
! [A2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_95_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_96_le__funD,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G5 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G5 @ X3 ) ) ) ).
% le_funD
thf(fact_97_le__funE,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G5 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G5 @ X3 ) ) ) ).
% le_funE
thf(fact_98_le__funI,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G5 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G5 ) ) ).
% le_funI
thf(fact_99_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G2: epistemic_fm_a > $o] :
! [X: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_fun_def
thf(fact_100_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_101_map__eq__map__tailrec,axiom,
map_Ep1988204514693272906c_fm_a = map_ta2019361550626295956c_fm_a ).
% map_eq_map_tailrec
thf(fact_102_map__eq__map__tailrec,axiom,
map_li3080774272885743684c_fm_a = map_ta7821446597547650702c_fm_a ).
% map_eq_map_tailrec
thf(fact_103_map__eq__map__tailrec,axiom,
map_Ep7084560364594560580c_fm_a = map_ta875960198429730446c_fm_a ).
% map_eq_map_tailrec
thf(fact_104_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_105_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_106_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_107_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_108_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_109_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_110_order__antisym__conv,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( ( ord_le2931979837777675552c_fm_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_111_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_112_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_113_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_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_115_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_116_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_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_118_ord__le__eq__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_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_119_ord__le__eq__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_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_121_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_122_ord__eq__le__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,F: ( epistemic_fm_a > $o ) > set_li769143395467472256c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_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_124_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_125_ord__eq__le__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_126_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: set_li769143395467472256c_fm_a > epistemic_fm_a > $o,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_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_127_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: set_li769143395467472256c_fm_a > set_Epistemic_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_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_128_ord__eq__le__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_129_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_130_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_131_order__eq__refl,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( X3 = Y )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_132_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_133_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_134_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_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_136_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_137_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst2,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_141_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_142_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_143_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: set_li769143395467472256c_fm_a > epistemic_fm_a > $o,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_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_145_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_146_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: set_li769143395467472256c_fm_a > set_Epistemic_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A2: set_li769143395467472256c_fm_a,F: ( epistemic_fm_a > $o ) > set_li769143395467472256c_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A2: set_li769143395467472256c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_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_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A2: set_li769143395467472256c_fm_a,F: set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] : ( Y3 = Z3 ) )
= ( ^ [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_151_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( Y3 = Z3 ) )
= ( ^ [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_152_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_li769143395467472256c_fm_a,Z3: set_li769143395467472256c_fm_a] : ( Y3 = Z3 ) )
= ( ^ [A4: set_li769143395467472256c_fm_a,B2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A4 @ B2 )
& ( ord_le2931979837777675552c_fm_a @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_153_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_154_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_155_antisym,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_156_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_157_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_158_dual__order_Otrans,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ C @ B )
=> ( ord_le2931979837777675552c_fm_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_159_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_160_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_161_dual__order_Oantisym,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_162_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] : ( Y3 = Z3 ) )
= ( ^ [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_163_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( Y3 = Z3 ) )
= ( ^ [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_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_li769143395467472256c_fm_a,Z3: set_li769143395467472256c_fm_a] : ( Y3 = Z3 ) )
= ( ^ [A4: set_li769143395467472256c_fm_a,B2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B2 @ A4 )
& ( ord_le2931979837777675552c_fm_a @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_order__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z4 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Z4 ) ) ) ).
% order_trans
thf(fact_166_order__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Z4 ) ) ) ).
% order_trans
thf(fact_167_order__trans,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ Y )
=> ( ( ord_le2931979837777675552c_fm_a @ Y @ Z4 )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Z4 ) ) ) ).
% order_trans
thf(fact_168_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_169_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_170_order_Otrans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_171_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_172_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_173_order__antisym,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ Y )
=> ( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_174_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_175_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_176_ord__le__eq__trans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_177_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_178_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_179_ord__eq__le__trans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( A2 = B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_180_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] : ( Y3 = Z3 ) )
= ( ^ [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_181_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( Y3 = Z3 ) )
= ( ^ [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_182_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_li769143395467472256c_fm_a,Z3: set_li769143395467472256c_fm_a] : ( Y3 = Z3 ) )
= ( ^ [X: set_li769143395467472256c_fm_a,Y4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X @ Y4 )
& ( ord_le2931979837777675552c_fm_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a,R: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ R ) ) ) ) ) ).
% K_DisE
thf(fact_191_conjunct_Osimps_I1_J,axiom,
( ( stalnaker_conjunct_a @ nil_Epistemic_fm_a )
= ( epistemic_Imp_a @ epistemic_FF_a @ epistemic_FF_a ) ) ).
% conjunct.simps(1)
thf(fact_192_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_193_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_194_Greatest__equality,axiom,
! [P4: set_li769143395467472256c_fm_a > $o,X3: set_li769143395467472256c_fm_a] :
( ( P4 @ X3 )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ Y2 @ X3 ) )
=> ( ( order_3649805357607789401c_fm_a @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_195_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_196_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_197_GreatestI2__order,axiom,
! [P4: set_li769143395467472256c_fm_a > $o,X3: set_li769143395467472256c_fm_a,Q4: set_li769143395467472256c_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ Y2 @ X3 ) )
=> ( ! [X2: set_li769143395467472256c_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_li769143395467472256c_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le2931979837777675552c_fm_a @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_3649805357607789401c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_198_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs ) )
=> ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_199_member__rec_I1_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( member6038508265109909045c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member6038508265109909045c_fm_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_200_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_201_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_202_le__rel__bool__arg__iff,axiom,
( ord_le378634792081309101c_fm_a
= ( ^ [X4: $o > set_li769143395467472256c_fm_a,Y6: $o > set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2931979837777675552c_fm_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_203_verit__comp__simplify1_I2_J,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_204_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_205_verit__comp__simplify1_I2_J,axiom,
! [A2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_206_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_207_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_208_map__is__Nil__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_209_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_210_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_211_Nil__is__map__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( map_li3080774272885743684c_fm_a @ F @ Xs ) )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_212_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_213_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ A2 )
= nil_li2451196919128234278c_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_214_list_Omap__disc__iff,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A2: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ A2 )
= nil_li2451196919128234278c_fm_a )
= ( A2 = nil_li2451196919128234278c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_215_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ A2 )
= nil_Epistemic_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_216_map__eq__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Xs ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X )
= ( G5 @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_217_map__eq__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Xs ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X )
= ( G5 @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_218_map__eq__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Xs ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X )
= ( G5 @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_219_subset__code_I1_J,axiom,
! [Xs: list_s580375451141968640c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) @ B3 )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( member536094252920883875c_fm_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_220_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_221_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ B3 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_222_subset__code_I1_J,axiom,
! [Xs: list_l6083326122719238310c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( set_li8442223810127165109c_fm_a @ Xs ) @ B3 )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( member5906877432388582473c_fm_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_223_in__set__member,axiom,
! [X3: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
= ( member378061430901374997c_fm_a @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_224_in__set__member,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_225_in__set__member,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( member6038508265109909045c_fm_a @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_226_member__rec_I2_J,axiom,
! [Y: epistemic_fm_a] :
~ ( member6038508265109909045c_fm_a @ nil_Epistemic_fm_a @ Y ) ).
% member_rec(2)
thf(fact_227_transpose_Ocases,axiom,
! [X3: list_l6083326122719238310c_fm_a] :
( ( X3 != nil_li2451196919128234278c_fm_a )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ~ ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_228_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a,X21: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ X22 ) )
=> ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_229_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a2 @ Y @ ( set_a2 @ X22 ) )
=> ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_230_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_a,X22: list_Epistemic_fm_a,X21: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ X22 ) )
=> ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_231_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a] : ( member536094252920883875c_fm_a @ X21 @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_232_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_233_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X21 @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_234_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A2: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ A2 ) )
=> ( ! [Z22: list_s580375451141968640c_fm_a] :
( A2
!= ( cons_s4394448069393209776c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: list_s580375451141968640c_fm_a] :
( ( A2
= ( cons_s4394448069393209776c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_235_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a2 @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z22: list_a] :
( A2
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_236_list_Oset__cases,axiom,
! [E: epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ A2 ) )
=> ( ! [Z22: list_Epistemic_fm_a] :
( A2
!= ( cons_Epistemic_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: list_Epistemic_fm_a] :
( ( A2
= ( cons_Epistemic_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_237_set__ConsD,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_238_set__ConsD,axiom,
! [Y: a,X3: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_239_set__ConsD,axiom,
! [Y: epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_240_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_241_set__subset__Cons,axiom,
! [Xs: list_l6083326122719238310c_fm_a,X3: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( set_li8442223810127165109c_fm_a @ Xs ) @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_242_ex__map__conv,axiom,
! [Ys: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( ? [Xs3: list_Epistemic_fm_a] :
( Ys
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs3 ) ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ? [Y4: epistemic_fm_a] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_243_ex__map__conv,axiom,
! [Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( ? [Xs3: list_l6083326122719238310c_fm_a] :
( Ys
= ( map_li3080774272885743684c_fm_a @ F @ Xs3 ) ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ? [Y4: list_Epistemic_fm_a] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_244_ex__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ? [Xs3: list_Epistemic_fm_a] :
( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs3 ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ? [Y4: epistemic_fm_a] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_245_map__cong,axiom,
! [Xs: list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_246_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_247_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_248_map__idI,axiom,
! [Xs: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_249_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_250_map__idI,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_251_map__idI,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_252_map__ext,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_253_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_254_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_255_list_Omap__ident__strong,axiom,
! [T: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( set_se8855941934996405007c_fm_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_256_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_257_list_Omap__ident__strong,axiom,
! [T: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_258_list_Omap__ident__strong,axiom,
! [T: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_259_list_Oinj__map__strong,axiom,
! [X3: list_l6083326122719238310c_fm_a,Xa: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Fa: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a,Za: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X3 ) )
=> ( ( member5906877432388582473c_fm_a @ Za @ ( set_li8442223810127165109c_fm_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li3080774272885743684c_fm_a @ F @ X3 )
= ( map_li3080774272885743684c_fm_a @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_260_list_Oinj__map__strong,axiom,
! [X3: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Fa: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Ep1988204514693272906c_fm_a @ F @ X3 )
= ( map_Ep1988204514693272906c_fm_a @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_261_list_Oinj__map__strong,axiom,
! [X3: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Fa: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_262_list_Omap__cong0,axiom,
! [X3: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X3 ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ X3 )
= ( map_li3080774272885743684c_fm_a @ G5 @ X3 ) ) ) ).
% list.map_cong0
thf(fact_263_list_Omap__cong0,axiom,
! [X3: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ X3 )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ X3 ) ) ) ).
% list.map_cong0
thf(fact_264_list_Omap__cong0,axiom,
! [X3: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ X3 ) ) ) ).
% list.map_cong0
thf(fact_265_list_Omap__cong,axiom,
! [X3: list_l6083326122719238310c_fm_a,Ya: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( X3 = Ya )
=> ( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ X3 )
= ( map_li3080774272885743684c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_266_list_Omap__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( X3 = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ X3 )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_267_list_Omap__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( X3 = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_268_list_Odistinct_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( nil_Epistemic_fm_a
!= ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_269_list_OdiscI,axiom,
! [List: list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( List
= ( cons_Epistemic_fm_a @ X21 @ X22 ) )
=> ( List != nil_Epistemic_fm_a ) ) ).
% list.discI
thf(fact_270_list_Oexhaust,axiom,
! [Y: list_Epistemic_fm_a] :
( ( Y != nil_Epistemic_fm_a )
=> ~ ! [X212: epistemic_fm_a,X222: list_Epistemic_fm_a] :
( Y
!= ( cons_Epistemic_fm_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_271_remdups__adj_Ocases,axiom,
! [X3: list_Epistemic_fm_a] :
( ( X3 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_272_neq__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
= ( ? [Y4: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_273_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_Epistemic_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_Epistemic_fm_a @ nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys3: list_Epistemic_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys3 ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Y2: epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( P4 @ Xs2 @ Ys3 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys3 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_274_list__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xs2 != nil_Epistemic_fm_a )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_275_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% list.simps(8)
thf(fact_276_list_Osimps_I8_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% list.simps(8)
thf(fact_277_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% list.simps(8)
thf(fact_278_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_279_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_280_imply_Osimps_I1_J,axiom,
! [Q: epistemic_fm_a] :
( ( epistemic_imply_a @ nil_Epistemic_fm_a @ Q )
= Q ) ).
% imply.simps(1)
thf(fact_281_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_282_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_283_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_284_eval_Osimps_I3_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
| ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_285_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_286_subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ X2 @ B3 ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ B3 ) ) ).
% subsetI
thf(fact_287_subsetI,axiom,
! [A: set_a,B3: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_a2 @ X2 @ B3 ) )
=> ( ord_less_eq_set_a @ A @ B3 ) ) ).
% subsetI
thf(fact_288_subsetI,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ X2 @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ).
% subsetI
thf(fact_289_subsetI,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( member5906877432388582473c_fm_a @ X2 @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ).
% subsetI
thf(fact_290_subset__antisym,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_291_subset__antisym,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( ord_le2931979837777675552c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_292_the__elem__set,axiom,
! [X3: epistemic_fm_a] :
( ( the_el2173195877760541071c_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_293_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G: 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 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte5333283044364550848c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_294_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X5: a] :
( ( member_a2 @ X5 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte295617885132580260cs_a_a @ M3 @ X5 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_295_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G: 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 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_296_fm_Oexhaust,axiom,
! [Y: epistemic_fm_a] :
( ( Y != epistemic_FF_a )
=> ( ! [X23: list_char] :
( Y
!= ( epistemic_Pro_a @ X23 ) )
=> ( ! [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_297_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_298_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 ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_299_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_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X5: a] :
( ( member_a2 @ X5 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X5 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_300_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 ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_301_subset__code_I3_J,axiom,
~ ( ord_le3275665582123262618c_fm_a @ ( coset_Epistemic_fm_a @ nil_Epistemic_fm_a ) @ ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% subset_code(3)
thf(fact_302_subset__code_I3_J,axiom,
~ ( ord_le2931979837777675552c_fm_a @ ( coset_1387336592765253011c_fm_a @ nil_li2451196919128234278c_fm_a ) @ ( set_li8442223810127165109c_fm_a @ nil_li2451196919128234278c_fm_a ) ) ).
% subset_code(3)
thf(fact_303_fm_Oinject_I1_J,axiom,
! [X24: list_char,Y23: list_char] :
( ( ( epistemic_Pro_a @ X24 )
= ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.inject(1)
thf(fact_304_fm_Odistinct_I15_J,axiom,
! [X24: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_305_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_306_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X24 ) ) ).
% fm.distinct(1)
thf(fact_307_fm_Odistinct_I13_J,axiom,
! [X24: list_char,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_308_fm_Odistinct_I11_J,axiom,
! [X24: list_char,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_309_eval_Osimps_I2_J,axiom,
! [G5: list_char > $o,Uw: epistemic_fm_a > $o,X3: list_char] :
( ( epistemic_eval_a @ G5 @ Uw @ ( epistemic_Pro_a @ X3 ) )
= ( G5 @ X3 ) ) ).
