TPTP Problem File: SLH0551^1.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_00326_011016__6649474_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1507 ( 641 unt; 219 typ; 0 def)
% Number of atoms : 3624 (1179 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 11819 ( 398 ~; 49 |; 228 &;9699 @)
% ( 0 <=>;1445 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 1085 (1085 >; 0 *; 0 +; 0 <<)
% Number of symbols : 203 ( 200 usr; 16 con; 0-4 aty)
% Number of variables : 3805 ( 441 ^;3289 !; 75 ?;3805 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:21.317
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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thf(ty_n_tf__i,type,
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% Explicit typings (200)
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
bot_bo4781621276559889198c_fm_i: set_se7339729205154126530c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__i_J,type,
bot_bot_set_i: set_i ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
ord_le190830114487426235fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le5706303257367236315fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
ord_le3843937902494030498c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le1209247765174941992c_fm_i: set_li8269783359719514504c_fm_i > set_li8269783359719514504c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
ord_le1241842563113145096c_fm_i: set_li2934481837907835752c_fm_i > set_li2934481837907835752c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le5389487502678872194c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
ord_le2772500130360476258c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__i_J,type,
ord_less_eq_set_i: set_i > set_i > $o ).
thf(sy_c_Partial__Function_Oflat__lub_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
partia6511039473171299566c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Partial__Function_Oflat__lub_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
partia2237093934842463950c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
collec4904205187116291597c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec5759350117362652307c_fm_i: ( list_Epistemic_fm_i > $o ) > set_li8269783359719514504c_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2200325868635342451c_fm_i: ( list_s8081015415394010888c_fm_i > $o ) > set_li2934481837907835752c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec3087743281813070829c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2761885132556217805c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_OCollect_001tf__i,type,
collect_i: ( i > $o ) > set_i ).
thf(sy_c_Set_Ofilter_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
filter6053540608743173075c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
filter8450032351001387443c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
image_120052958407486725c_fm_i: ( epistemic_fm_i > epistemic_fm_i ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_2913337242412145381c_fm_i: ( epistemic_fm_i > set_Epistemic_fm_i ) > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_8661220427522574853c_fm_i: ( list_Epistemic_fm_i > list_Epistemic_fm_i ) > set_li8269783359719514504c_fm_i > set_li8269783359719514504c_fm_i ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
image_7977712360212772357c_fm_i: ( list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ) > set_li2934481837907835752c_fm_i > set_li2934481837907835752c_fm_i ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
image_809639422248493861c_fm_i: ( set_Epistemic_fm_i > epistemic_fm_i ) > set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_2698589095351095557c_fm_i: ( set_Epistemic_fm_i > set_Epistemic_fm_i ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
insert7817948997695205106c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
insert7698009978809854162c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
insert7111703059596746418c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_Oinsert_001tf__i,type,
insert_i: i > set_i > set_i ).
thf(sy_c_Set_Ois__empty_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_emp7643256682846039796c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
is_emp8832846348672524500c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Set_Ois__singleton_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_sin4923117429174228502c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
is_sin2358732406619996150c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Set_Oremove_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove5552923401211475591c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
remove1392734035364174951c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Othe__elem_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
the_el2173195912186371479c_fm_i: set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
the_el1853918686291641207c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__i,type,
stalnaker_Ax_2_i: epistemic_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__i,type,
stalnaker_conjunct_i: list_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
member6642669606046002379c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member6475149752759350353c_fm_i: list_Epistemic_fm_i > set_li8269783359719514504c_fm_i > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member605545779687026225c_fm_i: list_s8081015415394010888c_fm_i > set_li2934481837907835752c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1104366573291651755c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_member_001tf__i,type,
member_i: i > set_i > $o ).
thf(sy_v_A,type,
a: epistemic_fm_i > $o ).
thf(sy_v_S_H____,type,
s: list_Epistemic_fm_i ).
thf(sy_v_U,type,
u: set_Epistemic_fm_i ).
thf(sy_v_V,type,
v: set_Epistemic_fm_i ).
thf(sy_v_W,type,
w: set_Epistemic_fm_i ).
thf(sy_v_i,type,
i2: i ).
% Relevant facts (1278)
thf(fact_0_assms_I1_J,axiom,
! [P: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P )
=> ( a @ P ) ) ).
% assms(1)
thf(fact_1__C_K_C_I3_J,axiom,
finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ s ) ).
% "*"(3)
thf(fact_2__C_K_C_I1_J,axiom,
epistemic_AK_i @ a @ ( epistemic_imply_i @ s @ epistemic_FF_i ) ).
% "*"(1)
thf(fact_3__C_K_C_I2_J,axiom,
( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ s )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ u ) ) ) ) ).
% "*"(2)
thf(fact_4_assms_I9_J,axiom,
( member1104366573291651755c_fm_i @ u
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ v ) ) ) ) ) ).
% assms(9)
thf(fact_5_assms_I8_J,axiom,
( member1104366573291651755c_fm_i @ w
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ v ) ) ) ) ) ).
% assms(8)
thf(fact_6_set__filter,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,Xs: list_s8635415444435394792c_fm_i] :
( ( set_se5076022966804059383c_fm_i @ ( filter2303706508003015903c_fm_i @ P3 @ Xs ) )
= ( collec2761885132556217805c_fm_i
@ ^ [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( set_se5076022966804059383c_fm_i @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_7_set__filter,axiom,
! [P3: i > $o,Xs: list_i] :
( ( set_i2 @ ( filter_i @ P3 @ Xs ) )
= ( collect_i
@ ^ [X: i] :
( ( member_i @ X @ ( set_i2 @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_8_set__filter,axiom,
! [P3: list_Epistemic_fm_i > $o,Xs: list_l4360594050116504750c_fm_i] :
( ( set_li9010496130497932989c_fm_i @ ( filter1009751957495906981c_fm_i @ P3 @ Xs ) )
= ( collec5759350117362652307c_fm_i
@ ^ [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_9_set__filter,axiom,
! [P3: list_s8081015415394010888c_fm_i > $o,Xs: list_l8939085672862518926c_fm_i] :
( ( set_li1699392436468070557c_fm_i @ ( filter251276092929125509c_fm_i @ P3 @ Xs ) )
= ( collec2200325868635342451c_fm_i
@ ^ [X: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X @ ( set_li1699392436468070557c_fm_i @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_10_set__filter,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) )
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_11_set__filter,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) )
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P3 @ X ) ) ) ) ).
% set_filter
thf(fact_12_filter__True,axiom,
! [Xs: list_l4360594050116504750c_fm_i,P3: list_Epistemic_fm_i > $o] :
( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( filter1009751957495906981c_fm_i @ P3 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_13_filter__True,axiom,
! [Xs: list_l8939085672862518926c_fm_i,P3: list_s8081015415394010888c_fm_i > $o] :
( ! [X2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( filter251276092929125509c_fm_i @ P3 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_14_filter__True,axiom,
! [Xs: list_s8081015415394010888c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( filter3188398074982218495c_fm_i @ P3 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_15_filter__True,axiom,
! [Xs: list_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( filter7636273843821131039c_fm_i @ P3 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_16__092_060open_062_092_060not_062_Aconsistent_AA_A_Iknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_J_092_060close_062,axiom,
~ ( episte2285483198712856234tent_i @ a
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ u ) ) ) ) ).
% \<open>\<not> consistent A (known W i \<union> known U i)\<close>
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_H_O_A_092_060lbrakk_062A_A_092_060turnstile_062_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_059_Aset_AS_H_A_092_060subseteq_062_Aknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_059_Afinite_A_Iset_AS_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ a @ ( epistemic_imply_i @ S @ epistemic_FF_i ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ u ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S ) ) ) ) ).
% \<open>\<And>thesis. (\<And>S'. \<lbrakk>A \<turnstile> S' \<^bold>\<leadsto> \<^bold>\<bottom>; set S' \<subseteq> known W i \<union> known U i; finite (set S')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_filter__filter,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter7636273843821131039c_fm_i @ P3 @ ( filter7636273843821131039c_fm_i @ Q @ Xs ) )
= ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( Q @ X )
& ( P3 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_19_filter__filter,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( filter3188398074982218495c_fm_i @ P3 @ ( filter3188398074982218495c_fm_i @ Q @ Xs ) )
= ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( Q @ X )
& ( P3 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_20_assms_I6_J,axiom,
episte2285483198712856234tent_i @ a @ u ).
% assms(6)
thf(fact_21_assms_I4_J,axiom,
episte2285483198712856234tent_i @ a @ w ).
% assms(4)
thf(fact_22_fm_Oinject_I5_J,axiom,
! [X61: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i,Y61: set_Epistemic_fm_i,Y62: episte4276422356747265950c_fm_i] :
( ( ( episte3581967596712343542c_fm_i @ X61 @ X62 )
= ( episte3581967596712343542c_fm_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_23_fm_Oinject_I5_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,Y61: epistemic_fm_i,Y62: episte1308613106011496894c_fm_i] :
( ( ( episte5657488666450005526c_fm_i @ X61 @ X62 )
= ( episte5657488666450005526c_fm_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_24_fm_Oinject_I5_J,axiom,
! [X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( ( epistemic_K_i @ X61 @ X62 )
= ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_25_UnCI,axiom,
! [C: list_s8081015415394010888c_fm_i,B: set_li2934481837907835752c_fm_i,A: set_li2934481837907835752c_fm_i] :
( ( ~ ( member605545779687026225c_fm_i @ C @ B )
=> ( member605545779687026225c_fm_i @ C @ A ) )
=> ( member605545779687026225c_fm_i @ C @ ( sup_su6123556432952969532c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_26_UnCI,axiom,
! [C: list_Epistemic_fm_i,B: set_li8269783359719514504c_fm_i,A: set_li8269783359719514504c_fm_i] :
( ( ~ ( member6475149752759350353c_fm_i @ C @ B )
=> ( member6475149752759350353c_fm_i @ C @ A ) )
=> ( member6475149752759350353c_fm_i @ C @ ( sup_su33170788683693916c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_27_UnCI,axiom,
! [C: i,B: set_i,A: set_i] :
( ( ~ ( member_i @ C @ B )
=> ( member_i @ C @ A ) )
=> ( member_i @ C @ ( sup_sup_set_i @ A @ B ) ) ) ).
% UnCI
thf(fact_28_UnCI,axiom,
! [C: set_se3485332733965609186c_fm_i,B: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i] :
( ( ~ ( member1461078328125707403c_fm_i @ C @ B )
=> ( member1461078328125707403c_fm_i @ C @ A ) )
=> ( member1461078328125707403c_fm_i @ C @ ( sup_su2317974602288960150c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_29_UnCI,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ A ) )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_30_UnCI,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ A ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_31_Un__iff,axiom,
! [C: list_s8081015415394010888c_fm_i,A: set_li2934481837907835752c_fm_i,B: set_li2934481837907835752c_fm_i] :
( ( member605545779687026225c_fm_i @ C @ ( sup_su6123556432952969532c_fm_i @ A @ B ) )
= ( ( member605545779687026225c_fm_i @ C @ A )
| ( member605545779687026225c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_32_Un__iff,axiom,
! [C: list_Epistemic_fm_i,A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( member6475149752759350353c_fm_i @ C @ ( sup_su33170788683693916c_fm_i @ A @ B ) )
= ( ( member6475149752759350353c_fm_i @ C @ A )
| ( member6475149752759350353c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_33_Un__iff,axiom,
! [C: i,A: set_i,B: set_i] :
( ( member_i @ C @ ( sup_sup_set_i @ A @ B ) )
= ( ( member_i @ C @ A )
| ( member_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_34_Un__iff,axiom,
! [C: set_se3485332733965609186c_fm_i,A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( member1461078328125707403c_fm_i @ C @ ( sup_su2317974602288960150c_fm_i @ A @ B ) )
= ( ( member1461078328125707403c_fm_i @ C @ A )
| ( member1461078328125707403c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_35_Un__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
| ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_36_Un__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
| ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_37_sup_Oidem,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_38_sup_Oidem,axiom,
! [A2: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_39_sup_Oidem,axiom,
! [A2: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_40_sup_Oidem,axiom,
! [A2: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_41_sup_Oidem,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_42_sup__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_43_sup__idem,axiom,
! [X3: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_44_sup__idem,axiom,
! [X3: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_45_sup__idem,axiom,
! [X3: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_46_sup__idem,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_47_assms_I2_J,axiom,
episte2285483198712856234tent_i @ a @ v ).
% assms(2)
thf(fact_48_subsetI,axiom,
! [A: set_li2934481837907835752c_fm_i,B: set_li2934481837907835752c_fm_i] :
( ! [X2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X2 @ A )
=> ( member605545779687026225c_fm_i @ X2 @ B ) )
=> ( ord_le1241842563113145096c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_49_subsetI,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ A )
=> ( member6475149752759350353c_fm_i @ X2 @ B ) )
=> ( ord_le1209247765174941992c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_50_subsetI,axiom,
! [A: set_i,B: set_i] :
( ! [X2: i] :
( ( member_i @ X2 @ A )
=> ( member_i @ X2 @ B ) )
=> ( ord_less_eq_set_i @ A @ B ) ) ).
% subsetI
thf(fact_51_subsetI,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A )
=> ( member1461078328125707403c_fm_i @ X2 @ B ) )
=> ( ord_le2772500130360476258c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_52_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( member1104366573291651755c_fm_i @ X2 @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_53_subsetI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( member6642669606046002379c_fm_i @ X2 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_54_subset__antisym,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_55_subset__antisym,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_56_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_57_sup__apply,axiom,
( sup_su2844414564337698183fm_i_o
= ( ^ [F: epistemic_fm_i > $o,G: epistemic_fm_i > $o,X: epistemic_fm_i] : ( sup_sup_o @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% sup_apply
thf(fact_58_sup__apply,axiom,
( sup_su6723579903606884263fm_i_o
= ( ^ [F: set_Epistemic_fm_i > $o,G: set_Epistemic_fm_i > $o,X: set_Epistemic_fm_i] : ( sup_sup_o @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% sup_apply
thf(fact_59_sup_Oright__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) @ B2 )
= ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_60_sup_Oright__idem,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) @ B2 )
= ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_61_sup_Oright__idem,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) @ B2 )
= ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_62_sup_Oright__idem,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) @ B2 )
= ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_63_sup_Oright__idem,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ B2 )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_64_sup__left__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) )
= ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_65_sup__left__idem,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) )
= ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_66_sup__left__idem,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) )
= ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_67_sup__left__idem,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) )
= ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_68_sup__left__idem,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_69_sup_Oleft__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A2 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) )
= ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_70_sup_Oleft__idem,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ A2 @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) )
= ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_71_sup_Oleft__idem,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ A2 @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) )
= ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_72_sup_Oleft__idem,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ A2 @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) )
= ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_73_sup_Oleft__idem,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_74_sup_Obounded__iff,axiom,
! [B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B2 @ C ) @ A2 )
= ( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
& ( ord_le5389487502678872194c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_75_sup_Obounded__iff,axiom,
! [B2: set_se7339729205154126530c_fm_i,C: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ B2 @ C ) @ A2 )
= ( ( ord_le2772500130360476258c_fm_i @ B2 @ A2 )
& ( ord_le2772500130360476258c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_76_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_77_sup_Obounded__iff,axiom,
! [B2: epistemic_fm_i > $o,C: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ B2 @ C ) @ A2 )
= ( ( ord_le190830114487426235fm_i_o @ B2 @ A2 )
& ( ord_le190830114487426235fm_i_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_78_sup_Obounded__iff,axiom,
! [B2: set_Epistemic_fm_i > $o,C: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ B2 @ C ) @ A2 )
= ( ( ord_le5706303257367236315fm_i_o @ B2 @ A2 )
& ( ord_le5706303257367236315fm_i_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_79_sup_Obounded__iff,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 )
= ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_80_le__sup__iff,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ Z )
= ( ( ord_le5389487502678872194c_fm_i @ X3 @ Z )
& ( ord_le5389487502678872194c_fm_i @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_81_le__sup__iff,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i,Z: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) @ Z )
= ( ( ord_le2772500130360476258c_fm_i @ X3 @ Z )
& ( ord_le2772500130360476258c_fm_i @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_82_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_83_le__sup__iff,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) @ Z )
= ( ( ord_le190830114487426235fm_i_o @ X3 @ Z )
& ( ord_le190830114487426235fm_i_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_84_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o,Z: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) @ Z )
= ( ( ord_le5706303257367236315fm_i_o @ X3 @ Z )
& ( ord_le5706303257367236315fm_i_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_85_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z )
& ( ord_le3843937902494030498c_fm_i @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_86_List_Ofinite__set,axiom,
! [Xs: list_s8635415444435394792c_fm_i] : ( finite9087983574947184523c_fm_i @ ( set_se5076022966804059383c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_87_List_Ofinite__set,axiom,
! [Xs: list_l4360594050116504750c_fm_i] : ( finite3106398504192302929c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_88_List_Ofinite__set,axiom,
! [Xs: list_l8939085672862518926c_fm_i] : ( finite5108596011607087409c_fm_i @ ( set_li1699392436468070557c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_89_List_Ofinite__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i] : ( finite7933139204641697195c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_90_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_i] : ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% List.finite_set
thf(fact_91_Un__subset__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
& ( ord_le5389487502678872194c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_92_Un__subset__iff,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,C2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le2772500130360476258c_fm_i @ A @ C2 )
& ( ord_le2772500130360476258c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_93_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_94_assms_I5_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ w ).
% assms(5)
thf(fact_95_assms_I7_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ u ).
% assms(7)
thf(fact_96_assms_I3_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ v ).
% assms(3)
thf(fact_97_Ax,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( epistemic_AK_i @ A @ P4 ) ) ).
% Ax
thf(fact_98_in__mono,axiom,
! [A: set_li2934481837907835752c_fm_i,B: set_li2934481837907835752c_fm_i,X3: list_s8081015415394010888c_fm_i] :
( ( ord_le1241842563113145096c_fm_i @ A @ B )
=> ( ( member605545779687026225c_fm_i @ X3 @ A )
=> ( member605545779687026225c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_99_in__mono,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,X3: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( member6475149752759350353c_fm_i @ X3 @ A )
=> ( member6475149752759350353c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_100_in__mono,axiom,
! [A: set_i,B: set_i,X3: i] :
( ( ord_less_eq_set_i @ A @ B )
=> ( ( member_i @ X3 @ A )
=> ( member_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_101_in__mono,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( member1461078328125707403c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_102_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_103_in__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_104_subsetD,axiom,
! [A: set_li2934481837907835752c_fm_i,B: set_li2934481837907835752c_fm_i,C: list_s8081015415394010888c_fm_i] :
( ( ord_le1241842563113145096c_fm_i @ A @ B )
=> ( ( member605545779687026225c_fm_i @ C @ A )
=> ( member605545779687026225c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_105_subsetD,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,C: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( member6475149752759350353c_fm_i @ C @ A )
=> ( member6475149752759350353c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_106_subsetD,axiom,
! [A: set_i,B: set_i,C: i] :
( ( ord_less_eq_set_i @ A @ B )
=> ( ( member_i @ C @ A )
=> ( member_i @ C @ B ) ) ) ).
% subsetD
thf(fact_107_subsetD,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( member1461078328125707403c_fm_i @ C @ A )
=> ( member1461078328125707403c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_108_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_109_subsetD,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_110_equalityE,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ~ ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_111_equalityE,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( A = B )
=> ~ ( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ~ ( ord_le2772500130360476258c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_112_equalityE,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_113_subset__eq,axiom,
( ord_le1241842563113145096c_fm_i
= ( ^ [A3: set_li2934481837907835752c_fm_i,B3: set_li2934481837907835752c_fm_i] :
! [X: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X @ A3 )
=> ( member605545779687026225c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_114_subset__eq,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i] :
! [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ A3 )
=> ( member6475149752759350353c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_115_subset__eq,axiom,
( ord_less_eq_set_i
= ( ^ [A3: set_i,B3: set_i] :
! [X: i] :
( ( member_i @ X @ A3 )
=> ( member_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_116_subset__eq,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A3: set_se7339729205154126530c_fm_i,B3: set_se7339729205154126530c_fm_i] :
! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ A3 )
=> ( member1461078328125707403c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_117_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
=> ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_118_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
=> ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_119_equalityD1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_120_equalityD1,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( A = B )
=> ( ord_le2772500130360476258c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_121_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_122_equalityD2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_123_equalityD2,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( A = B )
=> ( ord_le2772500130360476258c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_124_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_125_subset__iff,axiom,
( ord_le1241842563113145096c_fm_i
= ( ^ [A3: set_li2934481837907835752c_fm_i,B3: set_li2934481837907835752c_fm_i] :
! [T: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ T @ A3 )
=> ( member605545779687026225c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_126_subset__iff,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i] :
! [T: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ T @ A3 )
=> ( member6475149752759350353c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_127_subset__iff,axiom,
( ord_less_eq_set_i
= ( ^ [A3: set_i,B3: set_i] :
! [T: i] :
( ( member_i @ T @ A3 )
=> ( member_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_128_subset__iff,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A3: set_se7339729205154126530c_fm_i,B3: set_se7339729205154126530c_fm_i] :
! [T: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ T @ A3 )
=> ( member1461078328125707403c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_129_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A3 )
=> ( member1104366573291651755c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_130_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A3 )
=> ( member6642669606046002379c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_131_subset__refl,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_132_subset__refl,axiom,
! [A: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_133_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_134_Collect__mono,axiom,
! [P3: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ! [X2: list_Epistemic_fm_i] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1209247765174941992c_fm_i @ ( collec5759350117362652307c_fm_i @ P3 ) @ ( collec5759350117362652307c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_135_Collect__mono,axiom,
! [P3: list_s8081015415394010888c_fm_i > $o,Q: list_s8081015415394010888c_fm_i > $o] :
( ! [X2: list_s8081015415394010888c_fm_i] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1241842563113145096c_fm_i @ ( collec2200325868635342451c_fm_i @ P3 ) @ ( collec2200325868635342451c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_136_Collect__mono,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,Q: set_se3485332733965609186c_fm_i > $o] :
( ! [X2: set_se3485332733965609186c_fm_i] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2772500130360476258c_fm_i @ ( collec2761885132556217805c_fm_i @ P3 ) @ ( collec2761885132556217805c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_137_Collect__mono,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_138_Collect__mono,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_139_subset__trans,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_140_subset__trans,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,C2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ C2 )
=> ( ord_le2772500130360476258c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_141_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_142_set__eq__subset,axiom,
( ( ^ [Y2: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] : ( Y2 = Z2 ) )
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
& ( ord_le5389487502678872194c_fm_i @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_143_set__eq__subset,axiom,
( ( ^ [Y2: set_se7339729205154126530c_fm_i,Z2: set_se7339729205154126530c_fm_i] : ( Y2 = Z2 ) )
= ( ^ [A3: set_se7339729205154126530c_fm_i,B3: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A3 @ B3 )
& ( ord_le2772500130360476258c_fm_i @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_144_set__eq__subset,axiom,
( ( ^ [Y2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y2 = Z2 ) )
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_145_Collect__mono__iff,axiom,
! [P3: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ( ord_le1209247765174941992c_fm_i @ ( collec5759350117362652307c_fm_i @ P3 ) @ ( collec5759350117362652307c_fm_i @ Q ) )
= ( ! [X: list_Epistemic_fm_i] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_146_Collect__mono__iff,axiom,
! [P3: list_s8081015415394010888c_fm_i > $o,Q: list_s8081015415394010888c_fm_i > $o] :
( ( ord_le1241842563113145096c_fm_i @ ( collec2200325868635342451c_fm_i @ P3 ) @ ( collec2200325868635342451c_fm_i @ Q ) )
= ( ! [X: list_s8081015415394010888c_fm_i] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_147_Collect__mono__iff,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,Q: set_se3485332733965609186c_fm_i > $o] :
( ( ord_le2772500130360476258c_fm_i @ ( collec2761885132556217805c_fm_i @ P3 ) @ ( collec2761885132556217805c_fm_i @ Q ) )
= ( ! [X: set_se3485332733965609186c_fm_i] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_148_Collect__mono__iff,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_149_Collect__mono__iff,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q ) )
= ( ! [X: epistemic_fm_i] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_150_mem__Collect__eq,axiom,
! [A2: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( member1461078328125707403c_fm_i @ A2 @ ( collec2761885132556217805c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A2: i,P3: i > $o] :
( ( member_i @ A2 @ ( collect_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A2: list_Epistemic_fm_i,P3: list_Epistemic_fm_i > $o] :
( ( member6475149752759350353c_fm_i @ A2 @ ( collec5759350117362652307c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_153_mem__Collect__eq,axiom,
! [A2: list_s8081015415394010888c_fm_i,P3: list_s8081015415394010888c_fm_i > $o] :
( ( member605545779687026225c_fm_i @ A2 @ ( collec2200325868635342451c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_154_mem__Collect__eq,axiom,
! [A2: epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_155_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( collec2761885132556217805c_fm_i
@ ^ [X: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_157_Collect__mem__eq,axiom,
! [A: set_i] :
( ( collect_i
@ ^ [X: i] : ( member_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( collec5759350117362652307c_fm_i
@ ^ [X: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A: set_li2934481837907835752c_fm_i] :
( ( collec2200325868635342451c_fm_i
@ ^ [X: list_s8081015415394010888c_fm_i] : ( member605545779687026225c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_162_Collect__cong,axiom,
! [P3: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ! [X2: list_Epistemic_fm_i] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec5759350117362652307c_fm_i @ P3 )
= ( collec5759350117362652307c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_163_Collect__cong,axiom,
! [P3: list_s8081015415394010888c_fm_i > $o,Q: list_s8081015415394010888c_fm_i > $o] :
( ! [X2: list_s8081015415394010888c_fm_i] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec2200325868635342451c_fm_i @ P3 )
= ( collec2200325868635342451c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_164_Collect__cong,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec4904205187116291597c_fm_i @ P3 )
= ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_165_Collect__cong,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3087743281813070829c_fm_i @ P3 )
= ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_166_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A3: epistemic_fm_i > $o,S2: set_Epistemic_fm_i] :
~ ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ S2 )
& ( epistemic_AK_i @ A3 @ ( epistemic_imply_i @ Qs @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_167_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S3: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S3 )
=> ? [S: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ S3 )
& ( finite3304564979551393739c_fm_i @ S )
& ~ ( episte2285483198712856234tent_i @ A @ S ) ) ) ).
% inconsistent_finite
thf(fact_168_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S3: set_Epistemic_fm_i,S4: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S3 )
=> ( ( ord_le3843937902494030498c_fm_i @ S4 @ S3 )
=> ( episte2285483198712856234tent_i @ A @ S4 ) ) ) ).
% consistent_hereditary
thf(fact_169_Collect__subset,axiom,
! [A: set_i,P3: i > $o] :
( ord_less_eq_set_i
@ ( collect_i
@ ^ [X: i] :
( ( member_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_170_Collect__subset,axiom,
! [A: set_li8269783359719514504c_fm_i,P3: list_Epistemic_fm_i > $o] :
( ord_le1209247765174941992c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_171_Collect__subset,axiom,
! [A: set_li2934481837907835752c_fm_i,P3: list_s8081015415394010888c_fm_i > $o] :
( ord_le1241842563113145096c_fm_i
@ ( collec2200325868635342451c_fm_i
@ ^ [X: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_172_Collect__subset,axiom,
! [A: set_se7339729205154126530c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ord_le2772500130360476258c_fm_i
@ ( collec2761885132556217805c_fm_i
@ ^ [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_173_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_174_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_175_sup__set__def,axiom,
( sup_sup_set_i
= ( ^ [A3: set_i,B3: set_i] :
( collect_i
@ ( sup_sup_i_o
@ ^ [X: i] : ( member_i @ X @ A3 )
@ ^ [X: i] : ( member_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_176_sup__set__def,axiom,
( sup_su33170788683693916c_fm_i
= ( ^ [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i] :
( collec5759350117362652307c_fm_i
@ ( sup_su7027133776084382337fm_i_o
@ ^ [X: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X @ A3 )
@ ^ [X: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_177_sup__set__def,axiom,
( sup_su6123556432952969532c_fm_i
= ( ^ [A3: set_li2934481837907835752c_fm_i,B3: set_li2934481837907835752c_fm_i] :
( collec2200325868635342451c_fm_i
@ ( sup_su2317072595969462433fm_i_o
@ ^ [X: list_s8081015415394010888c_fm_i] : ( member605545779687026225c_fm_i @ X @ A3 )
@ ^ [X: list_s8081015415394010888c_fm_i] : ( member605545779687026225c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_178_sup__set__def,axiom,
( sup_su2317974602288960150c_fm_i
= ( ^ [A3: set_se7339729205154126530c_fm_i,B3: set_se7339729205154126530c_fm_i] :
( collec2761885132556217805c_fm_i
@ ( sup_su5020628091962035655fm_i_o
@ ^ [X: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X @ A3 )
@ ^ [X: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_179_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_180_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_181_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q2: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q2 ) ) ) ) ).
% K_imply_weaken
thf(fact_182_R2,axiom,
! [A: episte4276422356747265950c_fm_i > $o,P4: episte4276422356747265950c_fm_i,I: set_Epistemic_fm_i] :
( ( episte4709833913615054190c_fm_i @ A @ P4 )
=> ( episte4709833913615054190c_fm_i @ A @ ( episte3581967596712343542c_fm_i @ I @ P4 ) ) ) ).
% R2
thf(fact_183_R2,axiom,
! [A: episte1308613106011496894c_fm_i > $o,P4: episte1308613106011496894c_fm_i,I: epistemic_fm_i] :
( ( episte3601224874663357326c_fm_i @ A @ P4 )
=> ( episte3601224874663357326c_fm_i @ A @ ( episte5657488666450005526c_fm_i @ I @ P4 ) ) ) ).