% eval.simps(2)
thf(fact_310_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_311_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_312_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_313_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_314_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I: a] :
( ! [M2: episte6182337868402532512t_unit,X2: a] :
( ( member_a2 @ X2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X2 @ P ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_315_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_316_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_317_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_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_318_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_319_Collect__mono__iff,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_320_Collect__mono__iff,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) )
= ( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_321_Collect__mono__iff,axiom,
! [P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q4 ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_322_set__eq__subset,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( Y3 = Z3 ) )
= ( ^ [A3: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_323_set__eq__subset,axiom,
( ( ^ [Y3: set_li769143395467472256c_fm_a,Z3: set_li769143395467472256c_fm_a] : ( Y3 = Z3 ) )
= ( ^ [A3: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A3 @ B4 )
& ( ord_le2931979837777675552c_fm_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_324_subset__trans,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_325_subset__trans,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( ord_le2931979837777675552c_fm_a @ B3 @ C2 )
=> ( ord_le2931979837777675552c_fm_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_326_Collect__mono,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_327_Collect__mono,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_328_Collect__mono,axiom,
! [P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ! [X2: list_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_329_subset__refl,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_330_subset__refl,axiom,
! [A: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_331_subset__iff,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
! [T2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ T2 @ A3 )
=> ( member536094252920883875c_fm_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_332_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [T2: a] :
( ( member_a2 @ T2 @ A3 )
=> ( member_a2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_333_subset__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
! [T2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ T2 @ A3 )
=> ( member6642669571620171971c_fm_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_334_subset__iff,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
! [T2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ T2 @ A3 )
=> ( member5906877432388582473c_fm_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_335_equalityD2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( A = B3 )
=> ( ord_le3275665582123262618c_fm_a @ B3 @ A ) ) ).
% equalityD2
thf(fact_336_equalityD2,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( A = B3 )
=> ( ord_le2931979837777675552c_fm_a @ B3 @ A ) ) ).
% equalityD2
thf(fact_337_equalityD1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( A = B3 )
=> ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ).
% equalityD1
thf(fact_338_equalityD1,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( A = B3 )
=> ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ).
% equalityD1
thf(fact_339_subset__eq,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( member536094252920883875c_fm_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_340_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [X: a] :
( ( member_a2 @ X @ A3 )
=> ( member_a2 @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_341_subset__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( member6642669571620171971c_fm_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_342_subset__eq,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ A3 )
=> ( member5906877432388582473c_fm_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_343_equalityE,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( A = B3 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ~ ( ord_le3275665582123262618c_fm_a @ B3 @ A ) ) ) ).
% equalityE
thf(fact_344_equalityE,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( A = B3 )
=> ~ ( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ~ ( ord_le2931979837777675552c_fm_a @ B3 @ A ) ) ) ).
% equalityE
thf(fact_345_subsetD,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( member536094252920883875c_fm_a @ C @ A )
=> ( member536094252920883875c_fm_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_346_subsetD,axiom,
! [A: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_347_subsetD,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( member6642669571620171971c_fm_a @ C @ A )
=> ( member6642669571620171971c_fm_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_348_subsetD,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,C: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( member5906877432388582473c_fm_a @ C @ A )
=> ( member5906877432388582473c_fm_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_349_in__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_350_in__mono,axiom,
! [A: set_a,B3: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( member_a2 @ X3 @ A )
=> ( member_a2 @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_351_in__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_352_in__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,X3: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( member5906877432388582473c_fm_a @ X3 @ A )
=> ( member5906877432388582473c_fm_a @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_353_subset__code_I2_J,axiom,
! [A: set_se5208064806568342746c_fm_a,Ys: list_s580375451141968640c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ ( coset_774667214334347501c_fm_a @ Ys ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Ys ) )
=> ~ ( member536094252920883875c_fm_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_354_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_355_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( coset_Epistemic_fm_a @ Ys ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ~ ( member6642669571620171971c_fm_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_356_subset__code_I2_J,axiom,
! [A: set_li769143395467472256c_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ ( coset_1387336592765253011c_fm_a @ Ys ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ~ ( member5906877432388582473c_fm_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_357_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_358_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_359_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_360_strong__soundness_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_361_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_362_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_363_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_364_main_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_365_strong__completeness_092_060_094sub_062T,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_366_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_367_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_368_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_369_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_370_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_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_371_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_372_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_373_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_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_374_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_375_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_376_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_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_377_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_378_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_379_neg__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ 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_380_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_381_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_382_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_383_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_384_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_385_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_386_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_387_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_388_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_389_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_390_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_391_pos__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ 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_392_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_393_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_394_truth,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_395_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_396_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_397_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_398_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( 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_399_product__lists_Osimps_I1_J,axiom,
( ( produc1391074687832863881c_fm_a @ nil_li2451196919128234278c_fm_a )
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ nil_li2451196919128234278c_fm_a ) ) ).
% product_lists.simps(1)
thf(fact_400_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_401_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_402_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( 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_403_fm_Orel__cases,axiom,
! [R2: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ 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 @ R2 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 @ R2 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 @ R2 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 ) )
=> ( ( R2 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_404_map__ident,axiom,
( ( map_li3080774272885743684c_fm_a
@ ^ [X: list_Epistemic_fm_a] : X )
= ( ^ [Xs3: list_l6083326122719238310c_fm_a] : Xs3 ) ) ).
% map_ident
thf(fact_405_map__ident,axiom,
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X: epistemic_fm_a] : X )
= ( ^ [Xs3: list_Epistemic_fm_a] : Xs3 ) ) ).
% map_ident
thf(fact_406_list_Omap__ident,axiom,
! [T: list_l6083326122719238310c_fm_a] :
( ( map_li3080774272885743684c_fm_a
@ ^ [X: list_Epistemic_fm_a] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_407_list_Omap__ident,axiom,
! [T: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X: epistemic_fm_a] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_408_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
= ( ^ [Y3: epistemic_fm_a,Z3: epistemic_fm_a] : ( Y3 = Z3 ) ) ) ).
% fm.rel_eq
thf(fact_409_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_410_fm_Orel__mono,axiom,
! [R2: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R2 @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R2 ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_411_Collect__subset,axiom,
! [A: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_412_Collect__subset,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_413_Collect__subset,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_414_Collect__subset,axiom,
! [A: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_415_less__eq__set__def,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B4: 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 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_416_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ A3 )
@ ^ [X: a] : ( member_a2 @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_417_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B4: 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 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_418_less__eq__set__def,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ord_le5289359451089373629fm_a_o
@ ^ [X: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X @ A3 )
@ ^ [X: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_419_fm_Orel__inject_I5_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R2 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_420_fm_Orel__intros_I5_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_421_fm_Orel__inject_I6_J,axiom,
! [R2: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R2 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R2 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_422_fm_Orel__intros_I6_J,axiom,
! [R2: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R2 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_423_fm_Octr__transfer_I1_J,axiom,
! [R2: a > a > $o] : ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_424_fm_Orel__intros_I4_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_425_fm_Orel__inject_I4_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R2 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_426_fm_Orel__inject_I2_J,axiom,
! [R2: a > a > $o,X24: list_char,Y23: list_char] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.rel_inject(2)
thf(fact_427_fm_Orel__intros_I2_J,axiom,
! [X24: list_char,Y23: list_char,R2: a > a > $o] :
( ( X24 = Y23 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) ) ) ).
% fm.rel_intros(2)
thf(fact_428_fm_Orel__inject_I3_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R2 @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_429_fm_Orel__intros_I3_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_430_fm_Orel__distinct_I30_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_431_fm_Orel__distinct_I29_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_432_fm_Orel__distinct_I7_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_433_fm_Orel__distinct_I8_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_434_fm_Orel__distinct_I9_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_435_fm_Orel__distinct_I10_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_436_fm_Orel__distinct_I25_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_437_fm_Orel__distinct_I26_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_438_fm_Orel__distinct_I27_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_439_fm_Orel__distinct_I28_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_440_fm_Orel__distinct_I16_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(16)
thf(fact_441_fm_Orel__distinct_I15_J,axiom,
! [R2: a > a > $o,X24: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_442_fm_Orel__distinct_I22_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_443_fm_Orel__distinct_I21_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_444_fm_Orel__distinct_I18_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(18)
thf(fact_445_fm_Orel__distinct_I17_J,axiom,
! [R2: a > a > $o,X24: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_446_fm_Orel__distinct_I24_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_447_fm_Orel__distinct_I23_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_448_fm_Orel__distinct_I5_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_449_fm_Orel__distinct_I6_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_450_fm_Orel__distinct_I1_J,axiom,
! [R2: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Pro_a @ Y23 ) ) ).
% fm.rel_distinct(1)
thf(fact_451_fm_Orel__distinct_I2_J,axiom,
! [R2: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ Y23 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_452_fm_Orel__distinct_I4_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_453_fm_Orel__distinct_I3_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_454_fm_Orel__distinct_I14_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(14)
thf(fact_455_fm_Orel__distinct_I13_J,axiom,
! [R2: a > a > $o,X24: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_456_fm_Orel__distinct_I19_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_457_fm_Orel__distinct_I20_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_458_fm_Orel__distinct_I11_J,axiom,
! [R2: a > a > $o,X24: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_459_fm_Orel__distinct_I12_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(12)
thf(fact_460_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_461_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_462_strong__soundness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_463_strong__completeness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_464_main_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_465_S5__S5_H__assms,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( 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 ) @ G )
& ( 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_466_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_467_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( 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_468_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_469_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_470_fm_Orel__induct,axiom,
! [R2: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R2 @ 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] :
( ( R2 @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_471_pred__subset__eq,axiom,
! [R2: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( ord_le1568123225352354019fm_a_o
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ R2 )
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ S ) )
= ( ord_le7112219575281605754c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_472_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ R2 )
@ ^ [X: a] : ( member_a2 @ X @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_473_pred__subset__eq,axiom,
! [R2: set_li769143395467472256c_fm_a,S: set_li769143395467472256c_fm_a] :
( ( ord_le5289359451089373629fm_a_o
@ ^ [X: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X @ R2 )
@ ^ [X: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X @ S ) )
= ( ord_le2931979837777675552c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_474_pred__subset__eq,axiom,
! [R2: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ R2 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ S ) )
= ( ord_le3275665582123262618c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_475_map__rec,axiom,
( map_Ep1988204514693272906c_fm_a
= ( ^ [F2: epistemic_fm_a > list_Epistemic_fm_a] :
( rec_li2622076421192168746c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [X: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_476_map__rec,axiom,
( map_li3080774272885743684c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( rec_li5579148116651729200c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [X: list_Epistemic_fm_a,Uu2: list_l6083326122719238310c_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_477_map__rec,axiom,
( map_Ep7084560364594560580c_fm_a
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a] :
( rec_li4360243526537435568c_fm_a @ nil_Epistemic_fm_a
@ ^ [X: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_478_subseqs_Osimps_I1_J,axiom,
( ( subseq859285839621985007c_fm_a @ nil_Epistemic_fm_a )
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ nil_li2451196919128234278c_fm_a ) ) ).
% subseqs.simps(1)
thf(fact_479_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_480_not__in__set__insert,axiom,
! [X3: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X3 @ Xs )
= ( cons_s4394448069393209776c_fm_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_481_not__in__set__insert,axiom,
! [X3: a,Xs: list_a] :
( ~ ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X3 @ Xs )
= ( cons_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_482_not__in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_483_insert__Nil,axiom,
! [X3: epistemic_fm_a] :
( ( insert177310161492556854c_fm_a @ X3 @ nil_Epistemic_fm_a )
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% insert_Nil
thf(fact_484_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ S2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_485_in__set__insert,axiom,
! [X3: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_486_in__set__insert,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_487_in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_488_consistent__hereditary,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S )
=> ( episte2285483198712856226tent_a @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_489_Cons__in__subseqsD,axiom,
! [Y: epistemic_fm_a,Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Y @ Ys ) @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) )
=> ( member5906877432388582473c_fm_a @ Ys @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_490_list_Osimps_I7_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 @ ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ X22 ) ) ) ).
% list.simps(7)
thf(fact_491_list_Osimps_I6_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ nil_Epistemic_fm_a )
= F1 ) ).
% list.simps(6)
thf(fact_492_List_Oinsert__def,axiom,
( insert1868103382419917334c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] : ( if_lis3811715425919989702c_fm_a @ ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs3 ) ) @ Xs3 @ ( cons_s4394448069393209776c_fm_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_493_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X: a,Xs3: list_a] : ( if_list_a @ ( member_a2 @ X @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_494_List_Oinsert__def,axiom,
( insert177310161492556854c_fm_a
= ( ^ [X: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_495_rec__list__Nil__imp,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( F
= ( rec_li4360243526537435568c_fm_a @ F1 @ F22 ) )
=> ( ( F @ nil_Epistemic_fm_a )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_496_rec__list__Cons__imp,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( F
= ( rec_li4360243526537435568c_fm_a @ F1 @ F22 ) )
=> ( ( F @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( F22 @ X3 @ Xs @ ( F @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_497_AxB__symmetric_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ W3 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_498_Ax5__Euclidean,axiom,
! [A: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V: set_Epistemic_fm_a,W3: 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 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ V
@ ( 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 @ W3
@ ( 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 @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_499_Ax4__transitive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) )
=> ( ( 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 ) @ W3 ) ) ) ) )
=> ( 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 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_500_AxT__reflexive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_501_ax__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_502_consequent__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ V )
=> ( member6642669571620171971c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_503_deriv__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( epistemic_AK_a @ A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_504_maximal__extension,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V @ W4 )
=> ( ( episte2285483198712856226tent_a @ A @ W4 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) ) ) ).