% R2
thf(fact_184_R2,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( epistemic_AK_i @ A @ P4 )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% R2
thf(fact_185_finite__list,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ? [Xs2: list_s8635415444435394792c_fm_i] :
( ( set_se5076022966804059383c_fm_i @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_186_finite__list,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( finite3106398504192302929c_fm_i @ A )
=> ? [Xs2: list_l4360594050116504750c_fm_i] :
( ( set_li9010496130497932989c_fm_i @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_187_finite__list,axiom,
! [A: set_li2934481837907835752c_fm_i] :
( ( finite5108596011607087409c_fm_i @ A )
=> ? [Xs2: list_l8939085672862518926c_fm_i] :
( ( set_li1699392436468070557c_fm_i @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_188_finite__list,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ? [Xs2: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_189_finite__list,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ? [Xs2: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_190_sup_OcoboundedI2,axiom,
! [C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_191_sup_OcoboundedI2,axiom,
! [C: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ C @ B2 )
=> ( ord_le2772500130360476258c_fm_i @ C @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_192_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_193_sup_OcoboundedI2,axiom,
! [C: epistemic_fm_i > $o,B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ C @ B2 )
=> ( ord_le190830114487426235fm_i_o @ C @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_194_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ C @ B2 )
=> ( ord_le5706303257367236315fm_i_o @ C @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_195_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_196_sup_OcoboundedI1,axiom,
! [C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_197_sup_OcoboundedI1,axiom,
! [C: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ C @ A2 )
=> ( ord_le2772500130360476258c_fm_i @ C @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_198_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_199_sup_OcoboundedI1,axiom,
! [C: epistemic_fm_i > $o,A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ C @ A2 )
=> ( ord_le190830114487426235fm_i_o @ C @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_200_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ C @ A2 )
=> ( ord_le5706303257367236315fm_i_o @ C @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_201_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_202_sup_Oabsorb__iff2,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_203_sup_Oabsorb__iff2,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A4: set_se7339729205154126530c_fm_i,B4: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_204_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_205_sup_Oabsorb__iff2,axiom,
( ord_le190830114487426235fm_i_o
= ( ^ [A4: epistemic_fm_i > $o,B4: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_206_sup_Oabsorb__iff2,axiom,
( ord_le5706303257367236315fm_i_o
= ( ^ [A4: set_Epistemic_fm_i > $o,B4: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_207_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_208_sup_Oabsorb__iff1,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B4: set_se3485332733965609186c_fm_i,A4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_209_sup_Oabsorb__iff1,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [B4: set_se7339729205154126530c_fm_i,A4: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_210_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_211_sup_Oabsorb__iff1,axiom,
( ord_le190830114487426235fm_i_o
= ( ^ [B4: epistemic_fm_i > $o,A4: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_212_sup_Oabsorb__iff1,axiom,
( ord_le5706303257367236315fm_i_o
= ( ^ [B4: set_Epistemic_fm_i > $o,A4: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_213_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B4: set_Epistemic_fm_i,A4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_214_sup_Ocobounded2,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ B2 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_215_sup_Ocobounded2,axiom,
! [B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ B2 @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_216_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_217_sup_Ocobounded2,axiom,
! [B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ B2 @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_218_sup_Ocobounded2,axiom,
! [B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ B2 @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_219_sup_Ocobounded2,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_220_sup_Ocobounded1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A2 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_221_sup_Ocobounded1,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ A2 @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_222_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_223_sup_Ocobounded1,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ A2 @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_224_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ A2 @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_225_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_226_sup_Oorder__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B4: set_se3485332733965609186c_fm_i,A4: set_se3485332733965609186c_fm_i] :
( A4
= ( sup_su2582925890723967158c_fm_i @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_227_sup_Oorder__iff,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [B4: set_se7339729205154126530c_fm_i,A4: set_se7339729205154126530c_fm_i] :
( A4
= ( sup_su2317974602288960150c_fm_i @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_228_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_229_sup_Oorder__iff,axiom,
( ord_le190830114487426235fm_i_o
= ( ^ [B4: epistemic_fm_i > $o,A4: epistemic_fm_i > $o] :
( A4
= ( sup_su2844414564337698183fm_i_o @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_230_sup_Oorder__iff,axiom,
( ord_le5706303257367236315fm_i_o
= ( ^ [B4: set_Epistemic_fm_i > $o,A4: set_Epistemic_fm_i > $o] :
( A4
= ( sup_su6723579903606884263fm_i_o @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_231_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B4: set_Epistemic_fm_i,A4: set_Epistemic_fm_i] :
( A4
= ( sup_su1936195050962291414c_fm_i @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_232_sup_OboundedI,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_233_sup_OboundedI,axiom,
! [B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i,C: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ B2 @ A2 )
=> ( ( ord_le2772500130360476258c_fm_i @ C @ A2 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_234_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_235_sup_OboundedI,axiom,
! [B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o,C: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B2 @ A2 )
=> ( ( ord_le190830114487426235fm_i_o @ C @ A2 )
=> ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_236_sup_OboundedI,axiom,
! [B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o,C: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ B2 @ A2 )
=> ( ( ord_le5706303257367236315fm_i_o @ C @ A2 )
=> ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_237_sup_OboundedI,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_238_sup_OboundedE,axiom,
! [B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_239_sup_OboundedE,axiom,
! [B2: set_se7339729205154126530c_fm_i,C: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le2772500130360476258c_fm_i @ B2 @ A2 )
=> ~ ( ord_le2772500130360476258c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_240_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_241_sup_OboundedE,axiom,
! [B2: epistemic_fm_i > $o,C: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le190830114487426235fm_i_o @ B2 @ A2 )
=> ~ ( ord_le190830114487426235fm_i_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_242_sup_OboundedE,axiom,
! [B2: set_Epistemic_fm_i > $o,C: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le5706303257367236315fm_i_o @ B2 @ A2 )
=> ~ ( ord_le5706303257367236315fm_i_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_243_sup_OboundedE,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_244_sup__absorb2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_245_sup__absorb2,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ X3 @ Y )
=> ( ( sup_su2317974602288960150c_fm_i @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_246_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_247_sup__absorb2,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X3 @ Y )
=> ( ( sup_su2844414564337698183fm_i_o @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_248_sup__absorb2,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ X3 @ Y )
=> ( ( sup_su6723579903606884263fm_i_o @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_249_sup__absorb2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_250_sup__absorb1,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_251_sup__absorb1,axiom,
! [Y: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ Y @ X3 )
=> ( ( sup_su2317974602288960150c_fm_i @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_252_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_253_sup__absorb1,axiom,
! [Y: epistemic_fm_i > $o,X3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ Y @ X3 )
=> ( ( sup_su2844414564337698183fm_i_o @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_254_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ Y @ X3 )
=> ( ( sup_su6723579903606884263fm_i_o @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_255_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_256_sup_Oabsorb2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( ( sup_su2582925890723967158c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_257_sup_Oabsorb2,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ B2 )
=> ( ( sup_su2317974602288960150c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_258_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_259_sup_Oabsorb2,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A2 @ B2 )
=> ( ( sup_su2844414564337698183fm_i_o @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_260_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ A2 @ B2 )
=> ( ( sup_su6723579903606884263fm_i_o @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_261_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_262_sup_Oabsorb1,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ( ( sup_su2582925890723967158c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_263_sup_Oabsorb1,axiom,
! [B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ B2 @ A2 )
=> ( ( sup_su2317974602288960150c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_264_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_265_sup_Oabsorb1,axiom,
! [B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B2 @ A2 )
=> ( ( sup_su2844414564337698183fm_i_o @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_266_sup_Oabsorb1,axiom,
! [B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ B2 @ A2 )
=> ( ( sup_su6723579903606884263fm_i_o @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_267_sup_Oabsorb1,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_268_sup__unique,axiom,
! [F2: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y3 @ X2 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z3 @ X2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_269_sup__unique,axiom,
! [F2: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i] :
( ! [X2: set_se7339729205154126530c_fm_i,Y3: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_se7339729205154126530c_fm_i,Y3: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_se7339729205154126530c_fm_i,Y3: set_se7339729205154126530c_fm_i,Z3: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ Y3 @ X2 )
=> ( ( ord_le2772500130360476258c_fm_i @ Z3 @ X2 )
=> ( ord_le2772500130360476258c_fm_i @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su2317974602288960150c_fm_i @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_270_sup__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_271_sup__unique,axiom,
! [F2: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o,X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o,Z3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ Y3 @ X2 )
=> ( ( ord_le190830114487426235fm_i_o @ Z3 @ X2 )
=> ( ord_le190830114487426235fm_i_o @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su2844414564337698183fm_i_o @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_272_sup__unique,axiom,
! [F2: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i > $o,Y3: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_i > $o,Y3: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_i > $o,Y3: set_Epistemic_fm_i > $o,Z3: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ Y3 @ X2 )
=> ( ( ord_le5706303257367236315fm_i_o @ Z3 @ X2 )
=> ( ord_le5706303257367236315fm_i_o @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su6723579903606884263fm_i_o @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_273_sup__unique,axiom,
! [F2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X2 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y3 @ ( F2 @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_274_sup_OorderI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( A2
= ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) )
=> ( ord_le5389487502678872194c_fm_i @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_275_sup_OorderI,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( A2
= ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) )
=> ( ord_le2772500130360476258c_fm_i @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_276_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_277_sup_OorderI,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( A2
= ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) )
=> ( ord_le190830114487426235fm_i_o @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_278_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( A2
= ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) )
=> ( ord_le5706303257367236315fm_i_o @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_279_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_280_sup_OorderE,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ( A2
= ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_281_sup_OorderE,axiom,
! [B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ B2 @ A2 )
=> ( A2
= ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_282_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_283_sup_OorderE,axiom,
! [B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B2 @ A2 )
=> ( A2
= ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_284_sup_OorderE,axiom,
! [B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ B2 @ A2 )
=> ( A2
= ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_285_sup_OorderE,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_286_le__iff__sup,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_287_le__iff__sup,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [X: set_se7339729205154126530c_fm_i,Y4: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_288_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_289_le__iff__sup,axiom,
( ord_le190830114487426235fm_i_o
= ( ^ [X: epistemic_fm_i > $o,Y4: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_290_le__iff__sup,axiom,
( ord_le5706303257367236315fm_i_o
= ( ^ [X: set_Epistemic_fm_i > $o,Y4: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_291_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_292_sup__least,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_293_sup__least,axiom,
! [Y: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i,Z: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ Y @ X3 )
=> ( ( ord_le2772500130360476258c_fm_i @ Z @ X3 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_294_sup__least,axiom,
! [Y: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_295_sup__least,axiom,
! [Y: epistemic_fm_i > $o,X3: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ Y @ X3 )
=> ( ( ord_le190830114487426235fm_i_o @ Z @ X3 )
=> ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_296_sup__least,axiom,
! [Y: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o,Z: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ Y @ X3 )
=> ( ( ord_le5706303257367236315fm_i_o @ Z @ X3 )
=> ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_297_sup__least,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_298_sup__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) @ ( sup_su2582925890723967158c_fm_i @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_299_sup__mono,axiom,
! [A2: set_se7339729205154126530c_fm_i,C: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i,D: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ C )
=> ( ( ord_le2772500130360476258c_fm_i @ B2 @ D )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) @ ( sup_su2317974602288960150c_fm_i @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_300_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_301_sup__mono,axiom,
! [A2: epistemic_fm_i > $o,C: epistemic_fm_i > $o,B2: epistemic_fm_i > $o,D: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A2 @ C )
=> ( ( ord_le190830114487426235fm_i_o @ B2 @ D )
=> ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) @ ( sup_su2844414564337698183fm_i_o @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_302_sup__mono,axiom,
! [A2: set_Epistemic_fm_i > $o,C: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o,D: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ A2 @ C )
=> ( ( ord_le5706303257367236315fm_i_o @ B2 @ D )
=> ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) @ ( sup_su6723579903606884263fm_i_o @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_303_sup__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_304_sup_Omono,axiom,
! [C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ D @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ C @ D ) @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_305_sup_Omono,axiom,
! [C: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i,D: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ C @ A2 )
=> ( ( ord_le2772500130360476258c_fm_i @ D @ B2 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ C @ D ) @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_306_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_307_sup_Omono,axiom,
! [C: epistemic_fm_i > $o,A2: epistemic_fm_i > $o,D: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ C @ A2 )
=> ( ( ord_le190830114487426235fm_i_o @ D @ B2 )
=> ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ C @ D ) @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_308_sup_Omono,axiom,
! [C: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o,D: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ C @ A2 )
=> ( ( ord_le5706303257367236315fm_i_o @ D @ B2 )
=> ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ C @ D ) @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_309_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D ) @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_310_le__supI2,axiom,
! [X3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_311_le__supI2,axiom,
! [X3: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ X3 @ B2 )
=> ( ord_le2772500130360476258c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_312_le__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_313_le__supI2,axiom,
! [X3: epistemic_fm_i > $o,B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X3 @ B2 )
=> ( ord_le190830114487426235fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_314_le__supI2,axiom,
! [X3: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ X3 @ B2 )
=> ( ord_le5706303257367236315fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_315_le__supI2,axiom,
! [X3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_316_le__supI1,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_317_le__supI1,axiom,
! [X3: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ X3 @ A2 )
=> ( ord_le2772500130360476258c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_318_le__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_319_le__supI1,axiom,
! [X3: epistemic_fm_i > $o,A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X3 @ A2 )
=> ( ord_le190830114487426235fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_320_le__supI1,axiom,
! [X3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ X3 @ A2 )
=> ( ord_le5706303257367236315fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_321_le__supI1,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_322_sup__ge2,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_323_sup__ge2,axiom,
! [Y: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ Y @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_324_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_325_sup__ge2,axiom,
! [Y: epistemic_fm_i > $o,X3: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ Y @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_326_sup__ge2,axiom,
! [Y: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ Y @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_327_sup__ge2,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_328_sup__ge1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_329_sup__ge1,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_330_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_331_sup__ge1,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_332_sup__ge1,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_333_sup__ge1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_334_le__supI,axiom,
! [A2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_335_le__supI,axiom,
! [A2: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ X3 )
=> ( ( ord_le2772500130360476258c_fm_i @ B2 @ X3 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_336_le__supI,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_337_le__supI,axiom,
! [A2: epistemic_fm_i > $o,X3: epistemic_fm_i > $o,B2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A2 @ X3 )
=> ( ( ord_le190830114487426235fm_i_o @ B2 @ X3 )
=> ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_338_le__supI,axiom,
! [A2: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ A2 @ X3 )
=> ( ( ord_le5706303257367236315fm_i_o @ B2 @ X3 )
=> ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_339_le__supI,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_340_le__supE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ~ ( ord_le5389487502678872194c_fm_i @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_341_le__supE,axiom,
! [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le2772500130360476258c_fm_i @ A2 @ X3 )
=> ~ ( ord_le2772500130360476258c_fm_i @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_342_le__supE,axiom,
! [A2: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_343_le__supE,axiom,
! [A2: epistemic_fm_i > $o,B2: epistemic_fm_i > $o,X3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ ( sup_su2844414564337698183fm_i_o @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le190830114487426235fm_i_o @ A2 @ X3 )
=> ~ ( ord_le190830114487426235fm_i_o @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_344_le__supE,axiom,
! [A2: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o] :
( ( ord_le5706303257367236315fm_i_o @ ( sup_su6723579903606884263fm_i_o @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le5706303257367236315fm_i_o @ A2 @ X3 )
=> ~ ( ord_le5706303257367236315fm_i_o @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_345_le__supE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_346_inf__sup__ord_I3_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_347_inf__sup__ord_I3_J,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_348_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_349_inf__sup__ord_I3_J,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_350_inf__sup__ord_I3_J,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_351_inf__sup__ord_I3_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_352_inf__sup__ord_I4_J,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_353_inf__sup__ord_I4_J,axiom,
! [Y: set_se7339729205154126530c_fm_i,X3: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ Y @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_354_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_355_inf__sup__ord_I4_J,axiom,
! [Y: epistemic_fm_i > $o,X3: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ Y @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_356_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i > $o] : ( ord_le5706303257367236315fm_i_o @ Y @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_357_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_358_subset__code_I1_J,axiom,
! [Xs: list_i,B: set_i] :
( ( ord_less_eq_set_i @ ( set_i2 @ Xs ) @ B )
= ( ! [X: i] :
( ( member_i @ X @ ( set_i2 @ Xs ) )
=> ( member_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_359_subset__code_I1_J,axiom,
! [Xs: list_l4360594050116504750c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) @ B )
= ( ! [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( member6475149752759350353c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_360_subset__code_I1_J,axiom,
! [Xs: list_l8939085672862518926c_fm_i,B: set_li2934481837907835752c_fm_i] :
( ( ord_le1241842563113145096c_fm_i @ ( set_li1699392436468070557c_fm_i @ Xs ) @ B )
= ( ! [X: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ X @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( member605545779687026225c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_361_subset__code_I1_J,axiom,
! [Xs: list_s8635415444435394792c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( set_se5076022966804059383c_fm_i @ Xs ) @ B )
= ( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( set_se5076022966804059383c_fm_i @ Xs ) )
=> ( member1461078328125707403c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_362_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_363_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_364_fm_Odistinct_I9_J,axiom,
! [X61: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i] :
( episte5042310466404390337c_fm_i
!= ( episte3581967596712343542c_fm_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_365_fm_Odistinct_I9_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i] :
( episte5073044278343014369c_fm_i
!= ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_366_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_367_subset__Un__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_368_subset__Un__eq,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A3: set_se7339729205154126530c_fm_i,B3: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_369_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_370_subset__UnE,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
=> ~ ! [A5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A5 @ A )
=> ! [B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B5 @ B )
=> ( C2
!= ( sup_su2582925890723967158c_fm_i @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_371_subset__UnE,axiom,
! [C2: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ C2 @ ( sup_su2317974602288960150c_fm_i @ A @ B ) )
=> ~ ! [A5: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A5 @ A )
=> ! [B5: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ B5 @ B )
=> ( C2
!= ( sup_su2317974602288960150c_fm_i @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_372_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ~ ! [A5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ A )
=> ! [B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ B )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_373_Un__absorb2,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( sup_su2582925890723967158c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_374_Un__absorb2,axiom,
! [B: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ( sup_su2317974602288960150c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_375_Un__absorb2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_376_Un__absorb1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( sup_su2582925890723967158c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_377_Un__absorb1,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( sup_su2317974602288960150c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_378_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_379_Un__upper2,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ B @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_380_Un__upper2,axiom,
! [B: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ B @ ( sup_su2317974602288960150c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_381_Un__upper2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_382_Un__upper1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_383_Un__upper1,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] : ( ord_le2772500130360476258c_fm_i @ A @ ( sup_su2317974602288960150c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_384_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_385_Un__least,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_386_Un__least,axiom,
! [A: set_se7339729205154126530c_fm_i,C2: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ C2 )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ C2 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_387_Un__least,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_388_Un__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ D2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_389_Un__mono,axiom,
! [A: set_se7339729205154126530c_fm_i,C2: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,D2: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ C2 )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ D2 )
=> ( ord_le2772500130360476258c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) @ ( sup_su2317974602288960150c_fm_i @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_390_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_391_sup__fun__def,axiom,
( sup_su2844414564337698183fm_i_o
= ( ^ [F: epistemic_fm_i > $o,G: epistemic_fm_i > $o,X: epistemic_fm_i] : ( sup_sup_o @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% sup_fun_def
thf(fact_392_sup__fun__def,axiom,
( sup_su6723579903606884263fm_i_o
= ( ^ [F: set_Epistemic_fm_i > $o,G: set_Epistemic_fm_i > $o,X: set_Epistemic_fm_i] : ( sup_sup_o @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% sup_fun_def
thf(fact_393_sup__left__commute,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) )
= ( sup_su2582925890723967158c_fm_i @ Y @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_394_sup__left__commute,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ Y @ Z ) )
= ( sup_su2844414564337698183fm_i_o @ Y @ ( sup_su2844414564337698183fm_i_o @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_395_sup__left__commute,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o,Z: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ Y @ Z ) )
= ( sup_su6723579903606884263fm_i_o @ Y @ ( sup_su6723579903606884263fm_i_o @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_396_sup__left__commute,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i,Z: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ Y @ Z ) )
= ( sup_su2317974602288960150c_fm_i @ Y @ ( sup_su2317974602288960150c_fm_i @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_397_sup__left__commute,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) )
= ( sup_su1936195050962291414c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_398_sup_Oleft__commute,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ B2 @ ( sup_su2582925890723967158c_fm_i @ A2 @ C ) )
= ( sup_su2582925890723967158c_fm_i @ A2 @ ( sup_su2582925890723967158c_fm_i @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_399_sup_Oleft__commute,axiom,
! [B2: epistemic_fm_i > $o,A2: epistemic_fm_i > $o,C: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ B2 @ ( sup_su2844414564337698183fm_i_o @ A2 @ C ) )
= ( sup_su2844414564337698183fm_i_o @ A2 @ ( sup_su2844414564337698183fm_i_o @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_400_sup_Oleft__commute,axiom,
! [B2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i > $o,C: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ B2 @ ( sup_su6723579903606884263fm_i_o @ A2 @ C ) )
= ( sup_su6723579903606884263fm_i_o @ A2 @ ( sup_su6723579903606884263fm_i_o @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_401_sup_Oleft__commute,axiom,
! [B2: set_se7339729205154126530c_fm_i,A2: set_se7339729205154126530c_fm_i,C: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ B2 @ ( sup_su2317974602288960150c_fm_i @ A2 @ C ) )
= ( sup_su2317974602288960150c_fm_i @ A2 @ ( sup_su2317974602288960150c_fm_i @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_402_sup_Oleft__commute,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ C ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_403_sup__commute,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( sup_su2582925890723967158c_fm_i @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_404_sup__commute,axiom,
( sup_su2844414564337698183fm_i_o
= ( ^ [X: epistemic_fm_i > $o,Y4: epistemic_fm_i > $o] : ( sup_su2844414564337698183fm_i_o @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_405_sup__commute,axiom,
( sup_su6723579903606884263fm_i_o
= ( ^ [X: set_Epistemic_fm_i > $o,Y4: set_Epistemic_fm_i > $o] : ( sup_su6723579903606884263fm_i_o @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_406_sup__commute,axiom,
( sup_su2317974602288960150c_fm_i
= ( ^ [X: set_se7339729205154126530c_fm_i,Y4: set_se7339729205154126530c_fm_i] : ( sup_su2317974602288960150c_fm_i @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_407_sup__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_408_sup_Ocommute,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] : ( sup_su2582925890723967158c_fm_i @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_409_sup_Ocommute,axiom,
( sup_su2844414564337698183fm_i_o
= ( ^ [A4: epistemic_fm_i > $o,B4: epistemic_fm_i > $o] : ( sup_su2844414564337698183fm_i_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_410_sup_Ocommute,axiom,
( sup_su6723579903606884263fm_i_o
= ( ^ [A4: set_Epistemic_fm_i > $o,B4: set_Epistemic_fm_i > $o] : ( sup_su6723579903606884263fm_i_o @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_411_sup_Ocommute,axiom,
( sup_su2317974602288960150c_fm_i
= ( ^ [A4: set_se7339729205154126530c_fm_i,B4: set_se7339729205154126530c_fm_i] : ( sup_su2317974602288960150c_fm_i @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_412_sup_Ocommute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_413_sup__assoc,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ Z )
= ( sup_su2582925890723967158c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_414_sup__assoc,axiom,
! [X3: epistemic_fm_i > $o,Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] :
( ( sup_su2844414564337698183fm_i_o @ ( sup_su2844414564337698183fm_i_o @ X3 @ Y ) @ Z )
= ( sup_su2844414564337698183fm_i_o @ X3 @ ( sup_su2844414564337698183fm_i_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_415_sup__assoc,axiom,
! [X3: set_Epistemic_fm_i > $o,Y: set_Epistemic_fm_i > $o,Z: set_Epistemic_fm_i > $o] :
( ( sup_su6723579903606884263fm_i_o @ ( sup_su6723579903606884263fm_i_o @ X3 @ Y ) @ Z )
= ( sup_su6723579903606884263fm_i_o @ X3 @ ( sup_su6723579903606884263fm_i_o @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_416_sup__assoc,axiom,
! [X3: set_se7339729205154126530c_fm_i,Y: set_se7339729205154126530c_fm_i,Z: set_se7339729205154126530c_fm_i] :
( ( sup_su2317974602288960150c_fm_i @ ( sup_su2317974602288960150c_fm_i @ X3 @ Y ) @ Z )
= ( sup_su2317974602288960150c_fm_i @ X3 @ ( sup_su2317974602288960150c_fm_i @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_417_sup__assoc,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_418_sup_Oassoc,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ C )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_419_inf__sup__aci_I5_J,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_420_inf__sup__aci_I6_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_421_inf__sup__aci_I7_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) )
= ( sup_su1936195050962291414c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_422_inf__sup__aci_I8_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_423_Un__left__commute,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_424_Un__left__absorb,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_left_absorb
thf(fact_425_Un__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_426_Un__absorb,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ A )
= A ) ).
% Un_absorb
thf(fact_427_Un__assoc,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_428_ball__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( P3 @ X ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( P3 @ X ) )
& ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( P3 @ X ) ) ) ) ).
% ball_Un
thf(fact_429_bex__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
& ( P3 @ X ) ) )
= ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) )
| ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
& ( P3 @ X ) ) ) ) ).
% bex_Un
thf(fact_430_UnI2,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_431_UnI2,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_432_UnI1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_433_UnI1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_434_UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_435_UnE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_436_filter__is__subset,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% filter_is_subset
thf(fact_437_filter__is__subset,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% filter_is_subset
thf(fact_438_Collect__disj__eq,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
| ( Q @ X ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_439_Collect__disj__eq,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
| ( Q @ X ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_440_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_441_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_442_filter__id__conv,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( ( filter7636273843821131039c_fm_i @ P3 @ Xs )
= Xs )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P3 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_443_filter__id__conv,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( ( filter3188398074982218495c_fm_i @ P3 @ Xs )
= Xs )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( P3 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_444_filter__cong,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( ( P3 @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter7636273843821131039c_fm_i @ P3 @ Xs )
= ( filter7636273843821131039c_fm_i @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_445_filter__cong,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ( ( P3 @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter3188398074982218495c_fm_i @ P3 @ Xs )
= ( filter3188398074982218495c_fm_i @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_446_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_447_finite__Un,axiom,
! [F3: set_Epistemic_fm_i,G2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F3 @ G2 ) )
= ( ( finite3304564979551393739c_fm_i @ F3 )
& ( finite3304564979551393739c_fm_i @ G2 ) ) ) ).
% finite_Un
thf(fact_448__092_060open_062_092_060nexists_062X_O_Aconsistent_AA_AX_A_092_060and_062_Amaximal_AA_AX_A_092_060and_062_AX_A_092_060in_062_A_123Wa_O_Aknown_AW_Ai_A_092_060subseteq_062_AWa_125_A_092_060inter_062_A_123W_O_Aknown_AU_Ai_A_092_060subseteq_062_AW_125_092_060close_062,axiom,
~ ? [X4: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ X4 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ X4 )
& ( member1104366573291651755c_fm_i @ X4
@ ( inf_in161960956874937808c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ w ) ) ) )
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P2 ) @ u ) ) ) ) ) ) ) ).
% \<open>\<nexists>X. consistent A X \<and> maximal A X \<and> X \<in> {Wa. known W i \<subseteq> Wa} \<inter> {W. known U i \<subseteq> W}\<close>
thf(fact_449_set__union,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( union_4878195246919425020c_fm_i @ Xs @ Ys ) )
= ( sup_su2582925890723967158c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) ) ) ).
% set_union
thf(fact_450_set__union,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( union_Epistemic_fm_i @ Xs @ Ys ) )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ).
% set_union
thf(fact_451_finite__Collect__disjI,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
| ( Q @ X ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_452_finite__Collect__disjI,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
| ( Q @ X ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_453_finite__Collect__conjI,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_454_finite__Collect__conjI,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_455_order__refl,axiom,
! [X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ X3 ) ).
% order_refl
thf(fact_456_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_457_infinite__Un,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) ) )
= ( ~ ( finite3304564979551393739c_fm_i @ S3 )
| ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_Un
thf(fact_458_Un__infinite,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S3 )
=> ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) ) ) ).
% Un_infinite
thf(fact_459_finite__UnI,axiom,
! [F3: set_Epistemic_fm_i,G2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( finite3304564979551393739c_fm_i @ G2 )
=> ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F3 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_460_inf_Oidem,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_461_inf__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_462_inf_Oleft__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_463_inf__left__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_464_inf_Oright__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ B2 )
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_465_inf__right__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_466_IntI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_467_IntI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_468_Int__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_469_Int__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_470_inf_Obounded__iff,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) )
= ( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_471_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_472_le__inf__iff,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
& ( ord_le5389487502678872194c_fm_i @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_473_le__inf__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
& ( ord_le3843937902494030498c_fm_i @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_474_inf__sup__absorb,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_475_inf__sup__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_476_sup__inf__absorb,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_477_sup__inf__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_478_finite__Int,axiom,
! [F3: set_Epistemic_fm_i,G2: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ F3 )
| ( finite3304564979551393739c_fm_i @ G2 ) )
=> ( finite3304564979551393739c_fm_i @ ( inf_in3450601097109690352c_fm_i @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_479_finite__Int,axiom,
! [F3: set_se3485332733965609186c_fm_i,G2: set_se3485332733965609186c_fm_i] :
( ( ( finite7933139204641697195c_fm_i @ F3 )
| ( finite7933139204641697195c_fm_i @ G2 ) )
=> ( finite7933139204641697195c_fm_i @ ( inf_in161960956874937808c_fm_i @ F3 @ G2 ) ) ) ).