% maximal_extension
thf(fact_505_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_506_subset__Collect__iff,axiom,
! [B3: set_a,A: set_a,P4: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ( ord_less_eq_set_a @ B3
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: a] :
( ( member_a2 @ X @ B3 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_507_subset__Collect__iff,axiom,
! [B3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ B3
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ B3 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_508_subset__Collect__iff,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ B3
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ B3 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_509_subset__Collect__iff,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ B3
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ B3 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_510_subset__CollectI,axiom,
! [B3: set_a,A: set_a,Q4: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ B3 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ B3 )
& ( Q4 @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_511_subset__CollectI,axiom,
! [B3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,Q4: set_Epistemic_fm_a > $o,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B3 @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B3 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ B3 )
& ( Q4 @ X ) ) )
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_512_subset__CollectI,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,Q4: epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B3 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ B3 )
& ( Q4 @ X ) ) )
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_513_subset__CollectI,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a,Q4: list_Epistemic_fm_a > $o,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ A )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ B3 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ B3 )
& ( Q4 @ X ) ) )
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_514_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_515_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_516_fm_Oset__cases,axiom,
! [E: a,A2: epistemic_fm_a] :
( ( member_a2 @ E @ ( epistemic_set_fm_a @ A2 ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ 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_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_517_Collect__restrict,axiom,
! [X6: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ X6 )
& ( P4 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_518_Collect__restrict,axiom,
! [X6: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ X6 )
& ( P4 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_519_Collect__restrict,axiom,
! [X6: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ X6 )
& ( P4 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_520_Collect__restrict,axiom,
! [X6: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ X6 )
& ( P4 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_521_prop__restrict,axiom,
! [X3: a,Z5: set_a,X6: set_a,P4: a > $o] :
( ( member_a2 @ X3 @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ X6 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_522_prop__restrict,axiom,
! [X3: set_Epistemic_fm_a,Z5: set_se5208064806568342746c_fm_a,X6: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X3 @ Z5 )
=> ( ( ord_le7112219575281605754c_fm_a @ Z5
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ X6 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_523_prop__restrict,axiom,
! [X3: epistemic_fm_a,Z5: set_Epistemic_fm_a,X6: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X3 @ Z5 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z5
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ X6 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_524_prop__restrict,axiom,
! [X3: list_Epistemic_fm_a,Z5: set_li769143395467472256c_fm_a,X6: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( member5906877432388582473c_fm_a @ X3 @ Z5 )
=> ( ( ord_le2931979837777675552c_fm_a @ Z5
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ X6 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_525_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_526_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_527_fm_Oset__intros_I5_J,axiom,
! [Ye: a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( member_a2 @ Ye @ ( epistemic_set_fm_a @ X51 ) )
=> ( member_a2 @ Ye @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_528_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_529_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_530_fm_Oset__intros_I6_J,axiom,
! [Yf: a,X52: epistemic_fm_a,X51: epistemic_fm_a] :
( ( member_a2 @ Yf @ ( epistemic_set_fm_a @ X52 ) )
=> ( member_a2 @ Yf @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_531_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_532_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_533_fm_Oset__intros_I7_J,axiom,
! [X61: a,X62: epistemic_fm_a] : ( member_a2 @ X61 @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_534_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_535_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_536_fm_Oset__intros_I8_J,axiom,
! [Yg: a,X62: epistemic_fm_a,X61: a] :
( ( member_a2 @ Yg @ ( epistemic_set_fm_a @ X62 ) )
=> ( member_a2 @ Yg @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_537_fm_Orel__refl__strong,axiom,
! [X3: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( Ra @ Z @ Z ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_538_fm_Orel__refl__strong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( Ra @ Z @ Z ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_539_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X3 ) )
=> ( Ra @ Z @ Z ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_540_fm_Orel__mono__strong,axiom,
! [R2: 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 @ R2 @ X3 @ Y )
=> ( ! [Z: epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_541_fm_Orel__mono__strong,axiom,
! [R2: 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 @ R2 @ X3 @ Y )
=> ( ! [Z: epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte4430291480098810304c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_542_fm_Orel__mono__strong,axiom,
! [R2: 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 @ R2 @ X3 @ Y )
=> ( ! [Z: set_Epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte6543039790609369600c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_543_fm_Orel__mono__strong,axiom,
! [R2: 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 @ R2 @ X3 @ Y )
=> ( ! [Z: set_Epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_544_fm_Orel__mono__strong,axiom,
! [R2: epistemic_fm_a > a > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R2 @ X3 @ Y )
=> ( ! [Z: epistemic_fm_a,Yb: a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_545_fm_Orel__mono__strong,axiom,
! [R2: set_Epistemic_fm_a > a > $o,X3: episte5999154429349999510c_fm_a,Y: epistemic_fm_a,Ra: set_Epistemic_fm_a > a > $o] :
( ( episte7682788968413056228fm_a_a @ R2 @ X3 @ Y )
=> ( ! [Z: set_Epistemic_fm_a,Yb: a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte7682788968413056228fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_546_fm_Orel__mono__strong,axiom,
! [R2: a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R2 @ X3 @ Y )
=> ( ! [Z: a,Yb: epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_547_fm_Orel__mono__strong,axiom,
! [R2: a > set_Epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte5999154429349999510c_fm_a,Ra: a > set_Epistemic_fm_a > $o] :
( ( episte6665775379439401564c_fm_a @ R2 @ X3 @ Y )
=> ( ! [Z: a,Yb: set_Epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( episte6665775379439401564c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_548_fm_Orel__mono__strong,axiom,
! [R2: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R2 @ X3 @ Y )
=> ( ! [Z: a,Yb: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_549_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R2: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R2 @ X3 @ Y )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_550_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R2: epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte4430291480098810304c_fm_a @ R2 @ X3 @ Y )
= ( episte4430291480098810304c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_551_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R2: set_Epistemic_fm_a > epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: set_Epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte6543039790609369600c_fm_a @ R2 @ X3 @ Y )
= ( episte6543039790609369600c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_552_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R2: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: set_Epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte8246226516352692192c_fm_a @ R2 @ X3 @ Y )
= ( episte8246226516352692192c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_553_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: epistemic_fm_a,Yb: a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R2 @ X3 @ Y )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_554_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: set_Epistemic_fm_a > a > $o,Ra: set_Epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: set_Epistemic_fm_a,Yb: a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte7682788968413056228fm_a_a @ R2 @ X3 @ Y )
= ( episte7682788968413056228fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_555_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R2: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb: epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R2 @ X3 @ Y )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_556_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R2: a > set_Epistemic_fm_a > $o,Ra: a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb: set_Epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( episte6665775379439401564c_fm_a @ R2 @ X3 @ Y )
= ( episte6665775379439401564c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_557_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X3 @ Y )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_558_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_559_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_560_fm_Oset__intros_I4_J,axiom,
! [Yd: a,X42: epistemic_fm_a,X41: epistemic_fm_a] :
( ( member_a2 @ Yd @ ( epistemic_set_fm_a @ X42 ) )
=> ( member_a2 @ Yd @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_561_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_562_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_563_fm_Oset__intros_I3_J,axiom,
! [Yc: a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( member_a2 @ Yc @ ( epistemic_set_fm_a @ X41 ) )
=> ( member_a2 @ Yc @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_564_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_565_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_566_fm_Oset__intros_I1_J,axiom,
! [Ya: a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( member_a2 @ Ya @ ( epistemic_set_fm_a @ X31 ) )
=> ( member_a2 @ Ya @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_567_fm_Oset__intros_I2_J,axiom,
! [Yb2: epistemic_fm_a,X32: episte740340785640729014c_fm_a,X31: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ X32 ) )
=> ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_568_fm_Oset__intros_I2_J,axiom,
! [Yb2: set_Epistemic_fm_a,X32: episte5999154429349999510c_fm_a,X31: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ X32 ) )
=> ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_569_fm_Oset__intros_I2_J,axiom,
! [Yb2: a,X32: epistemic_fm_a,X31: epistemic_fm_a] :
( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ X32 ) )
=> ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_570_conj__subset__def,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_571_conj__subset__def,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_572_conj__subset__def,axiom,
! [A: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ A
@ ( collec5191077796991884427c_fm_a
@ ^ [X: list_Epistemic_fm_a] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le2931979837777675552c_fm_a @ A @ ( collec5191077796991884427c_fm_a @ P4 ) )
& ( ord_le2931979837777675552c_fm_a @ A @ ( collec5191077796991884427c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_573_concat__map__singleton,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_Ep4939040189629079504c_fm_a
@ ^ [X: epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F @ X ) @ nil_li2451196919128234278c_fm_a )
@ Xs ) )
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_574_concat__map__singleton,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ^ [X: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F @ X ) @ nil_li2451196919128234278c_fm_a )
@ Xs ) )
= ( map_li3080774272885743684c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_575_concat__map__singleton,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ^ [X: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F @ X ) @ nil_Epistemic_fm_a )
@ Xs ) )
= ( map_li5516866612190465598c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_576_concat__map__singleton,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_Ep1988204514693272906c_fm_a
@ ^ [X: epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F @ X ) @ nil_Epistemic_fm_a )
@ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_577_list__ex1__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o] :
~ ( list_e2031426293596896995c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list_ex1_simps(1)
thf(fact_578_subset__subseqs,axiom,
! [X6: set_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X6 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member536094252920883875c_fm_a @ X6 @ ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_579_subset__subseqs,axiom,
! [X6: set_li769143395467472256c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X6 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( member2517045796615263401c_fm_a @ X6 @ ( image_7951968421222759263c_fm_a @ set_li8442223810127165109c_fm_a @ ( set_li647609056391172027c_fm_a @ ( subseq3170398737635186805c_fm_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_580_transpose__empty,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_581_list_Oset__map,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,V2: list_l6083326122719238310c_fm_a] :
( ( set_li8442223810127165109c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ V2 ) )
= ( image_7387318756023638021c_fm_a @ F @ ( set_li8442223810127165109c_fm_a @ V2 ) ) ) ).
% list.set_map
thf(fact_582_list_Oset__map,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,V2: list_Epistemic_fm_a] :
( ( set_li8442223810127165109c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ V2 ) )
= ( image_4430763730358542731c_fm_a @ F @ ( set_Epistemic_fm_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_583_list_Oset__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,V2: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( map_Ep7084560364594560580c_fm_a @ F @ V2 ) )
= ( image_4449434806354059013c_fm_a @ F @ ( set_Epistemic_fm_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_584_concat__eq__Nil__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( ( concat2780896542826320603c_fm_a @ Xss2 )
= nil_Epistemic_fm_a )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_585_Nil__eq__concat__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( nil_Epistemic_fm_a
= ( concat2780896542826320603c_fm_a @ Xss2 ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_586_image__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ ( image_4449434806354059013c_fm_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_587_image__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( image_4430763730358542731c_fm_a @ F @ A ) @ ( image_4430763730358542731c_fm_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_588_image__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_7959425827855735423c_fm_a @ F @ A ) @ ( image_7959425827855735423c_fm_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_589_image__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ F @ A ) @ ( image_7387318756023638021c_fm_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_590_image__subsetI,axiom,
! [A: set_a,F: a > a,B3: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_591_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > a,B3: set_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_592_image__subsetI,axiom,
! [A: set_a,F: a > epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_1898462159489238305c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_593_image__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > a,B3: set_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_4497856574827002249fm_a_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_594_image__subsetI,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_595_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_596_image__subsetI,axiom,
! [A: set_a,F: a > list_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member5906877432388582473c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( image_5290880835153512871c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_597_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_598_image__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_599_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member5906877432388582473c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( image_4430763730358542731c_fm_a @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_600_subset__imageE,axiom,
! [B3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B3
!= ( image_4449434806354059013c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_601_subset__imageE,axiom,
! [B3: set_Epistemic_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_7959425827855735423c_fm_a @ F @ A ) )
=> ~ ! [C3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C3 @ A )
=> ( B3
!= ( image_7959425827855735423c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_602_subset__imageE,axiom,
! [B3: set_li769143395467472256c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ ( image_4430763730358542731c_fm_a @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B3
!= ( image_4430763730358542731c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_603_subset__imageE,axiom,
! [B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ ( image_7387318756023638021c_fm_a @ F @ A ) )
=> ~ ! [C3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C3 @ A )
=> ( B3
!= ( image_7387318756023638021c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_604_image__subset__iff,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B3 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_605_image__subset__iff,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ F @ A ) @ B3 )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ A )
=> ( member5906877432388582473c_fm_a @ ( F @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_606_subset__image__iff,axiom,
! [B3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B3
= ( image_4449434806354059013c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_607_subset__image__iff,axiom,
! [B3: set_Epistemic_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_7959425827855735423c_fm_a @ F @ A ) )
= ( ? [AA: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ AA @ A )
& ( B3
= ( image_7959425827855735423c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_608_subset__image__iff,axiom,
! [B3: set_li769143395467472256c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ ( image_4430763730358542731c_fm_a @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B3
= ( image_4430763730358542731c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_609_subset__image__iff,axiom,
! [B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ ( image_7387318756023638021c_fm_a @ F @ A ) )
= ( ? [AA: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ AA @ A )
& ( B3
= ( image_7387318756023638021c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_610_map__concat,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat6109960377493484129c_fm_a @ ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_611_map__concat,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ ( concat6109960377493484129c_fm_a @ Xs ) )
= ( concat6109960377493484129c_fm_a @ ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_612_map__concat,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_613_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_614_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > a,B3: set_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_4497856574827002249fm_a_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_615_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_616_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > a,B3: set_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_617_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_618_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_619_image__Collect__subsetI,axiom,
! [P4: list_Epistemic_fm_a > $o,F: list_Epistemic_fm_a > list_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: list_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member5906877432388582473c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ F @ ( collec5191077796991884427c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_620_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > list_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member5906877432388582473c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( image_1284921450708845483c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_621_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > list_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member5906877432388582473c_fm_a @ ( F @ X2 ) @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( image_4430763730358542731c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_622_transpose__map__map,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ Xs ) )
= ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ ( transp7189280393105805488c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_623_transpose__map__map,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ Xs ) )
= ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ ( transp2882070860200649898c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_624_transpose__map__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ Xs ) )
= ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( transp2882070860200649898c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_625_transpose_Osimps_I1_J,axiom,
( ( transp2882070860200649898c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% transpose.simps(1)
thf(fact_626_image__set,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( image_7387318756023638021c_fm_a @ F @ ( set_li8442223810127165109c_fm_a @ Xs ) )
= ( set_li8442223810127165109c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_627_image__set,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( image_4430763730358542731c_fm_a @ F @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( set_li8442223810127165109c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_628_image__set,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( image_4449434806354059013c_fm_a @ F @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( set_Epistemic_fm_a2 @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_629_concat_Osimps_I1_J,axiom,
( ( concat2780896542826320603c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_Epistemic_fm_a ) ).
% concat.simps(1)
thf(fact_630_list__ex1__iff,axiom,
( list_e3415556844334454979c_fm_a
= ( ^ [P6: set_Epistemic_fm_a > $o,Xs3: list_s580375451141968640c_fm_a] :
? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ Y4 @ ( set_se8855941934996405007c_fm_a @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_631_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P6: a > $o,Xs3: list_a] :
? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ ( set_a2 @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_632_list__ex1__iff,axiom,
( list_e2031426293596896995c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,Xs3: list_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y4 @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_633_transpose_Osimps_I2_J,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
= ( transp2882070860200649898c_fm_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_634_product__lists_Osimps_I2_J,axiom,
! [Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( produc1391074687832863881c_fm_a @ ( cons_l8134865115577406678c_fm_a @ Xs @ Xss2 ) )
= ( concat6109960377493484129c_fm_a
@ ( map_Ep4939040189629079504c_fm_a
@ ^ [X: epistemic_fm_a] : ( map_li3080774272885743684c_fm_a @ ( cons_Epistemic_fm_a @ X ) @ ( produc1391074687832863881c_fm_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_635_can__select__set__list__ex1,axiom,
! [P4: epistemic_fm_a > $o,A: list_Epistemic_fm_a] :
( ( can_se5173380710277125655c_fm_a @ P4 @ ( set_Epistemic_fm_a2 @ A ) )
= ( list_e2031426293596896995c_fm_a @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_636_all__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ( P4 @ B4 ) ) )
= ( ! [B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_637_all__subset__image,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,A: set_li769143395467472256c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_7959425827855735423c_fm_a @ F @ A ) )
=> ( P4 @ B4 ) ) )
= ( ! [B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ A )
=> ( P4 @ ( image_7959425827855735423c_fm_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_638_all__subset__image,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_li769143395467472256c_fm_a > $o] :
( ( ! [B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ ( image_4430763730358542731c_fm_a @ F @ A ) )
=> ( P4 @ B4 ) ) )
= ( ! [B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( P4 @ ( image_4430763730358542731c_fm_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_639_all__subset__image,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,P4: set_li769143395467472256c_fm_a > $o] :
( ( ! [B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ ( image_7387318756023638021c_fm_a @ F @ A ) )
=> ( P4 @ B4 ) ) )
= ( ! [B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ A )
=> ( P4 @ ( image_7387318756023638021c_fm_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_640_set__concat,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( set_Epistemic_fm_a2 @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ Xs ) ) ) ) ).