% finite_Int
thf(fact_480_Int__subset__iff,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
& ( ord_le5389487502678872194c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_481_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
& ( ord_le3843937902494030498c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_482_Int__Un__eq_I4_J,axiom,
! [T2: set_se3485332733965609186c_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ T2 @ ( inf_in161960956874937808c_fm_i @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_483_Int__Un__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ T2 @ ( inf_in3450601097109690352c_fm_i @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_484_Int__Un__eq_I3_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ S3 @ ( inf_in161960956874937808c_fm_i @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_485_Int__Un__eq_I3_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ S3 @ ( inf_in3450601097109690352c_fm_i @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_486_Int__Un__eq_I2_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_487_Int__Un__eq_I2_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_488_Int__Un__eq_I1_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_489_Int__Un__eq_I1_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_490_Un__Int__eq_I4_J,axiom,
! [T2: set_se3485332733965609186c_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ T2 @ ( sup_su2582925890723967158c_fm_i @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_491_Un__Int__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ T2 @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_492_Un__Int__eq_I3_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ S3 @ ( sup_su2582925890723967158c_fm_i @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_493_Un__Int__eq_I3_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ S3 @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_494_Un__Int__eq_I2_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_495_Un__Int__eq_I2_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_496_Un__Int__eq_I1_J,axiom,
! [S3: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_497_Un__Int__eq_I1_J,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_498_inf__sup__aci_I4_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_499_inf__sup__aci_I3_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( inf_in161960956874937808c_fm_i @ Y @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_500_inf__sup__aci_I2_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Z )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_501_inf__sup__aci_I1_J,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_502_inf_Oassoc,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_503_inf__assoc,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Z )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_504_inf_Ocommute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_505_inf__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_506_inf_Oleft__commute,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ B2 @ ( inf_in161960956874937808c_fm_i @ A2 @ C ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_507_inf__left__commute,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( inf_in161960956874937808c_fm_i @ Y @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_508_IntE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_509_IntE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_510_IntD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_511_IntD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_512_IntD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_513_IntD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_514_Int__assoc,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_515_Int__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ A )
= A ) ).
% Int_absorb
thf(fact_516_Int__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_517_Int__left__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Int_left_absorb
thf(fact_518_Int__left__commute,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ B @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_519_Collect__conj__eq,axiom,
! [P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) )
= ( inf_in3450601097109690352c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_520_Collect__conj__eq,axiom,
! [P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) )
= ( inf_in161960956874937808c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_521_Int__Collect,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P3 ) ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P3 @ X3 ) ) ) ).
% Int_Collect
thf(fact_522_Int__Collect,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P3 ) ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P3 @ X3 ) ) ) ).
% Int_Collect
thf(fact_523_Int__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
& ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_524_Int__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
& ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_525_inf_OcoboundedI2,axiom,
! [B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_526_inf_OcoboundedI2,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_527_inf_OcoboundedI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_528_inf_OcoboundedI1,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_529_inf_Oabsorb__iff2,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B4: set_se3485332733965609186c_fm_i,A4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_530_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B4: set_Epistemic_fm_i,A4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_531_inf_Oabsorb__iff1,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_532_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_533_inf_Ocobounded2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_534_inf_Ocobounded2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_535_inf_Ocobounded1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_536_inf_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_537_inf_Oorder__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( A4
= ( inf_in161960956874937808c_fm_i @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_538_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( A4
= ( inf_in3450601097109690352c_fm_i @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_539_inf__greatest,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ Z )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_540_inf__greatest,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_541_inf_OboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_542_inf_OboundedI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_543_inf_OboundedE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_544_inf_OboundedE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_545_inf__absorb2,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_546_inf__absorb2,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_547_inf__absorb1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_548_inf__absorb1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_549_inf_Oabsorb2,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_550_inf_Oabsorb2,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_551_inf_Oabsorb1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_552_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_553_le__iff__inf,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_554_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_555_inf__unique,axiom,
! [F2: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F2 @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F2 @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y3 )
=> ( ( ord_le5389487502678872194c_fm_i @ X2 @ Z3 )
=> ( ord_le5389487502678872194c_fm_i @ X2 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_556_inf__unique,axiom,
! [F2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= ( F2 @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_557_inf_OorderI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_558_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_559_inf_OorderE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_560_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_561_le__infI2,axiom,
! [B2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_562_le__infI2,axiom,
! [B2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_563_le__infI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_564_le__infI1,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_565_inf__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ ( inf_in161960956874937808c_fm_i @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_566_inf__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_567_le__infI,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_568_le__infI,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_569_le__infE,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_570_le__infE,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_571_inf__le2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_572_inf__le2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_573_inf__le1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_574_inf__le1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_575_inf__sup__ord_I1_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_576_inf__sup__ord_I1_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_577_inf__sup__ord_I2_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_578_inf__sup__ord_I2_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_579_distrib__imp1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X2 @ ( sup_su2582925890723967158c_fm_i @ Y3 @ Z3 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X2 @ Y3 ) @ ( inf_in161960956874937808c_fm_i @ X2 @ Z3 ) ) )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_580_distrib__imp1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ Y3 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X2 @ Y3 ) @ ( inf_in3450601097109690352c_fm_i @ X2 @ Z3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_581_distrib__imp2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X2 @ ( inf_in161960956874937808c_fm_i @ Y3 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X2 @ Y3 ) @ ( sup_su2582925890723967158c_fm_i @ X2 @ Z3 ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_582_distrib__imp2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ ( inf_in3450601097109690352c_fm_i @ Y3 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y3 ) @ ( sup_su1936195050962291414c_fm_i @ X2 @ Z3 ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_583_inf__sup__distrib1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_584_inf__sup__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_585_inf__sup__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) @ X3 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ X3 ) @ ( inf_in161960956874937808c_fm_i @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_586_inf__sup__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_587_sup__inf__distrib1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_588_sup__inf__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_589_sup__inf__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) @ X3 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ X3 ) @ ( sup_su2582925890723967158c_fm_i @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_590_sup__inf__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_591_Int__Collect__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( P3 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P3 ) ) @ ( inf_in161960956874937808c_fm_i @ B @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_592_Int__Collect__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( P3 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P3 ) ) @ ( inf_in3450601097109690352c_fm_i @ B @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_593_Int__greatest,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ C2 @ B )
=> ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_594_Int__greatest,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_595_Int__absorb2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_596_Int__absorb2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_597_Int__absorb1,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_598_Int__absorb1,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_599_Int__lower2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_600_Int__lower2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_601_Int__lower1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_602_Int__lower1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_603_Int__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ D2 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_604_Int__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_605_Un__Int__crazy,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) )
= ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_606_Un__Int__crazy,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_607_Int__Un__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_608_Int__Un__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_609_Un__Int__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_610_Un__Int__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_611_Int__Un__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) @ A )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_612_Int__Un__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) @ A )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ A ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_613_Un__Int__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) @ A )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ A ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_614_Un__Int__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) @ A )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ A ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_615_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( A @ P4 )
=> ( member6642669606046002379c_fm_i @ P4 @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_616_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_617_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_618_distrib__inf__le,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) @ ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_619_distrib__inf__le,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z ) ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_620_distrib__sup__le,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) ) @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_621_distrib__sup__le,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_622_Un__Int__assoc__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) )
= ( ord_le5389487502678872194c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_623_Un__Int__assoc__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_624_inter__set__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_625_inter__set__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_626_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( epistemic_AK_i @ A @ P4 )
=> ( member6642669606046002379c_fm_i @ P4 @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_627_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ~ ! [W: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V @ W )
=> ( ( episte2285483198712856234tent_i @ A @ W )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) ) ) ).
% maximal_extension
thf(fact_628_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_629_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_630_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_631_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( X3 = Y )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_632_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F2 @ B2 ) @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_633_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_634_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y2 = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A4 @ B4 )
& ( ord_le3843937902494030498c_fm_i @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_635_antisym,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_636_dual__order_Otrans,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_637_dual__order_Oantisym,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_638_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y2 = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A4 )
& ( ord_le3843937902494030498c_fm_i @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_639_order__trans,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ Z )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_640_order_Otrans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_641_order__antisym,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_642_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_643_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2 = B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_644_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y2 = Z2 ) )
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y4 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_645_not__finite__existsD,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_646_not__finite__existsD,axiom,
! [P3: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
=> ? [X_1: epistemic_fm_i] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_647_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,R: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ? [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_648_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,R: epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ? [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_649_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( ord_le3843937902494030498c_fm_i @ X2 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_650_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( ord_le3843937902494030498c_fm_i @ A2 @ X2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_651_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_652_infinite__super,axiom,
! [S3: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S3 @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S3 )
=> ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_653_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_654_mcs__conjunction__mult,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,S3: list_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S3 ) @ V )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S3 ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ S3 ) @ V ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_655_sup__Un__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S3 ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( sup_su2582925890723967158c_fm_i @ R @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_656_sup__Un__eq,axiom,
! [R: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S3 ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ R @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_657_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ W2 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_658_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_659_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_660_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_661_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_662_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z ) @ X3 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ X3 ) @ ( inf_in161960956874937808c_fm_i @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_663_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_664_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ Z ) @ X3 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ X3 ) @ ( sup_su2582925890723967158c_fm_i @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_665_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ Z ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_666_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U: set_Epistemic_fm_i,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ U ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ U ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_667_inf__Int__eq,axiom,
! [R: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S3 ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ R @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_668_inf__Int__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( inf_in853329969422372621fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S3 ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( inf_in161960956874937808c_fm_i @ R @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_669_inf__set__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_670_inf__set__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( inf_in853329969422372621fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_671_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( A
= ( inf_in161960956874937808c_fm_i @ K @ A2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B2 )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_672_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( B
= ( inf_in161960956874937808c_fm_i @ K @ B2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_673_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Epistemic_fm_i,K: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B
= ( sup_su1936195050962291414c_fm_i @ K @ B2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_674_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Epistemic_fm_i,K: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A
= ( sup_su1936195050962291414c_fm_i @ K @ A2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_675_pred__subset__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S3 ) )
= ( ord_le5389487502678872194c_fm_i @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_676_pred__subset__eq,axiom,
! [R: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S3 ) )
= ( ord_le3843937902494030498c_fm_i @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_677_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i,U: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ W2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_678_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_679_dual__reach,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I: i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V )
=> ? [W: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) )
& ( member6642669606046002379c_fm_i @ P4 @ W ) ) ) ) ) ).
% dual_reach
thf(fact_680_reach__dualK,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) ) )
=> ! [P: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P @ W2 )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_681_K5__L,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ P4 ) ) ) ) ).
% K5_L
thf(fact_682_inter__Set__filter,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= ( filter6053540608743173075c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_683_inter__Set__filter,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= ( filter8450032351001387443c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_684_member__filter,axiom,
! [X3: epistemic_fm_i,P3: epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( filter6053540608743173075c_fm_i @ P3 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P3 @ X3 ) ) ) ).
% member_filter
thf(fact_685_member__filter,axiom,
! [X3: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( filter8450032351001387443c_fm_i @ P3 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P3 @ X3 ) ) ) ).
% member_filter
thf(fact_686_Ax4_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A2 )
=> ~ ! [I2: i,P5: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P5 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P5 ) ) ) ) ) ).
% Ax4.cases
thf(fact_687_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A4: epistemic_fm_i] :
? [I3: i,P2: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P2 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_688_Ax4_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P4 ) ) ) ) ).
% Ax4.intros
thf(fact_689_AxT_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxT_i @ A2 )
=> ~ ! [I2: i,P5: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P5 ) @ P5 ) ) ) ).
% AxT.cases
thf(fact_690_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A4: epistemic_fm_i] :
? [I3: i,P2: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ P2 ) ) ) ) ).
% AxT.simps
thf(fact_691_AxT_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ P4 ) ) ).
% AxT.intros
thf(fact_692_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Imp_i @ X51 @ X52 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_693_Set_Ofilter__def,axiom,
( filter6053540608743173075c_fm_i
= ( ^ [P6: epistemic_fm_i > $o,A3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [A4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A4 @ A3 )
& ( P6 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_694_Set_Ofilter__def,axiom,
( filter8450032351001387443c_fm_i
= ( ^ [P6: set_Epistemic_fm_i > $o,A3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A3 )
& ( P6 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_695_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_696_K__imp__trans_H,axiom,
! [A: epistemic_fm_i > $o,Q2: epistemic_fm_i,R2: epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q2 @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ Q2 ) @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_697_K__imp__trans,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q2 @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_698_K__trans,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ Q2 ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ Q2 @ R2 ) @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_trans
thf(fact_699_R1,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P4 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q2 ) )
=> ( epistemic_AK_i @ A @ Q2 ) ) ) ).
% R1
thf(fact_700_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ Q2 ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q2 ) ) ) ) ).
% K_A2'
thf(fact_701_K__map,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i,I: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q2 ) ) ) ) ).
% K_map
thf(fact_702_K__right__mp,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P4 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ ( epistemic_Imp_i @ P4 @ Q2 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q2 ) ) ) ) ).
% K_right_mp
thf(fact_703_finite__filter,axiom,
! [S3: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ S3 )
=> ( finite3304564979551393739c_fm_i @ ( filter6053540608743173075c_fm_i @ P3 @ S3 ) ) ) ).
% finite_filter
thf(fact_704_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ Q2 ) @ V )
=> ( member6642669606046002379c_fm_i @ Q2 @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_705_T__L,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_706_K4__L,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_707_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P4 ) ) ) ) ) ) ).
% S5'_4
thf(fact_708_S5_H__S5,axiom,
! [P4: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ P4 )
=> ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ P4 ) ) ).
% S5'_S5
thf(fact_709_S5__S5_H,axiom,
! [P4: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ P4 )
=> ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ P4 ) ) ).
% S5_S5'
thf(fact_710_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_711_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_712_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% K_L_dual
thf(fact_713_K__LK,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_714_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_715_Ax5_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_716_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A4: epistemic_fm_i] :
? [I3: i,P2: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_717_Ax5_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A2 )
=> ~ ! [I2: i,P5: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_718_AxB_Ointros,axiom,
! [P4: epistemic_fm_i,I: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_719_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A4: epistemic_fm_i] :
? [P2: epistemic_fm_i,I3: i] :
( A4
= ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_720_AxB_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxB_i @ A2 )
=> ~ ! [P5: epistemic_fm_i,I2: i] :
( A2
!= ( epistemic_Imp_i @ P5 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_721_K__imply__conjunct,axiom,
! [A: epistemic_fm_i > $o,G2: list_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ P4 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G2 ) @ P4 ) ) ) ).
% K_imply_conjunct
thf(fact_722_K__conjunct__imply,axiom,
! [A: epistemic_fm_i > $o,G2: list_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G2 ) @ P4 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ P4 ) ) ) ).
% K_conjunct_imply
thf(fact_723_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ A2 )
=> ~ ! [I2: i,P5: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P5 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_724_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_i
= ( ^ [A4: epistemic_fm_i] :
? [I3: i,P2: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_725_Ax__2_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( stalnaker_Ax_2_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax_2.intros
thf(fact_726_filter__set,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter6053540608743173075c_fm_i @ P3 @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) ).
% filter_set
thf(fact_727_filter__set,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( filter8450032351001387443c_fm_i @ P3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) ).
% filter_set
thf(fact_728_S5__S5_H__assms,axiom,
! [G2: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G2 )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs @ P4 ) ) ) )
= ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G2 )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ ( epistemic_imply_i @ Qs @ P4 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_729_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs2: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs2 ) )
=> ! [G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R2 ) @ ( epistemic_imply_i @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_730_conj__subset__def,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P3 ) )
& ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_731_conj__subset__def,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P3 ) )
& ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_732_prop__restrict,axiom,
! [X3: set_Epistemic_fm_i,Z4: set_se3485332733965609186c_fm_i,X5: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X3 @ Z4 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z4
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X5 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X3 ) ) ) ).
% prop_restrict
thf(fact_733_prop__restrict,axiom,
! [X3: epistemic_fm_i,Z4: set_Epistemic_fm_i,X5: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X3 @ Z4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z4
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X5 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X3 ) ) ) ).
% prop_restrict
thf(fact_734_Collect__restrict,axiom,
! [X5: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X5 )
& ( P3 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_735_Collect__restrict,axiom,
! [X5: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X5 )
& ( P3 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_736_subset__CollectI,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q: set_Epistemic_fm_i > $o,P3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P3 @ X2 ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B )
& ( Q @ X ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_737_subset__CollectI,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q: epistemic_fm_i > $o,P3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P3 @ X2 ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
& ( Q @ X ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_738_subset__Collect__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_739_subset__Collect__iff,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_740_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H2 @ ( epistemic_K_i @ I @ P4 ) )
= ( H2 @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% eval.simps(6)
thf(fact_741_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_i > $o] :
~ ( epistemic_eval_i @ Uu @ Uv @ epistemic_FF_i ) ).
% eval.simps(1)
thf(fact_742_A1,axiom,
! [P4: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ P4 )
=> ( epistemic_AK_i @ A @ P4 ) ) ).
% A1
thf(fact_743_duality__taut,axiom,
! [I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_744_imply__conjunct,axiom,
! [G2: list_Epistemic_fm_i,P4: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ G2 @ P4 ) @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G2 ) @ P4 ) ) ) ).
% imply_conjunct
thf(fact_745_conjunct__imply,axiom,
! [G2: list_Epistemic_fm_i,P4: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G2 ) @ P4 ) @ ( epistemic_imply_i @ G2 @ P4 ) ) ) ).
% conjunct_imply
thf(fact_746_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A3: epistemic_fm_i > $o,A4: epistemic_fm_i] :
( ? [P2: epistemic_fm_i] :
( ( A4 = P2 )
& ! [G: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G @ H4 @ P2 ) )
| ? [I3: i,P2: epistemic_fm_i,Q3: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ Q3 ) ) ) @ ( epistemic_K_i @ I3 @ Q3 ) ) )
| ? [P2: epistemic_fm_i] :
( ( A4 = P2 )
& ( A3 @ P2 ) )
| ? [P2: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( A4 = Q3 )
& ( epistemic_AK_i @ A3 @ P2 )
& ( epistemic_AK_i @ A3 @ ( epistemic_Imp_i @ P2 @ Q3 ) ) )
| ? [P2: epistemic_fm_i,I3: i] :
( ( A4
= ( epistemic_K_i @ I3 @ P2 ) )
& ( epistemic_AK_i @ A3 @ P2 ) ) ) ) ) ).
% AK.simps
thf(fact_747_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A2 )
=> ( ~ ! [G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ A2 )
=> ( ! [I2: i,P5: epistemic_fm_i,Q4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P5 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P5 @ Q4 ) ) ) @ ( epistemic_K_i @ I2 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P5: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P5 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P5 @ A2 ) ) )
=> ~ ! [P5: epistemic_fm_i] :
( ? [I2: i] :
( A2
= ( epistemic_K_i @ I2 @ P5 ) )
=> ~ ( epistemic_AK_i @ A @ P5 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_748_K__thm,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P4 @ Q2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_749_K__Boole,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ G2 ) @ epistemic_FF_i ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ P4 ) ) ) ).
% K_Boole
thf(fact_750_union__coset__filter,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( coset_1342939534705115381c_fm_i @ Xs ) @ A )
= ( coset_1342939534705115381c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_751_union__coset__filter,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( coset_Epistemic_fm_i @ Xs ) @ A )
= ( coset_Epistemic_fm_i
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_752_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_i,X22: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ X22 ) )
=> ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_753_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i,X21: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ X22 ) )
=> ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_754_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_755_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] : ( member1104366573291651755c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_756_list_Oset__cases,axiom,
! [E: epistemic_fm_i,A2: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ A2 ) )
=> ( ! [Z22: list_Epistemic_fm_i] :
( A2
!= ( cons_Epistemic_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: list_Epistemic_fm_i] :
( ( A2
= ( cons_Epistemic_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_757_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A2: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ A2 ) )
=> ( ! [Z22: list_s8081015415394010888c_fm_i] :
( A2
!= ( cons_s4962720389763977656c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z22: list_s8081015415394010888c_fm_i] :
( ( A2
= ( cons_s4962720389763977656c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_758_set__ConsD,axiom,
! [Y: epistemic_fm_i,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_759_set__ConsD,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_760_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_761_filter_Osimps_I2_J,axiom,
! [P3: epistemic_fm_i > $o,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( P3 @ X3 )
=> ( ( filter7636273843821131039c_fm_i @ P3 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) )
= ( cons_Epistemic_fm_i @ X3 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) )
& ( ~ ( P3 @ X3 )
=> ( ( filter7636273843821131039c_fm_i @ P3 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_762_filter_Osimps_I2_J,axiom,
! [P3: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( ( P3 @ X3 )
=> ( ( filter3188398074982218495c_fm_i @ P3 @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) )
= ( cons_s4962720389763977656c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) )
& ( ~ ( P3 @ X3 )
=> ( ( filter3188398074982218495c_fm_i @ P3 @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) )
= ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_763_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_764_set__subset__Cons,axiom,
! [Xs: list_s8081015415394010888c_fm_i,X3: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_765_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X3: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_766_imply_Osimps_I2_J,axiom,
! [P4: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q2 )
= ( epistemic_Imp_i @ P4 @ ( epistemic_imply_i @ Ps @ Q2 ) ) ) ).
% imply.simps(2)
thf(fact_767_K__imply__multi,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ B2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ C ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Con_i @ B2 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_768_K__multi__imply,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Imp_i @ B2 @ C ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ A2 @ B2 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_769_K__imply__head,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Ps: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ P4 ) ) ).
% K_imply_head
thf(fact_770_K__imply__Cons,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q2: epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q2 ) ) ) ).
% K_imply_Cons
thf(fact_771_K__swap,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i,G2: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ ( cons_Epistemic_fm_i @ Q2 @ G2 ) ) @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q2 @ ( cons_Epistemic_fm_i @ P4 @ G2 ) ) @ R2 ) ) ) ).
% K_swap
thf(fact_772_mcs__conjunction,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( member6642669606046002379c_fm_i @ P4 @ V )
& ( member6642669606046002379c_fm_i @ Q2 @ V ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P4 @ Q2 ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_773_K__conjunction__in,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q2 ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q2 ) ) ) ) ).
% K_conjunction_in
thf(fact_774_K__conjunction__out,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q2 ) ) @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q2 ) ) ) ) ).
% K_conjunction_out
thf(fact_775_K__conj__imply__factor,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q2 ) ) @ R2 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q2 ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_776_A2,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ Q2 ) ) ) @ ( epistemic_K_i @ I @ Q2 ) ) ) ).
% A2
thf(fact_777_K__ImpI,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G2: list_Epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ G2 ) @ Q2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ ( epistemic_Imp_i @ P4 @ Q2 ) ) ) ) ).
% K_ImpI
thf(fact_778_K__mp,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q2: epistemic_fm_i,G2: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P4 @ Q2 ) @ G2 ) ) @ Q2 ) ) ).
% K_mp
thf(fact_779_subset__code_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Ys ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ~ ( member1104366573291651755c_fm_i @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_780_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( coset_Epistemic_fm_i @ Ys ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ~ ( member6642669606046002379c_fm_i @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_781_not__in__set__insert,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X3 @ Xs )
= ( cons_Epistemic_fm_i @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_782_not__in__set__insert,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X3 @ Xs )
= ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_783_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [V2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V2 ) @ V )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ V2 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_784_empty__Collect__eq,axiom,
! [P3: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P3 ) )
= ( ! [X: epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_785_empty__Collect__eq,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P3 ) )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_786_Collect__empty__eq,axiom,
! [P3: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P3 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_787_Collect__empty__eq,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P3 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_788_all__not__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ! [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ A ) )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% all_not_in_conv
thf(fact_789_all__not__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ! [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ A ) )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% all_not_in_conv
thf(fact_790_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_791_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_792_insert__iff,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_793_insert__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_794_insertCI,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertCI
thf(fact_795_insertCI,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) ) ).
% insertCI
thf(fact_796_subset__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_empty
thf(fact_797_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_798_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X3 )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_left
thf(fact_799_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_right
thf(fact_800_inf__bot__right,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_right
thf(fact_801_inf__bot__left,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X3 )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_left
thf(fact_802_sup__bot_Oright__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_803_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_804_sup__bot_Oleft__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_805_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_806_sup__eq__bot__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_eq_bot_iff
thf(fact_807_bot__eq__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y = bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_eq_sup_iff
thf(fact_808_sup__bot__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= X3 ) ).
% sup_bot_right
thf(fact_809_sup__bot__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_810_singletonI,axiom,
! [A2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_811_singletonI,axiom,
! [A2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singletonI
thf(fact_812_finite__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ).
% finite_insert
thf(fact_813_insert__subset,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( ( member1104366573291651755c_fm_i @ X3 @ B )
& ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_814_insert__subset,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( ( member6642669606046002379c_fm_i @ X3 @ B )
& ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_815_Un__empty,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Un_empty
thf(fact_816_Int__insert__right__if1,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_817_Int__insert__right__if1,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_818_Int__insert__right__if0,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_819_Int__insert__right__if0,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_820_insert__inter__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_821_Int__insert__left__if1,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_822_Int__insert__left__if1,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_823_Int__insert__left__if0,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_824_Int__insert__left__if0,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_825_Un__insert__right,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_826_Un__insert__left,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_827_singleton__conv2,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y2: epistemic_fm_i,Z2: epistemic_fm_i] : ( Y2 = Z2 )
@ A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_828_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y2 = Z2 )
@ A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_829_singleton__conv,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv
thf(fact_830_singleton__conv,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv
thf(fact_831_in__set__insert,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_832_in__set__insert,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_833_singleton__insert__inj__eq,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_834_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A2 = B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_835_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_836_list_Osimps_I15_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) )
= ( insert7698009978809854162c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ X22 ) ) ) ).
% list.simps(15)
thf(fact_837_disjoint__insert_I2_J,axiom,
! [A: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B2 @ A )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_838_disjoint__insert_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) )
= ( ~ ( member1104366573291651755c_fm_i @ B2 @ A )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_839_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ B @ A )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_840_disjoint__insert_I1_J,axiom,
! [B: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ A ) )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ B @ A )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_841_insert__disjoint_I2_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_842_insert__disjoint_I2_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_843_insert__disjoint_I1_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_844_insert__disjoint_I1_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_845_List_Oset__insert,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( insert177310195918387262c_fm_i @ X3 @ Xs ) )
= ( insert7817948997695205106c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_846_List_Oset__insert,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( insert2436375702790685214c_fm_i @ X3 @ Xs ) )
= ( insert7698009978809854162c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ).
% List.set_insert
thf(fact_847_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
=> ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_848_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_849_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_850_finite_Ocases,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( A2 != bot_bo4194595901900360558c_fm_i )
=> ~ ! [A6: set_Epistemic_fm_i] :
( ? [A7: epistemic_fm_i] :
( A2
= ( insert7817948997695205106c_fm_i @ A7 @ A6 ) )
=> ~ ( finite3304564979551393739c_fm_i @ A6 ) ) ) ) ).
% finite.cases
thf(fact_851_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
( ( A4 = bot_bo4194595901900360558c_fm_i )
| ? [A3: set_Epistemic_fm_i,B4: epistemic_fm_i] :
( ( A4
= ( insert7817948997695205106c_fm_i @ B4 @ A3 ) )
& ( finite3304564979551393739c_fm_i @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_852_finite__induct,axiom,
! [F3: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,F4: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F4 )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X2 @ F4 ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_853_finite__induct,axiom,
! [F3: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X2 @ F4 ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_854_finite__ne__induct,axiom,
! [F3: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( F3 != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i] : ( P3 @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) )
=> ( ! [X2: set_Epistemic_fm_i,F4: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F4 )
=> ( ( F4 != bot_bo145720340923748686c_fm_i )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X2 @ F4 ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_855_finite__ne__induct,axiom,
! [F3: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i] : ( P3 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X2: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( F4 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X2 @ F4 ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_856_infinite__finite__induct,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ! [A6: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A6 )
=> ( P3 @ A6 ) )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,F4: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F4 )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X2 @ F4 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_857_infinite__finite__induct,axiom,
! [P3: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A6: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A6 )
=> ( P3 @ A6 ) )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X2 @ F4 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_858_singleton__Un__iff,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_859_Un__singleton__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_860_insert__is__Un,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% insert_is_Un
thf(fact_861_insert__Collect,axiom,
! [A2: epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P3 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U2: epistemic_fm_i] :
( ( U2 != A2 )
=> ( P3 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_862_insert__Collect,axiom,
! [A2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P3 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U2: set_Epistemic_fm_i] :
( ( U2 != A2 )
=> ( P3 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_863_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A4 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_864_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A4 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_865_mk__disjoint__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ? [B6: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A2 @ B6 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_866_mk__disjoint__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [B6: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ A2 @ B6 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_867_Collect__conv__if2,axiom,
! [P3: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_868_Collect__conv__if2,axiom,
! [P3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_869_Collect__conv__if,axiom,
! [P3: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_870_Collect__conv__if,axiom,
! [P3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_871_singleton__iff,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_872_singleton__iff,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_873_insert__eq__iff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B2 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B2 @ C3 )
& ( B
= ( insert7817948997695205106c_fm_i @ A2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_874_insert__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ B2 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ B2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ B2 @ C3 )
& ( B
= ( insert7698009978809854162c_fm_i @ A2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_875_insert__absorb,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_876_insert__absorb,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_877_insert__ident,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X3 @ A )
= ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_878_insert__ident,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ X3 @ A )
= ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_879_singletonD,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_880_singletonD,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_881_Set_Oset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ! [B6: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ B6 ) )
=> ( member6642669606046002379c_fm_i @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_882_Set_Oset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ~ ! [B6: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ X3 @ B6 ) )
=> ( member1104366573291651755c_fm_i @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_883_ex__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) )
= ( A != bot_bo4194595901900360558c_fm_i ) ) ).
% ex_in_conv
thf(fact_884_ex__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) )
= ( A != bot_bo145720340923748686c_fm_i ) ) ).
% ex_in_conv
thf(fact_885_insertI2,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertI2
thf(fact_886_insertI2,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) ) ).
% insertI2
thf(fact_887_insertI1,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_888_insertI1,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_889_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_890_equals0I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [Y3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ Y3 @ A )
=> ( A = bot_bo145720340923748686c_fm_i ) ) ).