% set_concat
thf(fact_641_concat__map__maps,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat2780896542826320603c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) )
= ( maps_E6072454603808916545c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_642_concat__map__maps,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ Xs ) )
= ( maps_l2078986973461115323c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_643_maps__def,axiom,
( maps_E6072454603808916545c_fm_a
= ( ^ [F2: epistemic_fm_a > list_Epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_644_maps__def,axiom,
( maps_l2078986973461115323c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_645_can__select__def,axiom,
( can_se5173380710277125655c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,A3: set_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
& ( P6 @ X )
& ! [Y4: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_646_can__select__def,axiom,
( can_se8336582597686632439c_fm_a
= ( ^ [P6: set_Epistemic_fm_a > $o,A3: set_se5208064806568342746c_fm_a] :
? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ( P6 @ X )
& ! [Y4: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_647_can__select__def,axiom,
( can_select_a
= ( ^ [P6: a > $o,A3: set_a] :
? [X: a] :
( ( member_a2 @ X @ A3 )
& ( P6 @ X )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_648_maps__simps_I2_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( maps_E6072454603808916545c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% maps_simps(2)
thf(fact_649_transpose_Osimps_I3_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Xss2 ) )
= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X3
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_650_transpose_Oelims,axiom,
! [X3: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ X3 )
= Y )
=> ( ( ( X3 = nil_li2451196919128234278c_fm_a )
=> ( Y != nil_li2451196919128234278c_fm_a ) )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( ( X3
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ( Y
!= ( transp2882070860200649898c_fm_a @ Xss ) ) )
=> ~ ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( X3
= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X2
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs2
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_651_SUP__subset__mono,axiom,
! [A: set_a,B3: set_a,F: a > set_Epistemic_fm_a,G5: a > set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_652_SUP__subset__mono,axiom,
! [A: set_a,B3: set_a,F: a > epistemic_fm_a > $o,G5: a > epistemic_fm_a > $o] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_653_SUP__subset__mono,axiom,
! [A: set_a,B3: set_a,F: a > set_li769143395467472256c_fm_a,G5: a > set_li769143395467472256c_fm_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_654_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G5: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_655_SUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G5: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_656_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,G5: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_657_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,G5: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_658_SUP__subset__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > set_Epistemic_fm_a,G5: list_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_659_SUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,G5: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_660_SUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,G5: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_661_UN__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G5: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_662_UN__mono,axiom,
! [A: set_a,B3: set_a,F: a > set_Epistemic_fm_a,G5: a > set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_663_UN__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,G5: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_664_UN__mono,axiom,
! [A: set_a,B3: set_a,F: a > set_li769143395467472256c_fm_a,G5: a > set_li769143395467472256c_fm_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_665_UN__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G5: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_666_UN__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,G5: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_667_UN__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > set_Epistemic_fm_a,G5: list_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_668_UN__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > set_li769143395467472256c_fm_a,G5: list_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_1280919820078628965c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_1280919820078628965c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_669_UN__least,axiom,
! [A: set_Epistemic_fm_a,B3: epistemic_fm_a > set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_670_UN__least,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a > set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_671_UN__least,axiom,
! [A: set_a,B3: a > set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_672_UN__least,axiom,
! [A: set_Epistemic_fm_a,B3: epistemic_fm_a > set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_673_UN__least,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_674_UN__least,axiom,
! [A: set_a,B3: a > set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ X2 ) @ C2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ B3 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_675_UN__upper,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B3: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ A2 ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_676_UN__upper,axiom,
! [A2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ A2 ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_677_UN__upper,axiom,
! [A2: a,A: set_a,B3: a > set_Epistemic_fm_a] :
( ( member_a2 @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( B3 @ A2 ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_678_UN__upper,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B3: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ A2 ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_679_UN__upper,axiom,
! [A2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ A2 ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_680_UN__upper,axiom,
! [A2: a,A: set_a,B3: a > set_li769143395467472256c_fm_a] :
( ( member_a2 @ A2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( B3 @ A2 ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ B3 @ A ) ) ) ) ).
% UN_upper
thf(fact_681_list_Odisc__eq__case_I2_J,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
= ( case_l2021589626981272552c_fm_a @ $false
@ ^ [Uu2: epistemic_fm_a,Uv2: list_Epistemic_fm_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_682_list_Odisc__eq__case_I1_J,axiom,
! [List: list_Epistemic_fm_a] :
( ( List = nil_Epistemic_fm_a )
= ( case_l2021589626981272552c_fm_a @ $true
@ ^ [Uu2: epistemic_fm_a,Uv2: list_Epistemic_fm_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_683_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] :
( ! [Z6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Z6 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Z6 @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_684_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] :
( ! [Z6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Z6 @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_685_Sup__eqI,axiom,
! [A: set_se4933744490392479200c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ! [Y2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Y2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ Y2 @ X3 ) )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ! [Z6: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Z6 @ A )
=> ( ord_le2931979837777675552c_fm_a @ Z6 @ Y2 ) )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Y2 ) )
=> ( ( comple943813931376279277c_fm_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_686_Sup__mono,axiom,
! [A: set_Epistemic_fm_a_o,B3: set_Epistemic_fm_a_o] :
( ! [A5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A5 @ A )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ B3 )
& ( ord_le4043730696559282883fm_a_o @ A5 @ X5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B3 ) ) ) ).
% Sup_mono
thf(fact_687_Sup__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [A5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A5 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ A5 @ X5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B3 ) ) ) ).
% Sup_mono
thf(fact_688_Sup__mono,axiom,
! [A: set_se4933744490392479200c_fm_a,B3: set_se4933744490392479200c_fm_a] :
( ! [A5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ A5 @ A )
=> ? [X5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X5 @ B3 )
& ( ord_le2931979837777675552c_fm_a @ A5 @ X5 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ ( comple943813931376279277c_fm_a @ B3 ) ) ) ).
% Sup_mono
thf(fact_689_Sup__least,axiom,
! [A: set_Epistemic_fm_a_o,Z4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z4 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_690_Sup__least,axiom,
! [A: set_se5208064806568342746c_fm_a,Z4: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_691_Sup__least,axiom,
! [A: set_se4933744490392479200c_fm_a,Z4: set_li769143395467472256c_fm_a] :
( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ X2 @ Z4 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_692_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_693_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_694_Sup__upper,axiom,
! [X3: set_li769143395467472256c_fm_a,A: set_se4933744490392479200c_fm_a] :
( ( member2517045796615263401c_fm_a @ X3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ ( comple943813931376279277c_fm_a @ A ) ) ) ).
% Sup_upper
thf(fact_695_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_696_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_697_Sup__le__iff,axiom,
! [A: set_se4933744490392479200c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ B )
= ( ! [X: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X @ A )
=> ( ord_le2931979837777675552c_fm_a @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_698_Sup__upper2,axiom,
! [U2: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o,V2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ U2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ V2 @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ V2 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_699_Sup__upper2,axiom,
! [U2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,V2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ U2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ V2 @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ V2 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_700_Sup__upper2,axiom,
! [U2: set_li769143395467472256c_fm_a,A: set_se4933744490392479200c_fm_a,V2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ U2 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ V2 @ U2 )
=> ( ord_le2931979837777675552c_fm_a @ V2 @ ( comple943813931376279277c_fm_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_701_Union__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B3 ) ) ) ).
% Union_mono
thf(fact_702_Union__mono,axiom,
! [A: set_se4933744490392479200c_fm_a,B3: set_se4933744490392479200c_fm_a] :
( ( ord_le3241105215597788544c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ ( comple943813931376279277c_fm_a @ B3 ) ) ) ).
% Union_mono
thf(fact_703_Union__least,axiom,
! [A: set_se5208064806568342746c_fm_a,C2: set_Epistemic_fm_a] :
( ! [X7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X7 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X7 @ C2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_704_Union__least,axiom,
! [A: set_se4933744490392479200c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ! [X7: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X7 @ A )
=> ( ord_le2931979837777675552c_fm_a @ X7 @ C2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_705_Union__upper,axiom,
! [B3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ B3 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Union_upper
thf(fact_706_Union__upper,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_se4933744490392479200c_fm_a] :
( ( member2517045796615263401c_fm_a @ B3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ B3 @ ( comple943813931376279277c_fm_a @ A ) ) ) ).
% Union_upper
thf(fact_707_Union__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ? [Y5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ X2 @ Y5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B3 ) ) ) ).
% Union_subsetI
thf(fact_708_Union__subsetI,axiom,
! [A: set_se4933744490392479200c_fm_a,B3: set_se4933744490392479200c_fm_a] :
( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
=> ? [Y5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Y5 @ B3 )
& ( ord_le2931979837777675552c_fm_a @ X2 @ Y5 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ ( comple943813931376279277c_fm_a @ B3 ) ) ) ).
% Union_subsetI
thf(fact_709_SUP__eq,axiom,
! [A: set_a,B3: set_a,F: a > set_Epistemic_fm_a,G5: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_710_SUP__eq,axiom,
! [A: set_a,B3: set_a,F: a > epistemic_fm_a > $o,G5: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_711_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B3: set_a,F: epistemic_fm_a > set_Epistemic_fm_a,G5: a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_712_SUP__eq,axiom,
! [A: set_a,B3: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a,G5: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_713_SUP__eq,axiom,
! [A: set_a,B3: set_a,F: a > set_li769143395467472256c_fm_a,G5: a > set_li769143395467472256c_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le2931979837777675552c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) )
= ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_714_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B3: set_a,F: epistemic_fm_a > epistemic_fm_a > $o,G5: a > epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_715_SUP__eq,axiom,
! [A: set_a,B3: set_Epistemic_fm_a,F: a > epistemic_fm_a > $o,G5: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B3 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_716_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G5: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B3 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_717_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G5: a > set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a2 @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B3 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_718_SUP__eq,axiom,
! [A: set_a,B3: set_se5208064806568342746c_fm_a,F: a > set_Epistemic_fm_a,G5: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G5 @ X5 ) ) ) )
=> ( ! [J: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J @ B3 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G5 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G5 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_719_Sup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B3: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B3 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_720_Sup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_721_Sup__subset__mono,axiom,
! [A: set_se4933744490392479200c_fm_a,B3: set_se4933744490392479200c_fm_a] :
( ( ord_le3241105215597788544c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ ( comple943813931376279277c_fm_a @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_722_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_723_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_724_SUP__upper2,axiom,
! [I: a,A: set_a,U2: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ 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_725_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_726_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_727_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ 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_728_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: set_li769143395467472256c_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_729_SUP__upper2,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,U2: set_li769143395467472256c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_730_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_li769143395467472256c_fm_a,F: a > set_li769143395467472256c_fm_a] :
( ( member_a2 @ I @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_731_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_732_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_733_SUP__upper,axiom,
! [I: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_734_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_735_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_736_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_737_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_738_SUP__upper,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_739_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_li769143395467472256c_fm_a] :
( ( member_a2 @ I @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_740_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_741_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_742_SUP__least,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ 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_743_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_744_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_745_SUP__least,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ 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_746_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,U2: set_li769143395467472256c_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_747_SUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,U2: set_li769143395467472256c_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_748_SUP__least,axiom,
! [A: set_a,F: a > set_li769143395467472256c_fm_a,U2: set_li769143395467472256c_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_749_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] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_750_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] :
( ! [I4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_751_SUP__eqI,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_752_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] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_753_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] :
( ! [I4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_754_SUP__eqI,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_755_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Y2 ) )
=> ( ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_756_SUP__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ! [I4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I4 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Y2 ) )
=> ( ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_757_SUP__eqI,axiom,
! [A: set_a,F: a > set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ Y2 ) )
=> ( ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_758_transpose_Opsimps_I3_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Xss2 ) )
=> ( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Xss2 ) )
= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X3
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_759_set__list__bind,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( bind_E8451893407412458119c_fm_a @ Xs @ F ) )
= ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( set_Epistemic_fm_a2 @ ( F @ X ) )
@ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_760_cSup__eq__maximum,axiom,
! [Z4: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ Z4 @ X6 )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z4 ) )
=> ( ( comple310706794042174646fm_a_o @ X6 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_761_cSup__eq__maximum,axiom,
! [Z4: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ X6 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z4 ) )
=> ( ( comple7868773486375872231c_fm_a @ X6 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_762_cSup__eq__maximum,axiom,
! [Z4: set_li769143395467472256c_fm_a,X6: set_se4933744490392479200c_fm_a] :
( ( member2517045796615263401c_fm_a @ Z4 @ X6 )
=> ( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ X2 @ Z4 ) )
=> ( ( comple943813931376279277c_fm_a @ X6 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_763_transpose_Opinduct,axiom,
! [A0: list_l6083326122719238310c_fm_a,P4: list_l6083326122719238310c_fm_a > $o] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ A0 )
=> ( ( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( P4 @ nil_li2451196919128234278c_fm_a ) )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ( ( P4 @ Xss )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) ) ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) )
=> ( ( P4
@ ( cons_l8134865115577406678c_fm_a @ Xs2
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss ) ) ) )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) ) ) )
=> ( P4 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_764_bind__simps_I1_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( bind_E8451893407412458119c_fm_a @ nil_Epistemic_fm_a @ F )
= nil_Epistemic_fm_a ) ).
% bind_simps(1)
thf(fact_765_transpose_Opsimps_I1_J,axiom,
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( ( transp2882070860200649898c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ) ).
% transpose.psimps(1)
thf(fact_766_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
=> ( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
= ( transp2882070860200649898c_fm_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_767_List_Obind__def,axiom,
( bind_E8451893407412458119c_fm_a
= ( ^ [Xs3: list_Epistemic_fm_a,F2: epistemic_fm_a > list_Epistemic_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_768_List_Obind__def,axiom,
( bind_l9085712471134790913c_fm_a
= ( ^ [Xs3: list_l6083326122719238310c_fm_a,F2: list_Epistemic_fm_a > list_Epistemic_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_769_accp__subset,axiom,
! [R1: epistemic_fm_a > epistemic_fm_a > $o,R22: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ R1 @ R22 )
=> ( ord_le4043730696559282883fm_a_o @ ( accp_Epistemic_fm_a @ R22 ) @ ( accp_Epistemic_fm_a @ R1 ) ) ) ).