% equals0I
thf(fact_891_equals0D,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_892_equals0D,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( A = bot_bo145720340923748686c_fm_i )
=> ~ ( member1104366573291651755c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_893_insertE,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_894_insertE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_895_emptyE,axiom,
! [A2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_896_emptyE,axiom,
! [A2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_897_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_898_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_899_subset__singletonD,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A = bot_bo4194595901900360558c_fm_i )
| ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_900_subset__singleton__iff,axiom,
! [X5: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X5 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( X5 = bot_bo4194595901900360558c_fm_i )
| ( X5
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_901_finite__subset__induct_H,axiom,
! [F3: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( ord_le5389487502678872194c_fm_i @ F3 @ A )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i,F4: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F4 )
=> ( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ F4 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A7 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ A7 @ F4 ) ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_902_finite__subset__induct_H,axiom,
! [F3: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ F4 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A7 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ A7 @ F4 ) ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_903_finite__subset__induct,axiom,
! [F3: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( ord_le5389487502678872194c_fm_i @ F3 @ A )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i,F4: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F4 )
=> ( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A7 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ A7 @ F4 ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_904_finite__subset__induct,axiom,
! [F3: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A7 @ F4 )
=> ( ( P3 @ F4 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ A7 @ F4 ) ) ) ) ) )
=> ( P3 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_905_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_906_finite_OinsertI,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_907_insert__subsetI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X5: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X5 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_908_insert__subsetI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,X5: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X5 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_909_insert__mono,axiom,
! [C2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ C2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_910_subset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_911_subset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_912_subset__insertI,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% subset_insertI
thf(fact_913_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_914_Int__insert__right,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_915_Int__insert__right,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_916_Int__insert__left,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_917_Int__insert__left,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_918_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_919_infinite__imp__nonempty,axiom,
! [S3: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S3 )
=> ( S3 != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_920_subset__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i ) ) ).
% subset_emptyI
thf(fact_921_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_922_disjoint__iff__not__equal,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_923_Int__empty__right,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_right
thf(fact_924_Int__empty__left,axiom,
! [B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ B )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_left
thf(fact_925_disjoint__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ~ ( member6642669606046002379c_fm_i @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_926_disjoint__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ~ ( member1104366573291651755c_fm_i @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_927_Int__emptyI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ B ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_928_Int__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ B ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ).
% Int_emptyI
thf(fact_929_Un__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Un_empty_right
thf(fact_930_Un__empty__left,axiom,
! [B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ B )
= B ) ).
% Un_empty_left
thf(fact_931_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_932_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_933_maximal__def,axiom,
! [A: epistemic_fm_i > $o,S3: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S3 )
= ( ! [P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ S3 ) )
=> ( member6642669606046002379c_fm_i @ P2 @ S3 ) ) ) ) ).
% maximal_def
thf(fact_934_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_935_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_936_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q2 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_937_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ P4 @ Q2 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_938_List_Oinsert__def,axiom,
( insert177310195918387262c_fm_i
= ( ^ [X: epistemic_fm_i,Xs3: list_Epistemic_fm_i] : ( if_lis3446954105117697518c_fm_i @ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_i @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_939_List_Oinsert__def,axiom,
( insert2436375702790685214c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Xs3: list_s8081015415394010888c_fm_i] : ( if_lis2088983353317256142c_fm_i @ ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs3 ) ) @ Xs3 @ ( cons_s4962720389763977656c_fm_i @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_940_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,V: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [W: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ W )
=> ( ( finite3304564979551393739c_fm_i @ W )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_941_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G2: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G2 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ P4 ) ) ) ).
% inconsistent_imply
thf(fact_942_extendS__def,axiom,
! [A: epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S3 @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S3 @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_943_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_944_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_945_bot__empty__eq,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_946_bot__empty__eq,axiom,
( bot_bo6089404950617257231fm_i_o
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ bot_bo145720340923748686c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_947_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_948_is__singletonI_H,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( member1104366573291651755c_fm_i @ Y3 @ A )
=> ( X2 = Y3 ) ) )
=> ( is_sin2358732406619996150c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_949_set__rec,axiom,
( set_Epistemic_fm_i2
= ( rec_li5834409330748533462c_fm_i @ bot_bo4194595901900360558c_fm_i
@ ^ [X: epistemic_fm_i,Uu2: list_Epistemic_fm_i] : ( insert7817948997695205106c_fm_i @ X ) ) ) ).
% set_rec
thf(fact_950_set__rec,axiom,
( set_se200842218512397079c_fm_i
= ( rec_li386023649751468758c_fm_i @ bot_bo145720340923748686c_fm_i
@ ^ [X: set_Epistemic_fm_i,Uu2: list_s8081015415394010888c_fm_i] : ( insert7698009978809854162c_fm_i @ X ) ) ) ).
% set_rec
thf(fact_951_Sup__fin_Oinsert,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_952_Inf__fin_Oinsert,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_953_inf__Sup__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_954_inf__Sup__absorb,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ ( lattic5920028816461561881c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_955_sup__Inf__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_956_Inf__fin__le__Sup__fin,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_957_Inf__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_958_Sup__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_959_Inf__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_960_Inf__fin_Oin__idem,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_961_Sup__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_962_Sup__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_963_Inf__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_964_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A7 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_965_Sup__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_966_Inf__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ A7 ) )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_967_Inf__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_968_Inf__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_969_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_970_Inf__fin_Osubset,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ B ) @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_971_Sup__fin_Osubset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ B ) @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_972_Inf__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X2 @ Y3 ) @ ( insert7698009978809854162c_fm_i @ X2 @ ( insert7698009978809854162c_fm_i @ Y3 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_973_Inf__fin_Oclosed,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ ( inf_in161960956874937808c_fm_i @ X2 @ Y3 ) @ ( insert7111703059596746418c_fm_i @ X2 @ ( insert7111703059596746418c_fm_i @ Y3 @ bot_bo4781621276559889198c_fm_i ) ) )
=> ( member1461078328125707403c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_974_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_975_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ~ ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_976_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_977_Sup__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y3 ) @ ( insert7698009978809854162c_fm_i @ X2 @ ( insert7698009978809854162c_fm_i @ Y3 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_978_Inf__fin_Ounion,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( finite9087983574947184523c_fm_i @ B )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ ( lattic7860428167660714879c_fm_i @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_979_Sup__fin_Ounion,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_980_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P4 @ Q2 ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_981_subset__singleton__iff__Uniq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [A4: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A4 @ bot_bo145720340923748686c_fm_i ) ) )
= ( uniq_s5891897140743797767c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_982_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A4: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_983_member__remove,axiom,
! [X3: epistemic_fm_i,Y: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( remove5552923401211475591c_fm_i @ Y @ A ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_984_member__remove,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( remove1392734035364174951c_fm_i @ Y @ A ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_985_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Dis_i @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_986_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_987_K__DisL,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q2: epistemic_fm_i,P7: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P7 @ Ps ) @ Q2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Dis_i @ P4 @ P7 ) @ Ps ) @ Q2 ) ) ) ) ).
% K_DisL
thf(fact_988_K__DisE,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G2: list_Epistemic_fm_i,R2: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ G2 ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q2 @ G2 ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ ( epistemic_Dis_i @ P4 @ Q2 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G2 @ R2 ) ) ) ) ) ).
% K_DisE
thf(fact_989_fm_Oexhaust,axiom,
! [Y: epistemic_fm_i] :
( ( Y != epistemic_FF_i )
=> ( ! [X23: list_char] :
( Y
!= ( epistemic_Pro_i @ X23 ) )
=> ( ! [X312: epistemic_fm_i,X322: epistemic_fm_i] :
( Y
!= ( epistemic_Dis_i @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_990_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_991_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X24 ) ) ).
% fm.distinct(1)
thf(fact_992_fm_Orel__cases,axiom,
! [R: i > i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ A2 @ B2 )
=> ( ( ( A2 = epistemic_FF_i )
=> ( B2 != epistemic_FF_i ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_i @ X2 ) )
=> ! [Y3: list_char] :
( ( B2
= ( epistemic_Pro_i @ Y3 ) )
=> ( X2 != Y3 ) ) )
=> ( ! [X1: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B2
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B2
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B2
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B2
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_993_fm_Orel__induct,axiom,
! [R: i > i > $o,X3: epistemic_fm_i,Y: epistemic_fm_i,Q: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R @ X3 @ Y )
=> ( ( Q @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q @ A31 @ B31 )
=> ( ( Q @ A32 @ B32 )
=> ( Q @ ( epistemic_Dis_i @ A31 @ A32 ) @ ( epistemic_Dis_i @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_i,A42: epistemic_fm_i,B41: epistemic_fm_i,B42: epistemic_fm_i] :
( ( Q @ A41 @ B41 )
=> ( ( Q @ A42 @ B42 )
=> ( Q @ ( epistemic_Con_i @ A41 @ A42 ) @ ( epistemic_Con_i @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_i,A52: epistemic_fm_i,B51: epistemic_fm_i,B52: epistemic_fm_i] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R @ A61 @ B61 )
=> ( ( Q @ A62 @ B62 )
=> ( Q @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_994_Sup__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_995_Sup__fin_Oinsert__remove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_996_DiffI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_997_DiffI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_998_Diff__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_999_Diff__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1000_finite__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1001_finite__Diff2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_Diff2
thf(fact_1002_insert__Diff1,axiom,
! [X3: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1003_insert__Diff1,axiom,
! [X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1004_Diff__insert0,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1005_Diff__insert0,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1006_Un__Diff__cancel2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ B @ A ) @ A )
= ( sup_su1936195050962291414c_fm_i @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1007_Un__Diff__cancel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1008_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1009_finite__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1010_Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ A ) )
= bot_bo145720340923748686c_fm_i ) ).
% Diff_disjoint
thf(fact_1011_fm_Octr__transfer_I1_J,axiom,
! [R: i > i > $o] : ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ epistemic_FF_i ) ).
% fm.ctr_transfer(1)
thf(fact_1012_Diff__infinite__finite,axiom,
! [T2: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S3 )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1013_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1014_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1015_double__diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B @ ( minus_1543973939109951465c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1016_DiffE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1017_DiffE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1018_DiffD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1019_DiffD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1020_DiffD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1021_DiffD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1022_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
& ~ ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1023_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
& ~ ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1024_fm_Orel__intros_I6_J,axiom,
! [R: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_1025_fm_Orel__inject_I6_J,axiom,
! [R: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_1026_Un__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1027_Diff__Int__distrib2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ C2 )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1028_Diff__Int__distrib,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ C2 @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1029_Diff__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1030_Diff__Int2,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1031_Int__Diff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1032_insert__Diff__if,axiom,
! [X3: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( insert7817948997695205106c_fm_i @ X3 @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1033_insert__Diff__if,axiom,
! [X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( insert7698009978809854162c_fm_i @ X3 @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1034_fm_Orel__distinct_I10_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ Y61 @ Y62 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(10)
thf(fact_1035_fm_Orel__distinct_I9_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_1036_diff__shunt__var,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X3 @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_1037_insert__Diff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1038_insert__Diff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1039_Diff__insert__absorb,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1040_Diff__insert__absorb,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1041_subset__Diff__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) )
& ~ ( member1104366573291651755c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1042_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1043_Int__Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= bot_bo145720340923748686c_fm_i ) ).
% Int_Diff_disjoint
thf(fact_1044_Diff__triv,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
=> ( ( minus_1073301138667672009c_fm_i @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1045_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1046_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1047_Un__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1048_Un__Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1049_Int__Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1050_Int__Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1051_Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1052_Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1053_Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1054_Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1055_finite__empty__induct,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( P3 @ A )
=> ( ! [A7: set_Epistemic_fm_i,A6: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A6 )
=> ( ( member1104366573291651755c_fm_i @ A7 @ A6 )
=> ( ( P3 @ A6 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A6 @ ( insert7698009978809854162c_fm_i @ A7 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) )
=> ( P3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1056_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P3 @ A )
=> ( ! [A7: epistemic_fm_i,A6: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A6 )
=> ( ( member6642669606046002379c_fm_i @ A7 @ A6 )
=> ( ( P3 @ A6 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A6 @ ( insert7817948997695205106c_fm_i @ A7 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1057_infinite__coinduct,axiom,
! [X5: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( X5 @ A )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( X5 @ A6 )
=> ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A6 )
& ( ( X5 @ ( minus_1543973939109951465c_fm_i @ A6 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A6 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% infinite_coinduct
thf(fact_1058_infinite__remove,axiom,
! [S3: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S3 )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S3 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% infinite_remove
thf(fact_1059_subset__insert__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) @ B ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1060_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1061_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1062_minus__coset__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( coset_Epistemic_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1063_minus__coset__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1064_remove__induct,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,B: set_se3485332733965609186c_fm_i] :
( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ( ~ ( finite7933139204641697195c_fm_i @ B )
=> ( P3 @ B ) )
=> ( ! [A6: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A6 )
=> ( ( A6 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A6 @ B )
=> ( ! [X6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X6 @ A6 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A6 @ ( insert7698009978809854162c_fm_i @ X6 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P3 @ A6 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% remove_induct
thf(fact_1065_remove__induct,axiom,
! [P3: set_Epistemic_fm_i > $o,B: set_Epistemic_fm_i] :
( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B )
=> ( P3 @ B ) )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A6 )
=> ( ( A6 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A6 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A6 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A6 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P3 @ A6 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% remove_induct
thf(fact_1066_finite__remove__induct,axiom,
! [B: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A6 )
=> ( ( A6 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A6 @ B )
=> ( ! [X6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X6 @ A6 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A6 @ ( insert7698009978809854162c_fm_i @ X6 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P3 @ A6 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1067_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A6 )
=> ( ( A6 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A6 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A6 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A6 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P3 @ A6 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1068_set__minus__filter__out,axiom,
! [Xs: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ Y @ bot_bo4194595901900360558c_fm_i ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( X != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1069_set__minus__filter__out,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Y: set_Epistemic_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( insert7698009978809854162c_fm_i @ Y @ bot_bo145720340923748686c_fm_i ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1070_Inf__fin_Oinsert__remove,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1071_Inf__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1072_Inf__fin_Oremove,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= X3 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1073_set__removeAll,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X3 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1074_set__removeAll,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( remove6226062834164189966c_fm_i @ X3 @ Xs ) )
= ( minus_1073301138667672009c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1075_removeAll__id,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove817376155714978094c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1076_removeAll__id,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remove6226062834164189966c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1077_removeAll__filter__not,axiom,
! [P3: epistemic_fm_i > $o,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( P3 @ X3 )
=> ( ( remove817376155714978094c_fm_i @ X3 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_1078_removeAll__filter__not,axiom,
! [P3: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( P3 @ X3 )
=> ( ( remove6226062834164189966c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) )
= ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_1079_minus__set__def,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( minus_6519238007984220724fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1080_minus__set__def,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( minus_5671197887077578836fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1081_removeAll__filter__not__eq,axiom,
( remove817376155714978094c_fm_i
= ( ^ [X: epistemic_fm_i] :
( filter7636273843821131039c_fm_i
@ ^ [Y4: epistemic_fm_i] : ( X != Y4 ) ) ) ) ).