% accp_subset
thf(fact_770_accp__subset__induct,axiom,
! [D: epistemic_fm_a > $o,R2: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ D @ ( accp_Epistemic_fm_a @ R2 ) )
=> ( ! [X2: epistemic_fm_a,Z: epistemic_fm_a] :
( ( D @ X2 )
=> ( ( R2 @ Z @ X2 )
=> ( D @ Z ) ) )
=> ( ( D @ X3 )
=> ( ! [X2: epistemic_fm_a] :
( ( D @ X2 )
=> ( ! [Z6: epistemic_fm_a] :
( ( R2 @ Z6 @ X2 )
=> ( P4 @ Z6 ) )
=> ( P4 @ X2 ) ) )
=> ( P4 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_771_subseqs__powset,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) )
= ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_772_Pow__iff,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B3 ) )
= ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ).
% Pow_iff
thf(fact_773_Pow__iff,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ A @ ( pow_li7854932614473680829c_fm_a @ B3 ) )
= ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ).
% Pow_iff
thf(fact_774_PowI,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B3 ) ) ) ).
% PowI
thf(fact_775_PowI,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( member2517045796615263401c_fm_a @ A @ ( pow_li7854932614473680829c_fm_a @ B3 ) ) ) ).
% PowI
thf(fact_776_PowD,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ).
% PowD
thf(fact_777_PowD,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ A @ ( pow_li7854932614473680829c_fm_a @ B3 ) )
=> ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ).
% PowD
thf(fact_778_Pow__def,axiom,
( pow_Epistemic_fm_a
= ( ^ [A3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_779_Pow__def,axiom,
( pow_li7854932614473680829c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a] :
( collec4111825885563579627c_fm_a
@ ^ [B4: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ B4 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_780_Pow__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ord_le7112219575281605754c_fm_a @ ( pow_Epistemic_fm_a @ A ) @ ( pow_Epistemic_fm_a @ B3 ) ) ) ).
% Pow_mono
thf(fact_781_Pow__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ord_le3241105215597788544c_fm_a @ ( pow_li7854932614473680829c_fm_a @ A ) @ ( pow_li7854932614473680829c_fm_a @ B3 ) ) ) ).
% Pow_mono
thf(fact_782_image__Pow__mono,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B3 )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ ( image_4449434806354059013c_fm_a @ F ) @ ( pow_Epistemic_fm_a @ A ) ) @ ( pow_Epistemic_fm_a @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_783_image__Pow__mono,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ F @ A ) @ B3 )
=> ( ord_le3241105215597788544c_fm_a @ ( image_8805014153957852165c_fm_a @ ( image_7387318756023638021c_fm_a @ F ) @ ( pow_li7854932614473680829c_fm_a @ A ) ) @ ( pow_li7854932614473680829c_fm_a @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_784_list__ex1__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_e2031426293596896995c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( ( P4 @ X3 )
=> ( list_a5841931967666341838c_fm_a
@ ^ [Y4: epistemic_fm_a] :
( ~ ( P4 @ Y4 )
| ( X3 = Y4 ) )
@ Xs ) )
& ( ~ ( P4 @ X3 )
=> ( list_e2031426293596896995c_fm_a @ P4 @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_785_transpose__aux__filter__tail,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) )
= ( map_li3080774272885743684c_fm_a @ tl_Epistemic_fm_a
@ ( filter441479637125139101c_fm_a
@ ^ [Ys2: list_Epistemic_fm_a] : ( Ys2 != nil_Epistemic_fm_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_786_Shift__def,axiom,
( bNF_Gr8437504134799245625c_fm_a
= ( ^ [Kl: set_li769143395467472256c_fm_a,K: epistemic_fm_a] :
( collec5191077796991884427c_fm_a
@ ^ [Kl2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_787_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G: 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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_788_frame_Oext__inject,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W7: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W6 @ K2 @ More )
= ( episte2888590659910966568t_unit @ W7 @ K3 @ More2 ) )
= ( ( W6 = W7 )
& ( K2 = K3 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_789_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_790_filter__filter,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( filter7636273809395300631c_fm_a @ P4 @ ( filter7636273809395300631c_fm_a @ Q4 @ Xs ) )
= ( filter7636273809395300631c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( Q4 @ X )
& ( P4 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_791_filter__True,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X2 ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_792_list__all__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( P4 @ X3 )
& ( list_a5841931967666341838c_fm_a @ P4 @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_793_list_Opred__inject_I2_J,axiom,
! [P4: epistemic_fm_a > $o,A2: epistemic_fm_a,Aa: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ A2 @ Aa ) )
= ( ( P4 @ A2 )
& ( list_a5841931967666341838c_fm_a @ P4 @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_794_list__all__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o] : ( list_a5841931967666341838c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list_all_simps(2)
thf(fact_795_set__filter,axiom,
! [P4: a > $o,Xs: list_a] :
( ( set_a2 @ ( filter_a @ P4 @ Xs ) )
= ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_796_set__filter,axiom,
! [P4: set_Epistemic_fm_a > $o,Xs: list_s580375451141968640c_fm_a] :
( ( set_se8855941934996405007c_fm_a @ ( filter2620125754611450615c_fm_a @ P4 @ Xs ) )
= ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_797_set__filter,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) )
= ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_798_filter__False,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X2 ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a ) ) ).
% filter_False
thf(fact_799_map__tl,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ ( tl_Epistemic_fm_a @ Xs ) )
= ( tl_lis3363300823944467983c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_800_map__tl,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ ( tl_lis3363300823944467983c_fm_a @ Xs ) )
= ( tl_lis3363300823944467983c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_801_map__tl,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( tl_Epistemic_fm_a @ Xs ) )
= ( tl_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_802_kripke_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W6 @ K2 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_803_list_Opred__True,axiom,
( ( list_a5841931967666341838c_fm_a
@ ^ [Uu2: epistemic_fm_a] : $true )
= ( ^ [Uu2: list_Epistemic_fm_a] : $true ) ) ).
% list.pred_True
thf(fact_804_kripke_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K4: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_805_frame_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K4: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K4 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_806_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_807_kripke_Ocases,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K4: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K4 @ ( episte8239586592105053771t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_808_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_a > list_char > $o,W: set_Epistemic_fm_a,W3: 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 @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_809_filter__cong,axiom,
! [Xs: list_s580375451141968640c_fm_a,Ys: list_s580375451141968640c_fm_a,P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( Xs = Ys )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter2620125754611450615c_fm_a @ P4 @ Xs )
= ( filter2620125754611450615c_fm_a @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_810_filter__cong,axiom,
! [Xs: list_a,Ys: list_a,P4: a > $o,Q4: a > $o] :
( ( Xs = Ys )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter_a @ P4 @ Xs )
= ( filter_a @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_811_filter__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= ( filter7636273809395300631c_fm_a @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_812_filter__id__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= Xs )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_813_filter_Osimps_I1_J,axiom,
! [P4: epistemic_fm_a > $o] :
( ( filter7636273809395300631c_fm_a @ P4 @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% filter.simps(1)
thf(fact_814_filter_Osimps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( P4 @ X3 )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( cons_Epistemic_fm_a @ X3 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) )
& ( ~ ( P4 @ X3 )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_815_list_Osel_I2_J,axiom,
( ( tl_Epistemic_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% list.sel(2)
thf(fact_816_list_Osel_I3_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( tl_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_817_list_Opred__inject_I1_J,axiom,
! [P4: epistemic_fm_a > $o] : ( list_a5841931967666341838c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list.pred_inject(1)
thf(fact_818_list__all__cong,axiom,
! [X3: list_s580375451141968640c_fm_a,Ya: list_s580375451141968640c_fm_a,P4: set_Epistemic_fm_a > $o,Pa: set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( set_se8855941934996405007c_fm_a @ Ya ) )
=> ( ( P4 @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_a5130230249644717486c_fm_a @ P4 @ X3 )
= ( list_a5130230249644717486c_fm_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_819_list__all__cong,axiom,
! [X3: list_a,Ya: list_a,P4: a > $o,Pa: a > $o] :
( ( X3 = Ya )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ Ya ) )
=> ( ( P4 @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_all_a @ P4 @ X3 )
= ( list_all_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_820_list__all__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( P4 @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
= ( list_a5841931967666341838c_fm_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_821_list_Opred__mono__strong,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: list_s580375451141968640c_fm_a,Pa: set_Epistemic_fm_a > $o] :
( ( list_a5130230249644717486c_fm_a @ P4 @ X3 )
=> ( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( set_se8855941934996405007c_fm_a @ X3 ) )
=> ( ( P4 @ Z )
=> ( Pa @ Z ) ) )
=> ( list_a5130230249644717486c_fm_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_822_list_Opred__mono__strong,axiom,
! [P4: a > $o,X3: list_a,Pa: a > $o] :
( ( list_all_a @ P4 @ X3 )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ X3 ) )
=> ( ( P4 @ Z )
=> ( Pa @ Z ) ) )
=> ( list_all_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_823_list_Opred__mono__strong,axiom,
! [P4: epistemic_fm_a > $o,X3: list_Epistemic_fm_a,Pa: epistemic_fm_a > $o] :
( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( P4 @ Z )
=> ( Pa @ Z ) ) )
=> ( list_a5841931967666341838c_fm_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_824_list_Opred__mono,axiom,
! [P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Pa )
=> ( ord_le5289359451089373629fm_a_o @ ( list_a5841931967666341838c_fm_a @ P4 ) @ ( list_a5841931967666341838c_fm_a @ Pa ) ) ) ).
% list.pred_mono
thf(fact_825_list_Omap__cong__pred,axiom,
! [X3: list_l6083326122719238310c_fm_a,Ya: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( X3 = Ya )
=> ( ( list_a5890043570083025364c_fm_a
@ ^ [Z2: list_Epistemic_fm_a] :
( ( F @ Z2 )
= ( G5 @ Z2 ) )
@ Ya )
=> ( ( map_li3080774272885743684c_fm_a @ F @ X3 )
= ( map_li3080774272885743684c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_826_list_Omap__cong__pred,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( X3 = Ya )
=> ( ( list_a5841931967666341838c_fm_a
@ ^ [Z2: epistemic_fm_a] :
( ( F @ Z2 )
= ( G5 @ Z2 ) )
@ Ya )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ X3 )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_827_list_Omap__cong__pred,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( X3 = Ya )
=> ( ( list_a5841931967666341838c_fm_a
@ ^ [Z2: epistemic_fm_a] :
( ( F @ Z2 )
= ( G5 @ Z2 ) )
@ Ya )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_828_filter__is__subset,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) @ ( set_Epistemic_fm_a2 @ Xs ) ) ).
% filter_is_subset
thf(fact_829_filter__is__subset,axiom,
! [P4: list_Epistemic_fm_a > $o,Xs: list_l6083326122719238310c_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( set_li8442223810127165109c_fm_a @ ( filter441479637125139101c_fm_a @ P4 @ Xs ) ) @ ( set_li8442223810127165109c_fm_a @ Xs ) ) ).
% filter_is_subset
thf(fact_830_filter__empty__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_831_empty__filter__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_832_frame_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W6 @ K2 @ More ) )
= W6 ) ).
% frame.select_convs(1)
thf(fact_833_Nil__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( tl_Epistemic_fm_a @ Xs ) )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ) ) ).
% Nil_tl
thf(fact_834_tl__Nil,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( tl_Epistemic_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ) ) ).
% tl_Nil
thf(fact_835_list_Oset__sel_I2_J,axiom,
! [A2: list_s580375451141968640c_fm_a,X3: set_Epistemic_fm_a] :
( ( A2 != nil_se102576565402781184c_fm_a )
=> ( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ ( tl_set2007602089164433257c_fm_a @ A2 ) ) )
=> ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_836_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X3: a] :
( ( A2 != nil_a )
=> ( ( member_a2 @ X3 @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_837_list_Oset__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( tl_Epistemic_fm_a @ A2 ) ) )
=> ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_838_filter__concat,axiom,
! [P: epistemic_fm_a > $o,Xs: list_l6083326122719238310c_fm_a] :
( ( filter7636273809395300631c_fm_a @ P @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( filter7636273809395300631c_fm_a @ P ) @ Xs ) ) ) ).
% filter_concat
thf(fact_839_list_Omap__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( tl_lis3363300823944467983c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ A2 ) )
= ( map_Ep1988204514693272906c_fm_a @ F @ ( tl_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_840_list_Omap__sel_I2_J,axiom,
! [A2: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( A2 != nil_li2451196919128234278c_fm_a )
=> ( ( tl_lis3363300823944467983c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ A2 ) )
= ( map_li3080774272885743684c_fm_a @ F @ ( tl_lis3363300823944467983c_fm_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_841_list_Omap__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ A2 ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ ( tl_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_842_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) )
@ V
@ P ) ) ) ) ).
% truth_lemma
thf(fact_843_tl__def,axiom,
( tl_Epistemic_fm_a
= ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [X213: epistemic_fm_a,X223: list_Epistemic_fm_a] : X223 ) ) ).
% tl_def
thf(fact_844_ShiftD,axiom,
! [Kl3: list_Epistemic_fm_a,Kl4: set_li769143395467472256c_fm_a,K5: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Kl3 @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K5 ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ K5 @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_845_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
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_846_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
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_847_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
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_848_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
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_849_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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_850_transpose__aux__filter__head,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) )
= ( map_li5516866612190465598c_fm_a @ hd_Epistemic_fm_a
@ ( filter441479637125139101c_fm_a
@ ^ [Ys2: list_Epistemic_fm_a] : ( Ys2 != nil_Epistemic_fm_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_851_empty__Shift,axiom,
! [Kl4: set_li4204741992506657632c_fm_a,K5: set_Epistemic_fm_a] :
( ( member2328277852289759785c_fm_a @ nil_se102576565402781184c_fm_a @ Kl4 )
=> ( ( member536094252920883875c_fm_a @ K5 @ ( bNF_Gr1807846264492387989c_fm_a @ Kl4 @ nil_se102576565402781184c_fm_a ) )
=> ( member2328277852289759785c_fm_a @ nil_se102576565402781184c_fm_a @ ( bNF_Gr7707547485936239385c_fm_a @ Kl4 @ K5 ) ) ) ) ).
% empty_Shift
thf(fact_852_empty__Shift,axiom,
! [Kl4: set_list_a,K5: a] :
( ( member_list_a @ nil_a @ Kl4 )
=> ( ( member_a2 @ K5 @ ( bNF_Greatest_Succ_a @ Kl4 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K5 ) ) ) ) ).
% empty_Shift
thf(fact_853_empty__Shift,axiom,
! [Kl4: set_li769143395467472256c_fm_a,K5: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ Kl4 )
=> ( ( member6642669571620171971c_fm_a @ K5 @ ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ nil_Epistemic_fm_a ) )
=> ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K5 ) ) ) ) ).
% empty_Shift
thf(fact_854_Succ__Shift,axiom,
! [Kl4: set_li769143395467472256c_fm_a,K5: epistemic_fm_a,Kl3: list_Epistemic_fm_a] :
( ( bNF_Gr1093487135560555701c_fm_a @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K5 ) @ Kl3 )
= ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ ( cons_Epistemic_fm_a @ K5 @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_855_empty__subsetI,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A ) ).
% empty_subsetI
thf(fact_856_empty__subsetI,axiom,
! [A: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ bot_bo8284763806281179116c_fm_a @ A ) ).
% empty_subsetI
thf(fact_857_subset__empty,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a )
= ( A = bot_bo3626323581529592678c_fm_a ) ) ).
% subset_empty
thf(fact_858_subset__empty,axiom,
! [A: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ bot_bo8284763806281179116c_fm_a )
= ( A = bot_bo8284763806281179116c_fm_a ) ) ).
% subset_empty
thf(fact_859_set__empty,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( set_Epistemic_fm_a2 @ Xs )
= bot_bo3626323581529592678c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% set_empty
thf(fact_860_set__empty2,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% set_empty2
thf(fact_861_list_Ocollapse,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
=> ( ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ List ) @ ( tl_Epistemic_fm_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_862_hd__Cons__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ Xs ) @ ( tl_Epistemic_fm_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_863_subset__emptyI,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le7112219575281605754c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) ).
% subset_emptyI
thf(fact_864_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_865_subset__emptyI,axiom,
! [A: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% subset_emptyI
thf(fact_866_subset__emptyI,axiom,
! [A: set_li769143395467472256c_fm_a] :
( ! [X2: list_Epistemic_fm_a] :
~ ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ A @ bot_bo8284763806281179116c_fm_a ) ) ).
% subset_emptyI
thf(fact_867_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_868_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_869_bot_Oextremum__uniqueI,axiom,
! [A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ bot_bo8284763806281179116c_fm_a )
=> ( A2 = bot_bo8284763806281179116c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_870_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_871_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_872_bot_Oextremum__unique,axiom,
! [A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ bot_bo8284763806281179116c_fm_a )
= ( A2 = bot_bo8284763806281179116c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_873_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ bot_bo6433428028861825271fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_874_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_875_bot_Oextremum,axiom,
! [A2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ bot_bo8284763806281179116c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_876_list_Osel_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( hd_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_877_hd__concat,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( Xs != nil_li2451196919128234278c_fm_a )
=> ( ( ( hd_lis6228752984540680715c_fm_a @ Xs )
!= nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( hd_Epistemic_fm_a @ ( hd_lis6228752984540680715c_fm_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_878_list_Oset__sel_I1_J,axiom,
! [A2: list_s580375451141968640c_fm_a] :
( ( A2 != nil_se102576565402781184c_fm_a )
=> ( member536094252920883875c_fm_a @ ( hd_set2739578467680395621c_fm_a @ A2 ) @ ( set_se8855941934996405007c_fm_a @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_879_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a2 @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_880_list_Oset__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( hd_Epistemic_fm_a @ A2 ) @ ( set_Epistemic_fm_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_881_hd__in__set,axiom,
! [Xs: list_s580375451141968640c_fm_a] :
( ( Xs != nil_se102576565402781184c_fm_a )
=> ( member536094252920883875c_fm_a @ ( hd_set2739578467680395621c_fm_a @ Xs ) @ ( set_se8855941934996405007c_fm_a @ Xs ) ) ) ).
% hd_in_set
thf(fact_882_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a2 @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_883_hd__in__set,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( hd_Epistemic_fm_a @ Xs ) @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_884_list_Omap__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( hd_lis6228752984540680715c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ A2 ) )
= ( F @ ( hd_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_885_list_Omap__sel_I1_J,axiom,
! [A2: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( A2 != nil_li2451196919128234278c_fm_a )
=> ( ( hd_lis6228752984540680715c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ A2 ) )
= ( F @ ( hd_lis6228752984540680715c_fm_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_886_list_Omap__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ A2 ) )
= ( F @ ( hd_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_887_hd__map,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( hd_lis6228752984540680715c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) )
= ( F @ ( hd_Epistemic_fm_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_888_hd__map,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs != nil_li2451196919128234278c_fm_a )
=> ( ( hd_lis6228752984540680715c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ Xs ) )
= ( F @ ( hd_lis6228752984540680715c_fm_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_889_hd__map,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( F @ ( hd_Epistemic_fm_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_890_less__eq__Sup,axiom,
! [A: set_Epistemic_fm_a_o,U2: epistemic_fm_a > $o] :
( ! [V4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ V4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ V4 ) )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_891_less__eq__Sup,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a] :
( ! [V4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ V4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ V4 ) )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_892_less__eq__Sup,axiom,
! [A: set_se4933744490392479200c_fm_a,U2: set_li769143395467472256c_fm_a] :
( ! [V4: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ V4 @ A )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ V4 ) )
=> ( ( A != bot_bo8499242532914112588c_fm_a )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_893_cSup__least,axiom,
! [X6: set_Epistemic_fm_a_o,Z4: epistemic_fm_a > $o] :
( ( X6 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z4 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ X6 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_894_cSup__least,axiom,
! [X6: set_se5208064806568342746c_fm_a,Z4: set_Epistemic_fm_a] :
( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ X6 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_895_cSup__least,axiom,
! [X6: set_se4933744490392479200c_fm_a,Z4: set_li769143395467472256c_fm_a] :
( ( X6 != bot_bo8499242532914112588c_fm_a )
=> ( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ X2 @ Z4 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ X6 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_896_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_897_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_898_cSup__eq__non__empty,axiom,
! [X6: set_se4933744490392479200c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( X6 != bot_bo8499242532914112588c_fm_a )
=> ( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ X2 @ A2 ) )
=> ( ! [Y2: set_li769143395467472256c_fm_a] :
( ! [X5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X5 @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ X5 @ Y2 ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ Y2 ) )
=> ( ( comple943813931376279277c_fm_a @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_899_empty__set,axiom,
( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% empty_set
thf(fact_900_list_Oexpand,axiom,
! [List: list_Epistemic_fm_a,List2: list_Epistemic_fm_a] :
( ( ( List = nil_Epistemic_fm_a )
= ( List2 = nil_Epistemic_fm_a ) )
=> ( ( ( List != nil_Epistemic_fm_a )
=> ( ( List2 != nil_Epistemic_fm_a )
=> ( ( ( hd_Epistemic_fm_a @ List )
= ( hd_Epistemic_fm_a @ List2 ) )
& ( ( tl_Epistemic_fm_a @ List )
= ( tl_Epistemic_fm_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_901_fm_Osimps_I90_J,axiom,
( ( episte9089240958480457552c_fm_a @ episte5073044243917183961c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(90)
thf(fact_902_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_903_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( episte9089240958480457552c_fm_a @ ( episte3759128466173231372c_fm_a @ X24 ) )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(91)
thf(fact_904_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( epistemic_set_fm_a @ ( epistemic_Pro_a @ X24 ) )
= bot_bot_set_a ) ).
% fm.simps(91)
thf(fact_905_hd__def,axiom,
( hd_Epistemic_fm_a
= ( case_l5911136813010175462c_fm_a @ undefi6158949259153642370c_fm_a
@ ^ [X213: epistemic_fm_a,X223: list_Epistemic_fm_a] : X213 ) ) ).
% hd_def
thf(fact_906_SUP__eq__iff,axiom,
! [I5: set_se5208064806568342746c_fm_a,C: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( I5 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_907_SUP__eq__iff,axiom,
! [I5: set_a,C: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_908_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_909_SUP__eq__iff,axiom,
! [I5: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( I5 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_910_SUP__eq__iff,axiom,
! [I5: set_a,C: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_911_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_912_SUP__eq__iff,axiom,
! [I5: set_se5208064806568342746c_fm_a,C: set_li769143395467472256c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( I5 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I5 )
=> ( ord_le2931979837777675552c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_913_SUP__eq__iff,axiom,
! [I5: set_a,C: set_li769143395467472256c_fm_a,F: a > set_li769143395467472256c_fm_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_le2931979837777675552c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_914_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: set_li769143395467472256c_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_le2931979837777675552c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_915_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_916_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_a2 @ 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_917_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_918_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_919_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_a2 @ 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_920_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_921_cSUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_922_cSUP__least,axiom,
! [A: set_a,F: a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_923_cSUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_924_list_Oexhaust__sel,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
=> ( List
= ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ List ) @ ( tl_Epistemic_fm_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_925_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 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_926_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W6: set_se5208064806568342746c_fm_a,K2: 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 @ W6 @ K2 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K2 @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_927_kripke_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W6 @ K2 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_928_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W6: set_se5208064806568342746c_fm_a,K2: 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 @ W6 @ K2 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K2 @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_929_min__bot,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_mi5245583150369600225c_fm_a @ bot_bo3626323581529592678c_fm_a @ X3 )
= bot_bo3626323581529592678c_fm_a ) ).
% min_bot
thf(fact_930_min__bot2,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_mi5245583150369600225c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% min_bot2
thf(fact_931_min__absorb2,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_mi8079229082941695292fm_a_o @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_932_min__absorb2,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_mi5245583150369600225c_fm_a @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_933_min__absorb2,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( ( ord_mi8068400396335842791c_fm_a @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_934_min__absorb1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_mi8079229082941695292fm_a_o @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_935_min__absorb1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_mi5245583150369600225c_fm_a @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_936_min__absorb1,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ Y )
=> ( ( ord_mi8068400396335842791c_fm_a @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_937_min__def,axiom,
( ord_mi8079229082941695292fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_938_min__def,axiom,
( ord_mi5245583150369600225c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_939_min__def,axiom,
( ord_mi8068400396335842791c_fm_a
= ( ^ [A4: set_li769143395467472256c_fm_a,B2: set_li769143395467472256c_fm_a] : ( if_set4000483370245493318c_fm_a @ ( ord_le2931979837777675552c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_940_min__def__raw,axiom,
( ord_mi8079229082941695292fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_941_min__def__raw,axiom,
( ord_mi5245583150369600225c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_942_min__def__raw,axiom,
( ord_mi8068400396335842791c_fm_a
= ( ^ [A4: set_li769143395467472256c_fm_a,B2: set_li769143395467472256c_fm_a] : ( if_set4000483370245493318c_fm_a @ ( ord_le2931979837777675552c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_943_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_944_kripke_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R3: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R3 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R3 ) )
=> ( ( ( episte2398645135750866164t_unit @ R )
= ( episte2398645135750866164t_unit @ R3 ) )
=> ( ( ( episte5479201149095757850t_unit @ R )
= ( episte5479201149095757850t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ) ).
% kripke.equality
thf(fact_945_image__Fpow__mono,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B3 )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ ( image_4449434806354059013c_fm_a @ F ) @ ( finite7994277033128312448c_fm_a @ A ) ) @ ( finite7994277033128312448c_fm_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_946_image__Fpow__mono,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ F @ A ) @ B3 )
=> ( ord_le3241105215597788544c_fm_a @ ( image_8805014153957852165c_fm_a @ ( image_7387318756023638021c_fm_a @ F ) @ ( finite2132446552880859014c_fm_a @ A ) ) @ ( finite2132446552880859014c_fm_a @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_947_cSUP__subset__mono,axiom,
! [A: set_a,G5: a > set_Epistemic_fm_a,B3: set_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) )
=> ( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_948_cSUP__subset__mono,axiom,
! [A: set_a,G5: a > epistemic_fm_a > $o,B3: set_a,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ G5 @ B3 ) )
=> ( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_949_cSUP__subset__mono,axiom,
! [A: set_a,G5: a > set_li769143395467472256c_fm_a,B3: set_a,F: a > set_li769143395467472256c_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit688303582726876132c_fm_a @ ( image_7867170089500699143c_fm_a @ G5 @ B3 ) )
=> ( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_950_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G5: epistemic_fm_a > set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_951_cSUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G5: set_Epistemic_fm_a > set_Epistemic_fm_a,B3: 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 @ G5 @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_952_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a > $o,B3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ G5 @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_953_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G5: epistemic_fm_a > set_li769143395467472256c_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ ( image_2304902010338030059c_fm_a @ G5 @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_954_cSUP__subset__mono,axiom,
! [A: set_li769143395467472256c_fm_a,G5: list_Epistemic_fm_a > set_Epistemic_fm_a,B3: set_li769143395467472256c_fm_a,F: list_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo8284763806281179116c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_971165786557580383c_fm_a @ G5 @ B3 ) )
=> ( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_955_cSUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G5: set_Epistemic_fm_a > epistemic_fm_a > $o,B3: 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 @ G5 @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_956_cSUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G5: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ ( image_4454716909333697547c_fm_a @ G5 @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ ( G5 @ X2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ G5 @ B3 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_957_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_958_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_959_bdd__above_OI,axiom,
! [A: set_se4933744490392479200c_fm_a,M: set_li769143395467472256c_fm_a] :
( ! [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ X2 @ M ) )
=> ( condit688303582726876132c_fm_a @ A ) ) ).
% bdd_above.I
thf(fact_960_bdd__above_Ounfold,axiom,
( condit6663374420580310783fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
? [M4: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_961_bdd__above_Ounfold,axiom,
( condit2203544419062987614c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
? [M4: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_962_bdd__above_Ounfold,axiom,
( condit688303582726876132c_fm_a
= ( ^ [A3: set_se4933744490392479200c_fm_a] :
? [M4: set_li769143395467472256c_fm_a] :
! [X: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X @ A3 )
=> ( ord_le2931979837777675552c_fm_a @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_963_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_964_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_965_bdd__above_OE,axiom,
! [A: set_se4933744490392479200c_fm_a] :
( ( condit688303582726876132c_fm_a @ A )
=> ~ ! [M2: set_li769143395467472256c_fm_a] :
~ ! [X5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X5 @ A )
=> ( ord_le2931979837777675552c_fm_a @ X5 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_966_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_967_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_968_bdd__above_OI2,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_969_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_970_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_971_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_972_bdd__above_OI2,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit688303582726876132c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_973_bdd__above_OI2,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit688303582726876132c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_974_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_li769143395467472256c_fm_a,M: set_li769143395467472256c_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit688303582726876132c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_975_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_976_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_977_cSup__upper2,axiom,
! [X3: set_li769143395467472256c_fm_a,X6: set_se4933744490392479200c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X3 @ X6 )
=> ( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( ( condit688303582726876132c_fm_a @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ Y @ ( comple943813931376279277c_fm_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_978_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_979_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_980_cSup__upper,axiom,
! [X3: set_li769143395467472256c_fm_a,X6: set_se4933744490392479200c_fm_a] :
( ( member2517045796615263401c_fm_a @ X3 @ X6 )
=> ( ( condit688303582726876132c_fm_a @ X6 )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ ( comple943813931376279277c_fm_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_981_frame_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W6 @ K2 @ More ) )
= K2 ) ).
% frame.select_convs(2)
thf(fact_982_Fpow__mono,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ord_le7112219575281605754c_fm_a @ ( finite7994277033128312448c_fm_a @ A ) @ ( finite7994277033128312448c_fm_a @ B3 ) ) ) ).
% Fpow_mono
thf(fact_983_Fpow__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ord_le3241105215597788544c_fm_a @ ( finite2132446552880859014c_fm_a @ A ) @ ( finite2132446552880859014c_fm_a @ B3 ) ) ) ).
% Fpow_mono
thf(fact_984_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_985_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_986_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ 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_987_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_988_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_989_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ 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_990_cSUP__upper,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( condit688303582726876132c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_991_cSUP__upper,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( condit688303582726876132c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_992_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > set_li769143395467472256c_fm_a] :
( ( member_a2 @ X3 @ A )
=> ( ( condit688303582726876132c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_993_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_994_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_995_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_a2 @ 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_996_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_997_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_998_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_a2 @ 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_999_cSUP__upper2,axiom,
! [F: epistemic_fm_a > set_li769143395467472256c_fm_a,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: set_li769143395467472256c_fm_a] :
( ( condit688303582726876132c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1000_cSUP__upper2,axiom,
! [F: set_Epistemic_fm_a > set_li769143395467472256c_fm_a,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,U2: set_li769143395467472256c_fm_a] :
( ( condit688303582726876132c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_4454716909333697547c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1001_cSUP__upper2,axiom,
! [F: a > set_li769143395467472256c_fm_a,A: set_a,X3: a,U2: set_li769143395467472256c_fm_a] :
( ( condit688303582726876132c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) )
=> ( ( member_a2 @ X3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le2931979837777675552c_fm_a @ U2 @ ( comple943813931376279277c_fm_a @ ( image_7867170089500699143c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1002_cSup__le__iff,axiom,
! [S: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( S != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ S )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ S ) @ A2 )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ S )
=> ( ord_le4043730696559282883fm_a_o @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1003_cSup__le__iff,axiom,
! [S: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( S != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ S ) @ A2 )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ S )
=> ( ord_le3275665582123262618c_fm_a @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1004_cSup__le__iff,axiom,
! [S: set_se4933744490392479200c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( S != bot_bo8499242532914112588c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ S )
=> ( ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ S ) @ A2 )
= ( ! [X: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X @ S )
=> ( ord_le2931979837777675552c_fm_a @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1005_cSup__mono,axiom,
! [B3: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ( B3 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ! [B5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ B5 @ B3 )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ B5 @ X5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ B3 ) @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1006_cSup__mono,axiom,
! [B3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( B3 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ! [B5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B5 @ B3 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ B5 @ X5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ B3 ) @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1007_cSup__mono,axiom,
! [B3: set_se4933744490392479200c_fm_a,A: set_se4933744490392479200c_fm_a] :
( ( B3 != bot_bo8499242532914112588c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ A )
=> ( ! [B5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ B5 @ B3 )
=> ? [X5: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X5 @ A )
& ( ord_le2931979837777675552c_fm_a @ B5 @ X5 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ B3 ) @ ( comple943813931376279277c_fm_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1008_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( ( member536094252920883875c_fm_a @ Z2 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
& ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Z2 ) ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_1009_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_1010_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_1011_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_1012_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_li769143395467472256c_fm_a,U2: set_li769143395467472256c_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) )
=> ( ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ ( image_2304902010338030059c_fm_a @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1013_cSup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B3: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ B3 )
=> ( ( ord_le4525777775358908921fm_a_o @ A @ B3 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B3 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1014_cSup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B3 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1015_cSup__subset__mono,axiom,
! [A: set_se4933744490392479200c_fm_a,B3: set_se4933744490392479200c_fm_a] :
( ( A != bot_bo8499242532914112588c_fm_a )
=> ( ( condit688303582726876132c_fm_a @ B3 )
=> ( ( ord_le3241105215597788544c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( comple943813931376279277c_fm_a @ A ) @ ( comple943813931376279277c_fm_a @ B3 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1016_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
= ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_1017_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( ( member536094252920883875c_fm_a @ Z2 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( member536094252920883875c_fm_a @ Z2 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_1018_frame_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte3309513806868946049t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_1019_frame_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R3: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R3 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R3 ) )
=> ( ( ( episte3309513806868946049t_unit @ R )
= ( episte3309513806868946049t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ).
% frame.equality
thf(fact_1020_Cons__in__shuffles__iff,axiom,
! [Z4: epistemic_fm_a,Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z4 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Xs )
= Z4 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Ys )
= Z4 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ ( tl_Epistemic_fm_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_1021_inconsistent__subset,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [V5: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V5 ) @ V )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V5 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_1022_le__sup__iff,axiom,
! [X3: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z4 )
= ( ( ord_less_eq_set_a @ X3 @ Z4 )
& ( ord_less_eq_set_a @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_1023_le__sup__iff,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) @ Z4 )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Z4 )
& ( ord_le4043730696559282883fm_a_o @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_1024_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) @ Z4 )
= ( ( ord_le3275665582123262618c_fm_a @ X3 @ Z4 )
& ( ord_le3275665582123262618c_fm_a @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_1025_le__sup__iff,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ X3 @ Y ) @ Z4 )
= ( ( ord_le2931979837777675552c_fm_a @ X3 @ Z4 )
& ( ord_le2931979837777675552c_fm_a @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_1026_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_1027_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_1028_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_1029_sup_Obounded__iff,axiom,
! [B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ B @ C ) @ A2 )
= ( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
& ( ord_le2931979837777675552c_fm_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1030_insert__subset,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X3 @ A ) @ B3 )
= ( ( member536094252920883875c_fm_a @ X3 @ B3 )
& ( ord_le7112219575281605754c_fm_a @ A @ B3 ) ) ) ).
% insert_subset
thf(fact_1031_insert__subset,axiom,
! [X3: a,A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X3 @ A ) @ B3 )
= ( ( member_a2 @ X3 @ B3 )
& ( ord_less_eq_set_a @ A @ B3 ) ) ) ).
% insert_subset
thf(fact_1032_insert__subset,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ A ) @ B3 )
= ( ( member6642669571620171971c_fm_a @ X3 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ) ).
% insert_subset
thf(fact_1033_insert__subset,axiom,
! [X3: list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( insert6567282714061225712c_fm_a @ X3 @ A ) @ B3 )
= ( ( member5906877432388582473c_fm_a @ X3 @ B3 )
& ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ) ).
% insert_subset
thf(fact_1034_Un__subset__iff,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1035_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ C2 )
= ( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1036_Un__subset__iff,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) @ C2 )
= ( ( ord_le2931979837777675552c_fm_a @ A @ C2 )
& ( ord_le2931979837777675552c_fm_a @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1037_Nil__in__shuffles,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( ( Xs = nil_Epistemic_fm_a )
& ( Ys = nil_Epistemic_fm_a ) ) ) ).
% Nil_in_shuffles
thf(fact_1038_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B: a] :
( ( ( insert_a2 @ A2 @ A )
= ( insert_a2 @ B @ bot_bot_set_a ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1039_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ A2 @ A )
= ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1040_singleton__insert__inj__eq_H,axiom,
! [A2: list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B: list_Epistemic_fm_a] :
( ( ( insert6567282714061225712c_fm_a @ A2 @ A )
= ( insert6567282714061225712c_fm_a @ B @ bot_bo8284763806281179116c_fm_a ) )
= ( ( A2 = B )
& ( ord_le2931979837777675552c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ B @ bot_bo8284763806281179116c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1041_singleton__insert__inj__eq,axiom,
! [B: a,A2: a,A: set_a] :
( ( ( insert_a2 @ B @ bot_bot_set_a )
= ( insert_a2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1042_singleton__insert__inj__eq,axiom,
! [B: epistemic_fm_a,A2: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a )
= ( insert7817948963269374698c_fm_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1043_singleton__insert__inj__eq,axiom,
! [B: list_Epistemic_fm_a,A2: list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ( insert6567282714061225712c_fm_a @ B @ bot_bo8284763806281179116c_fm_a )
= ( insert6567282714061225712c_fm_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le2931979837777675552c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ B @ bot_bo8284763806281179116c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1044_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1045_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( insert7817948963269374698c_fm_a @ X21 @ ( set_Epistemic_fm_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1046_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_1047_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a2 @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_1048_List_Oset__insert,axiom,
! [X3: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X3 @ Xs ) )
= ( insert_a2 @ X3 @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1049_List_Oset__insert,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( insert177310161492556854c_fm_a @ X3 @ Xs ) )
= ( insert7817948963269374698c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1050_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z4: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z4 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ Z4 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_1051_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z4: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z4 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Z4 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_1052_Nil__in__shufflesI,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs = nil_Epistemic_fm_a )
=> ( ( Ys = nil_Epistemic_fm_a )
=> ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_1053_set__shuffles,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( ( set_Epistemic_fm_a2 @ Zs3 )
= ( sup_su1367922730591523534c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1054_set__shuffles,axiom,
! [Zs3: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs3 @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( set_a2 @ Zs3 )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1055_filter__shuffles,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( image_7387318756023638021c_fm_a @ ( filter7636273809395300631c_fm_a @ P4 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( shuffl511019022169630517c_fm_a @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) @ ( filter7636273809395300631c_fm_a @ P4 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_1056_Un__Pow__subset,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] : ( ord_le7112219575281605754c_fm_a @ ( sup_su4305657963326700718c_fm_a @ ( pow_Epistemic_fm_a @ A ) @ ( pow_Epistemic_fm_a @ B3 ) ) @ ( pow_Epistemic_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% Un_Pow_subset
thf(fact_1057_Un__Pow__subset,axiom,
! [A: set_a,B3: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B3 ) ) @ ( pow_a @ ( sup_sup_set_a @ A @ B3 ) ) ) ).
% Un_Pow_subset
thf(fact_1058_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a2 @ A2 @ C2 ) @ ( insert_a2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1059_insert__mono,axiom,
! [C2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ A2 @ C2 ) @ ( insert7817948963269374698c_fm_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1060_insert__mono,axiom,
! [C2: set_li769143395467472256c_fm_a,D: set_li769143395467472256c_fm_a,A2: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C2 @ D )
=> ( ord_le2931979837777675552c_fm_a @ ( insert6567282714061225712c_fm_a @ A2 @ C2 ) @ ( insert6567282714061225712c_fm_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_1061_subset__insert,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ X3 @ B3 ) )
= ( ord_le7112219575281605754c_fm_a @ A @ B3 ) ) ) ).
% subset_insert
thf(fact_1062_subset__insert,axiom,
! [X3: a,A: set_a,B3: set_a] :
( ~ ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X3 @ B3 ) )
= ( ord_less_eq_set_a @ A @ B3 ) ) ) ).
% subset_insert
thf(fact_1063_subset__insert,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X3 @ B3 ) )
= ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ) ).
% subset_insert
thf(fact_1064_subset__insert,axiom,
! [X3: list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ~ ( member5906877432388582473c_fm_a @ X3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ X3 @ B3 ) )
= ( ord_le2931979837777675552c_fm_a @ A @ B3 ) ) ) ).
% subset_insert
thf(fact_1065_subset__insertI,axiom,
! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a2 @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1066_subset__insertI,axiom,
! [B3: set_Epistemic_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B3 @ ( insert7817948963269374698c_fm_a @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1067_subset__insertI,axiom,
! [B3: set_li769143395467472256c_fm_a,A2: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ B3 @ ( insert6567282714061225712c_fm_a @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_1068_subset__insertI2,axiom,
! [A: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1069_subset__insertI2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1070_subset__insertI2,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,B: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1071_insert__subsetI,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ X6 @ A )
=> ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1072_insert__subsetI,axiom,
! [X3: a,A: set_a,X6: set_a] :
( ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1073_insert__subsetI,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,X6: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1074_insert__subsetI,axiom,
! [X3: list_Epistemic_fm_a,A: set_li769143395467472256c_fm_a,X6: set_li769143395467472256c_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ X6 @ A )
=> ( ord_le2931979837777675552c_fm_a @ ( insert6567282714061225712c_fm_a @ X3 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1075_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1076_subset__Un__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1077_subset__Un__eq,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( sup_su1755902861286427476c_fm_a @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1078_subset__UnE,axiom,
! [C2: set_a,A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B3 ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B3 )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1079_subset__UnE,axiom,
! [C2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) )
=> ~ ! [A6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A6 @ A )
=> ! [B6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B6 @ B3 )
=> ( C2
!= ( sup_su1367922730591523534c_fm_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1080_subset__UnE,axiom,
! [C2: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C2 @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) )
=> ~ ! [A6: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A6 @ A )
=> ! [B6: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B6 @ B3 )
=> ( C2
!= ( sup_su1755902861286427476c_fm_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1081_Un__absorb2,axiom,
! [B3: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ( sup_sup_set_a @ A @ B3 )
= A ) ) ).
% Un_absorb2
thf(fact_1082_Un__absorb2,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B3 )
= A ) ) ).
% Un_absorb2
thf(fact_1083_Un__absorb2,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ A )
=> ( ( sup_su1755902861286427476c_fm_a @ A @ B3 )
= A ) ) ).
% Un_absorb2
thf(fact_1084_Un__absorb1,axiom,
! [A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( sup_sup_set_a @ A @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_1085_Un__absorb1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_1086_Un__absorb1,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( sup_su1755902861286427476c_fm_a @ A @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_1087_Un__upper2,axiom,
! [B3: set_a,A: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A @ B3 ) ) ).
% Un_upper2
thf(fact_1088_Un__upper2,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B3 @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ).
% Un_upper2
thf(fact_1089_Un__upper2,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ B3 @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) ) ).
% Un_upper2
thf(fact_1090_Un__upper1,axiom,
! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B3 ) ) ).
% Un_upper1
thf(fact_1091_Un__upper1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ).
% Un_upper1
thf(fact_1092_Un__upper1,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) ) ).
% Un_upper1
thf(fact_1093_Un__least,axiom,
! [A: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1094_Un__least,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1095_Un__least,axiom,
! [A: set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ C2 )
=> ( ( ord_le2931979837777675552c_fm_a @ B3 @ C2 )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1096_Un__mono,axiom,
! [A: set_a,C2: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1097_Un__mono,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ ( sup_su1367922730591523534c_fm_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1098_Un__mono,axiom,
! [A: set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a,D: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ C2 )
=> ( ( ord_le2931979837777675552c_fm_a @ B3 @ D )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A @ B3 ) @ ( sup_su1755902861286427476c_fm_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1099_Cons__shuffles__subset1,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ X3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_1100_Cons__shuffles__subset2,axiom,
! [Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ Y ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_1101_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_1102_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_1103_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_1104_sup_OcoboundedI2,axiom,
! [C: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C @ B )
=> ( ord_le2931979837777675552c_fm_a @ C @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1105_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_1106_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_1107_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_1108_sup_OcoboundedI1,axiom,
! [C: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C @ A2 )
=> ( ord_le2931979837777675552c_fm_a @ C @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1109_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_1110_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_1111_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_1112_sup_Oabsorb__iff2,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A4: set_li769143395467472256c_fm_a,B2: set_li769143395467472256c_fm_a] :
( ( sup_su1755902861286427476c_fm_a @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_1113_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_1114_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_1115_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_1116_sup_Oabsorb__iff1,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [B2: set_li769143395467472256c_fm_a,A4: set_li769143395467472256c_fm_a] :
( ( sup_su1755902861286427476c_fm_a @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1117_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_1118_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_1119_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_1120_sup_Ocobounded2,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ B @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1121_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_1122_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_1123_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_1124_sup_Ocobounded1,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A2 @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1125_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_1126_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_1127_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_1128_sup_Oorder__iff,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [B2: set_li769143395467472256c_fm_a,A4: set_li769143395467472256c_fm_a] :
( A4
= ( sup_su1755902861286427476c_fm_a @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_1129_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_1130_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_1131_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_1132_sup_OboundedI,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ C @ A2 )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1133_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_1134_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_1135_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_1136_sup_OboundedE,axiom,
! [B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ~ ( ord_le2931979837777675552c_fm_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1137_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_1138_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_1139_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_1140_sup__absorb2,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ Y )
=> ( ( sup_su1755902861286427476c_fm_a @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1141_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_1142_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_1143_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_1144_sup__absorb1,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( ( sup_su1755902861286427476c_fm_a @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_1145_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_1146_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_1147_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_1148_sup_Oabsorb2,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( sup_su1755902861286427476c_fm_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1149_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_1150_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_1151_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_1152_sup_Oabsorb1,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( sup_su1755902861286427476c_fm_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1153_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,Z: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1154_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,Z: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X2 )
=> ( ( ord_le4043730696559282883fm_a_o @ Z @ X2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y2 @ Z ) @ X2 ) ) )
=> ( ( sup_su6697315146409554831fm_a_o @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1155_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,Z: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y2 @ Z ) @ X2 ) ) )
=> ( ( sup_su1367922730591523534c_fm_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1156_sup__unique,axiom,
! [F: set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a > set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a,Z: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y2 @ X2 )
=> ( ( ord_le2931979837777675552c_fm_a @ Z @ X2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ Y2 @ Z ) @ X2 ) ) )
=> ( ( sup_su1755902861286427476c_fm_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1157_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_1158_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_1159_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_1160_sup_OorderI,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( A2
= ( sup_su1755902861286427476c_fm_a @ A2 @ B ) )
=> ( ord_le2931979837777675552c_fm_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1161_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_1162_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_1163_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_1164_sup_OorderE,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( A2
= ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1165_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_1166_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_1167_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_1168_le__iff__sup,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [X: set_li769143395467472256c_fm_a,Y4: set_li769143395467472256c_fm_a] :
( ( sup_su1755902861286427476c_fm_a @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1169_sup__least,axiom,
! [Y: set_a,X3: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ Z4 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X3 ) ) ) ).
% sup_least
thf(fact_1170_sup__least,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ Z4 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ Y @ Z4 ) @ X3 ) ) ) ).
% sup_least
thf(fact_1171_sup__least,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z4 @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ Y @ Z4 ) @ X3 ) ) ) ).
% sup_least
thf(fact_1172_sup__least,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y @ X3 )
=> ( ( ord_le2931979837777675552c_fm_a @ Z4 @ X3 )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ Y @ Z4 ) @ X3 ) ) ) ).
% sup_least
thf(fact_1173_sup__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1174_sup__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,D2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ D2 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ ( sup_su6697315146409554831fm_a_o @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1175_sup__mono,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1176_sup__mono,axiom,
! [A2: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,D2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ C )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ D2 )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) @ ( sup_su1755902861286427476c_fm_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1177_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1178_sup_Omono,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,D2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ D2 @ B )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ C @ D2 ) @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1179_sup_Omono,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,D2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ D2 @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1180_sup_Omono,axiom,
! [C: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,D2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ C @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ D2 @ B )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ C @ D2 ) @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1181_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_1182_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_1183_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_1184_le__supI2,axiom,
! [X3: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ B )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1185_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_1186_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_1187_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_1188_le__supI1,axiom,
! [X3: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ A2 )
=> ( ord_le2931979837777675552c_fm_a @ X3 @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1189_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_1190_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_1191_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_1192_sup__ge2,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ Y @ ( sup_su1755902861286427476c_fm_a @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_1193_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_1194_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_1195_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_1196_sup__ge1,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ X3 @ ( sup_su1755902861286427476c_fm_a @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_1197_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_1198_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_1199_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_1200_le__supI,axiom,
! [A2: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ X3 )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ X3 )
=> ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_1201_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_1202_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_1203_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_1204_le__supE,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( sup_su1755902861286427476c_fm_a @ A2 @ B ) @ X3 )
=> ~ ( ( ord_le2931979837777675552c_fm_a @ A2 @ X3 )
=> ~ ( ord_le2931979837777675552c_fm_a @ B @ X3 ) ) ) ).
% le_supE
thf(fact_1205_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_1206_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_1207_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_1208_inf__sup__ord_I3_J,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ X3 @ ( sup_su1755902861286427476c_fm_a @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1209_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_1210_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_1211_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_1212_inf__sup__ord_I4_J,axiom,
! [Y: set_li769143395467472256c_fm_a,X3: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ Y @ ( sup_su1755902861286427476c_fm_a @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1213_consistent__disjuncts,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Dis_a @ P @ Q ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
| ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_1214_partition__in__shuffles,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( member5906877432388582473c_fm_a @ Xs
@ ( shuffl511019022169630517c_fm_a @ ( filter7636273809395300631c_fm_a @ P4 @ Xs )
@ ( filter7636273809395300631c_fm_a
@ ^ [X: epistemic_fm_a] :
~ ( P4 @ X )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_1215_maximal__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( ! [P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P2 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ P2 @ S ) ) ) ) ).
% maximal_def
thf(fact_1216_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1217_subset__singleton__iff,axiom,
! [X6: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X6 @ ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) )
= ( ( X6 = bot_bo3626323581529592678c_fm_a )
| ( X6
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1218_subset__singleton__iff,axiom,
! [X6: set_li769143395467472256c_fm_a,A2: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X6 @ ( insert6567282714061225712c_fm_a @ A2 @ bot_bo8284763806281179116c_fm_a ) )
= ( ( X6 = bot_bo8284763806281179116c_fm_a )
| ( X6
= ( insert6567282714061225712c_fm_a @ A2 @ bot_bo8284763806281179116c_fm_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1219_subset__singletonD,axiom,
! [A: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X3 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1220_subset__singletonD,axiom,
! [A: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) )
=> ( ( A = bot_bo3626323581529592678c_fm_a )
| ( A
= ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singletonD
thf(fact_1221_subset__singletonD,axiom,
! [A: set_li769143395467472256c_fm_a,X3: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ X3 @ bot_bo8284763806281179116c_fm_a ) )
=> ( ( A = bot_bo8284763806281179116c_fm_a )
| ( A
= ( insert6567282714061225712c_fm_a @ X3 @ bot_bo8284763806281179116c_fm_a ) ) ) ) ).
% subset_singletonD
thf(fact_1222_consistent__consequent,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_1223_consistent__consequent_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_1224_shufflesE,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( ( ( Zs3 = Xs )
=> ( Ys != nil_Epistemic_fm_a ) )
=> ( ( ( Zs3 = Ys )
=> ( Xs != nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a,Xs4: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ X2 @ Xs4 ) )
=> ! [Z: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z @ Zs4 ) )
=> ( ( X2 = Z )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y2: epistemic_fm_a,Ys4: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Y2 @ Ys4 ) )
=> ! [Z: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z @ Zs4 ) )
=> ( ( Y2 = Z )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_set__rec,axiom,
( set_a2
= ( rec_list_set_a_a @ bot_bot_set_a
@ ^ [X: a,Uu2: list_a] : ( insert_a2 @ X ) ) ) ).
% set_rec
thf(fact_1232_set__rec,axiom,
( set_Epistemic_fm_a2
= ( rec_li8401250755714504918c_fm_a @ bot_bo3626323581529592678c_fm_a
@ ^ [X: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( insert7817948963269374698c_fm_a @ X ) ) ) ).
% set_rec
thf(fact_1233_union__coset__filter,axiom,
! [Xs: list_s580375451141968640c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( sup_su4305657963326700718c_fm_a @ ( coset_774667214334347501c_fm_a @ Xs ) @ A )
= ( coset_774667214334347501c_fm_a
@ ( filter2620125754611450615c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_1234_union__coset__filter,axiom,
! [Xs: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ ( coset_Epistemic_fm_a @ Xs ) @ A )
= ( coset_Epistemic_fm_a
@ ( filter7636273809395300631c_fm_a
@ ^ [X: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_1235_union__coset__filter,axiom,
! [Xs: list_a,A: set_a] :
( ( sup_sup_set_a @ ( coset_a @ Xs ) @ A )
= ( coset_a
@ ( filter_a
@ ^ [X: a] :
~ ( member_a2 @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_1236_frame_Oselect__convs_I3_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W6 @ K2 @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_1237_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: 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 @ G ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% inconsistent_imply
thf(fact_1238_set__union,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( union_Epistemic_fm_a @ Xs @ Ys ) )
= ( sup_su1367922730591523534c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1239_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1240_extendS__def,axiom,
! [A: epistemic_fm_a > $o,N: epistemic_fm_a,Prev: set_Epistemic_fm_a,S: 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 ) @ S @ 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 ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_1241_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V: 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 ) @ V ) )
=> ~ ! [W4: 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 ) @ W4 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W4 )
=> ( ( finite3304564945125563331c_fm_a @ W4 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1242_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_a] : ( finite3304564945125563331c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1243_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [B4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1244_finite__Collect__subsets,axiom,
! [A: set_li769143395467472256c_fm_a] :
( ( finite2538126183821535049c_fm_a @ A )
=> ( finite7020096028535324585c_fm_a
@ ( collec4111825885563579627c_fm_a
@ ^ [B4: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1245_shuffles_Oelims,axiom,
! [X3: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ( shuffl511019022169630517c_fm_a @ X3 @ Xa )
= Y )
=> ( ( ( X3 = nil_Epistemic_fm_a )
=> ( Y
!= ( insert6567282714061225712c_fm_a @ Xa @ bot_bo8284763806281179116c_fm_a ) ) )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ( Y
!= ( insert6567282714061225712c_fm_a @ X3 @ bot_bo8284763806281179116c_fm_a ) ) )
=> ~ ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( X3
= ( cons_Epistemic_fm_a @ X2 @ Xs2 ) )
=> ! [Y2: epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ Y2 @ Ys3 ) )
=> ( Y
!= ( sup_su1755902861286427476c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ X2 ) @ ( shuffl511019022169630517c_fm_a @ Xs2 @ ( cons_Epistemic_fm_a @ Y2 @ Ys3 ) ) ) @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ Y2 ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1246_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 )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1247_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 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1248_finite__has__maximal2,axiom,
! [A: set_se4933744490392479200c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( finite7020096028535324585c_fm_a @ A )
=> ( ( member2517045796615263401c_fm_a @ A2 @ A )
=> ? [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
& ( ord_le2931979837777675552c_fm_a @ A2 @ X2 )
& ! [Xa3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Xa3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1249_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 )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1250_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 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1251_finite__has__minimal2,axiom,
! [A: set_se4933744490392479200c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( finite7020096028535324585c_fm_a @ A )
=> ( ( member2517045796615263401c_fm_a @ A2 @ A )
=> ? [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
& ( ord_le2931979837777675552c_fm_a @ X2 @ A2 )
& ! [Xa3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Xa3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1252_rev__finite__subset,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( finite3304564945125563331c_fm_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1253_rev__finite__subset,axiom,
! [B3: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a] :
( ( finite2538126183821535049c_fm_a @ B3 )
=> ( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( finite2538126183821535049c_fm_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_1254_infinite__super,axiom,
! [S: set_Epistemic_fm_a,T3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S @ T3 )
=> ( ~ ( finite3304564945125563331c_fm_a @ S )
=> ~ ( finite3304564945125563331c_fm_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1255_infinite__super,axiom,
! [S: set_li769143395467472256c_fm_a,T3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ S @ T3 )
=> ( ~ ( finite2538126183821535049c_fm_a @ S )
=> ~ ( finite2538126183821535049c_fm_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1256_finite__subset,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( finite3304564945125563331c_fm_a @ B3 )
=> ( finite3304564945125563331c_fm_a @ A ) ) ) ).
% finite_subset
thf(fact_1257_finite__subset,axiom,
! [A: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B3 )
=> ( ( finite2538126183821535049c_fm_a @ B3 )
=> ( finite2538126183821535049c_fm_a @ A ) ) ) ).
% finite_subset
thf(fact_1258_finite__list,axiom,
! [A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A )
=> ? [Xs2: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_1259_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 )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1260_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 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1261_finite__has__minimal,axiom,
! [A: set_se4933744490392479200c_fm_a] :
( ( finite7020096028535324585c_fm_a @ A )
=> ( ( A != bot_bo8499242532914112588c_fm_a )
=> ? [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
& ! [Xa3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Xa3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1262_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 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1263_finite__has__maximal,axiom,
! [A: set_se4933744490392479200c_fm_a] :
( ( finite7020096028535324585c_fm_a @ A )
=> ( ( A != bot_bo8499242532914112588c_fm_a )
=> ? [X2: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ X2 @ A )
& ! [Xa3: set_li769143395467472256c_fm_a] :
( ( member2517045796615263401c_fm_a @ Xa3 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1264_lists__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( lists_Epistemic_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) )
= ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) ) ) ).
% lists_image
thf(fact_1265_map__lists__mono,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B3 )
=> ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) ) @ ( lists_Epistemic_fm_a @ B3 ) ) ) ).
% map_lists_mono
thf(fact_1266_map__lists__surjective,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ( image_4449434806354059013c_fm_a @ F @ A )
= B3 )
=> ( ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) )
= ( lists_Epistemic_fm_a @ B3 ) ) ) ).
% map_lists_surjective
thf(fact_1267_sorted__wrt__map,axiom,
! [R2: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( sorted73444913887879815c_fm_a @ R2 @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( sorted73444913887879815c_fm_a
@ ^ [X: epistemic_fm_a,Y4: epistemic_fm_a] : ( R2 @ ( F @ X ) @ ( F @ Y4 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1268_map__append,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) @ ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_1269_map__eq__append__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( append9179727413925872949c_fm_a @ Ys @ Zs3 ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Us @ Vs ) )
& ( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Us ) )
& ( Zs3
= ( map_Ep7084560364594560580c_fm_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1270_append__eq__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Ys @ Zs3 )
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Us @ Vs ) )
& ( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Us ) )
& ( Zs3
= ( map_Ep7084560364594560580c_fm_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1271_distinct__map__filter,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( distin5379319156202237459c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
=> ( distin5379319156202237459c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_1272_rotate1__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( rotate3540958558250653293c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ ( rotate3540958558250653293c_fm_a @ Xs ) ) ) ).
% rotate1_map
thf(fact_1273_length__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( size_s2789134727074377100c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( size_s2789134727074377100c_fm_a @ Xs ) ) ).
% length_map
thf(fact_1274_map__eq__imp__length__eq,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ys ) )
=> ( ( size_s2789134727074377100c_fm_a @ Xs )
= ( size_s2789134727074377100c_fm_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
% Helper facts (13)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $true @ X3 @ Y )
= X3 ) ).
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_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 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X3: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X3: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( if_set4000483370245493318c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X3: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( if_set4000483370245493318c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ i ) @ g ) ) @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ g ) ) ) ).
%------------------------------------------------------------------------------