% removeAll_filter_not_eq
thf(fact_1082_removeAll__filter__not__eq,axiom,
( remove6226062834164189966c_fm_i
= ( ^ [X: set_Epistemic_fm_i] :
( filter3188398074982218495c_fm_i
@ ^ [Y4: set_Epistemic_fm_i] : ( X != Y4 ) ) ) ) ).
% removeAll_filter_not_eq
thf(fact_1083_remove__code_I1_J,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( remove5552923401211475591c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X3 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1084_remove__code_I1_J,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( remove1392734035364174951c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i @ ( remove6226062834164189966c_fm_i @ X3 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1085_MCS_Omaximal__def,axiom,
! [Consistent: set_se3485332733965609186c_fm_i > $o,S3: set_se3485332733965609186c_fm_i] :
( ( maxima5540402109999912942c_fm_i @ Consistent )
=> ( ( maxima7741914846562630577c_fm_i @ Consistent @ S3 )
= ( ! [P2: set_Epistemic_fm_i] :
( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ P2 @ bot_bo145720340923748686c_fm_i ) @ S3 ) )
=> ( member1104366573291651755c_fm_i @ P2 @ S3 ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_1086_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S3: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S3 )
= ( ! [P2: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ S3 ) )
=> ( member6642669606046002379c_fm_i @ P2 @ S3 ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_1087_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_1088_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_1089_is__empty__set,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( is_emp7643256682846039796c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( null_Epistemic_fm_i @ Xs ) ) ).
% is_empty_set
thf(fact_1090_is__empty__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i] :
( ( is_emp8832846348672524500c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( null_s8880300773456351248c_fm_i @ Xs ) ) ).
% is_empty_set
thf(fact_1091_MCS__Lim__Ord_OextendS__def,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S3 @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S3 @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_1092_filter__shuffles__disjoint1_I2_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_1093_filter__shuffles__disjoint1_I2_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_1094_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S3: set_Epistemic_fm_i,S4: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( Consistent @ S3 )
=> ( ( ord_le3843937902494030498c_fm_i @ S4 @ S3 )
=> ( Consistent @ S4 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_1095_partition__in__shuffles,axiom,
! [Xs: list_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( member6475149752759350353c_fm_i @ Xs
@ ( shuffl511019056595460925c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 @ Xs )
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( P3 @ X )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_1096_partition__in__shuffles,axiom,
! [Xs: list_s8081015415394010888c_fm_i,P3: set_Epistemic_fm_i > $o] :
( member605545779687026225c_fm_i @ Xs
@ ( shuffl5790413406345785629c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs )
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( P3 @ X )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_1097_set__shuffles,axiom,
! [Zs: list_s8081015415394010888c_fm_i,Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( set_se200842218512397079c_fm_i @ Zs )
= ( sup_su2582925890723967158c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1098_set__shuffles,axiom,
! [Zs: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( set_Epistemic_fm_i2 @ Zs )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1099_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S3: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ~ ( Consistent @ S3 )
=> ? [S: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ S3 )
& ( finite3304564979551393739c_fm_i @ S )
& ~ ( Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1100_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1101_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1102_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1103_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1104_filter__shuffles__disjoint1_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_1105_filter__shuffles__disjoint1_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_1106_Set__filter__fold,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( filter6053540608743173075c_fm_i @ P3 @ A )
= ( finite6274917443777430184c_fm_i
@ ^ [X: epistemic_fm_i,A9: set_Epistemic_fm_i] : ( if_set2943457721768014536c_fm_i @ ( P3 @ X ) @ ( insert7817948997695205106c_fm_i @ X @ A9 ) @ A9 )
@ bot_bo4194595901900360558c_fm_i
@ A ) ) ) ).
% Set_filter_fold
thf(fact_1107_fm_Oset__cases,axiom,
! [E: epistemic_fm_i,A2: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ A2 ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z22: episte1308613106011496894c_fm_i] :
( A2
!= ( episte5657488666450005526c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte5657488666450005526c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1108_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A2: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ A2 ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte535002356104313075c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte535002356104313075c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte64184772710425087c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte64184772710425087c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte5768433346120044277c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte5768433346120044277c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z22: episte4276422356747265950c_fm_i] :
( A2
!= ( episte3581967596712343542c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte3581967596712343542c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1109_fm_Oset__cases,axiom,
! [E: i,A2: epistemic_fm_i] :
( ( member_i @ E @ ( epistemic_set_fm_i @ A2 ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_i] :
( A2
!= ( epistemic_K_i @ E @ Z22 ) )
=> ~ ! [Z1: i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1110_fm_Osimps_I95_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) )
= ( insert_i @ X61 @ ( epistemic_set_fm_i @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_1111_fold__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G5: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i] :
( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite2135258021436235976c_fm_i @ F2 @ Z @ A )
= ( finite2135258021436235976c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1112_fold__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G5: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite6274917443777430184c_fm_i @ F2 @ Z @ A )
= ( finite6274917443777430184c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1113_fold__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G5: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite4171219623613778664c_fm_i @ F2 @ Z @ A )
= ( finite4171219623613778664c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1114_fold__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G5: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite2237691025846545608c_fm_i @ F2 @ Z @ A )
= ( finite2237691025846545608c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1115_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_i,X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ X41 ) )
=> ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_1116_fm_Oset__intros_I3_J,axiom,
! [Yc: set_Epistemic_fm_i,X41: episte4276422356747265950c_fm_i,X42: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yc @ ( episte6795552051664633144c_fm_i @ X41 ) )
=> ( member1104366573291651755c_fm_i @ Yc @ ( episte6795552051664633144c_fm_i @ ( episte64184772710425087c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_1117_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_i,X42: episte1308613106011496894c_fm_i,X41: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ X42 ) )
=> ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_1118_fm_Oset__intros_I4_J,axiom,
! [Yd: set_Epistemic_fm_i,X42: episte4276422356747265950c_fm_i,X41: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yd @ ( episte6795552051664633144c_fm_i @ X42 ) )
=> ( member1104366573291651755c_fm_i @ Yd @ ( episte6795552051664633144c_fm_i @ ( episte64184772710425087c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_1119_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i] : ( member6642669606046002379c_fm_i @ X61 @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1120_fm_Oset__intros_I7_J,axiom,
! [X61: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i] : ( member1104366573291651755c_fm_i @ X61 @ ( episte6795552051664633144c_fm_i @ ( episte3581967596712343542c_fm_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1121_fm_Oset__intros_I7_J,axiom,
! [X61: i,X62: epistemic_fm_i] : ( member_i @ X61 @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1122_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,X61: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ X62 ) )
=> ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1123_fm_Oset__intros_I8_J,axiom,
! [Yg: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i,X61: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Yg @ ( episte6795552051664633144c_fm_i @ X62 ) )
=> ( member1104366573291651755c_fm_i @ Yg @ ( episte6795552051664633144c_fm_i @ ( episte3581967596712343542c_fm_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1124_fm_Oset__intros_I8_J,axiom,
! [Yg: i,X62: epistemic_fm_i,X61: i] :
( ( member_i @ Yg @ ( epistemic_set_fm_i @ X62 ) )
=> ( member_i @ Yg @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1125_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_i,X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ X51 ) )
=> ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_1126_fm_Oset__intros_I5_J,axiom,
! [Ye: set_Epistemic_fm_i,X51: episte4276422356747265950c_fm_i,X52: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Ye @ ( episte6795552051664633144c_fm_i @ X51 ) )
=> ( member1104366573291651755c_fm_i @ Ye @ ( episte6795552051664633144c_fm_i @ ( episte5768433346120044277c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_1127_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_i,X52: episte1308613106011496894c_fm_i,X51: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ X52 ) )
=> ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_1128_fm_Oset__intros_I6_J,axiom,
! [Yf: set_Epistemic_fm_i,X52: episte4276422356747265950c_fm_i,X51: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yf @ ( episte6795552051664633144c_fm_i @ X52 ) )
=> ( member1104366573291651755c_fm_i @ Yf @ ( episte6795552051664633144c_fm_i @ ( episte5768433346120044277c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_1129_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_i,X31: episte1308613106011496894c_fm_i,X32: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ X31 ) )
=> ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_1130_fm_Oset__intros_I1_J,axiom,
! [Ya: set_Epistemic_fm_i,X31: episte4276422356747265950c_fm_i,X32: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Ya @ ( episte6795552051664633144c_fm_i @ X31 ) )
=> ( member1104366573291651755c_fm_i @ Ya @ ( episte6795552051664633144c_fm_i @ ( episte535002356104313075c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_1131_fm_Oset__intros_I2_J,axiom,
! [Yb: epistemic_fm_i,X32: episte1308613106011496894c_fm_i,X31: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ X32 ) )
=> ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_1132_fm_Oset__intros_I2_J,axiom,
! [Yb: set_Epistemic_fm_i,X32: episte4276422356747265950c_fm_i,X31: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yb @ ( episte6795552051664633144c_fm_i @ X32 ) )
=> ( member1104366573291651755c_fm_i @ Yb @ ( episte6795552051664633144c_fm_i @ ( episte535002356104313075c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_1133_fm_Orel__refl__strong,axiom,
! [X3: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ! [Z3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( episte9089240992906287960c_fm_i @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_1134_fm_Orel__refl__strong,axiom,
! [X3: episte4276422356747265950c_fm_i,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ! [Z3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( episte6795552051664633144c_fm_i @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte2362472030976073184c_fm_i @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_1135_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_i > epistemic_fm_i > $o,X3: episte1308613106011496894c_fm_i,Y: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( episte3445413862081925600c_fm_i @ R @ X3 @ Y )
=> ( ! [Z3: epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( episte9089240992906287960c_fm_i @ X3 ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Y ) )
=> ( ( R @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_1136_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_i > set_Epistemic_fm_i > $o,X3: episte1308613106011496894c_fm_i,Y: episte4276422356747265950c_fm_i,Ra: epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( episte6079896185807049664c_fm_i @ R @ X3 @ Y )
=> ( ! [Z3: epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( episte9089240992906287960c_fm_i @ X3 ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Y ) )
=> ( ( R @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte6079896185807049664c_fm_i @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_1137_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_i > epistemic_fm_i > $o,X3: episte4276422356747265950c_fm_i,Y: episte1308613106011496894c_fm_i,Ra: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ( episte3976198365643398144c_fm_i @ R @ X3 @ Y )
=> ( ! [Z3: set_Epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( episte6795552051664633144c_fm_i @ X3 ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Y ) )
=> ( ( R @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte3976198365643398144c_fm_i @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_1138_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,X3: episte4276422356747265950c_fm_i,Y: episte4276422356747265950c_fm_i,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( episte2362472030976073184c_fm_i @ R @ X3 @ Y )
=> ( ! [Z3: set_Epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( episte6795552051664633144c_fm_i @ X3 ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Y ) )
=> ( ( R @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte2362472030976073184c_fm_i @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_1139_fm_Orel__cong,axiom,
! [X3: episte1308613106011496894c_fm_i,Ya: episte1308613106011496894c_fm_i,Y: episte1308613106011496894c_fm_i,Xa2: episte1308613106011496894c_fm_i,R: epistemic_fm_i > epistemic_fm_i > $o,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( episte9089240992906287960c_fm_i @ Ya ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Xa2 ) )
=> ( ( R @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte3445413862081925600c_fm_i @ R @ X3 @ Y )
= ( episte3445413862081925600c_fm_i @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1140_fm_Orel__cong,axiom,
! [X3: episte1308613106011496894c_fm_i,Ya: episte1308613106011496894c_fm_i,Y: episte4276422356747265950c_fm_i,Xa2: episte4276422356747265950c_fm_i,R: epistemic_fm_i > set_Epistemic_fm_i > $o,Ra: epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( episte9089240992906287960c_fm_i @ Ya ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Xa2 ) )
=> ( ( R @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte6079896185807049664c_fm_i @ R @ X3 @ Y )
= ( episte6079896185807049664c_fm_i @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1141_fm_Orel__cong,axiom,
! [X3: episte4276422356747265950c_fm_i,Ya: episte4276422356747265950c_fm_i,Y: episte1308613106011496894c_fm_i,Xa2: episte1308613106011496894c_fm_i,R: set_Epistemic_fm_i > epistemic_fm_i > $o,Ra: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: set_Epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( episte6795552051664633144c_fm_i @ Ya ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Xa2 ) )
=> ( ( R @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte3976198365643398144c_fm_i @ R @ X3 @ Y )
= ( episte3976198365643398144c_fm_i @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1142_fm_Orel__cong,axiom,
! [X3: episte4276422356747265950c_fm_i,Ya: episte4276422356747265950c_fm_i,Y: episte4276422356747265950c_fm_i,Xa2: episte4276422356747265950c_fm_i,R: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: set_Epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( episte6795552051664633144c_fm_i @ Ya ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Xa2 ) )
=> ( ( R @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte2362472030976073184c_fm_i @ R @ X3 @ Y )
= ( episte2362472030976073184c_fm_i @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1143_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_i @ epistemic_FF_i )
= bot_bot_set_i ) ).
% fm.simps(90)
thf(fact_1144_fm_Osimps_I94_J,axiom,
! [X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X51 ) @ ( episte9089240992906287960c_fm_i @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_1145_fm_Osimps_I93_J,axiom,
! [X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X41 ) @ ( episte9089240992906287960c_fm_i @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_1146_fm_Osimps_I92_J,axiom,
! [X31: episte1308613106011496894c_fm_i,X32: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X31 ) @ ( episte9089240992906287960c_fm_i @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_1147_union__fold__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( finite6274917443777430184c_fm_i @ insert7817948997695205106c_fm_i @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_1148_minus__fold__remove,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( minus_1543973939109951465c_fm_i @ B @ A )
= ( finite6274917443777430184c_fm_i @ remove5552923401211475591c_fm_i @ B @ A ) ) ) ).
% minus_fold_remove
thf(fact_1149_Inf__fin_Oeq__fold,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( finite436508220448094408c_fm_i @ inf_in161960956874937808c_fm_i @ X3 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1150_Sup__fin_Oeq__fold,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( finite2237691025846545608c_fm_i @ sup_su1936195050962291414c_fm_i @ X3 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1151_flat__lub__def,axiom,
( partia2237093934842463950c_fm_i
= ( ^ [B4: set_Epistemic_fm_i,A3: set_se3485332733965609186c_fm_i] :
( if_set2943457721768014536c_fm_i @ ( ord_le5389487502678872194c_fm_i @ A3 @ ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i ) ) @ B4
@ ( the_se2508142961706643081c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( minus_1073301138667672009c_fm_i @ A3 @ ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1152_flat__lub__def,axiom,
( partia6511039473171299566c_fm_i
= ( ^ [B4: epistemic_fm_i,A3: set_Epistemic_fm_i] :
( if_Epistemic_fm_i @ ( ord_le3843937902494030498c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i ) ) @ B4
@ ( the_Epistemic_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( minus_1543973939109951465c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1153_fold__graph__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G5: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i] :
( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite4898207529229619782c_fm_i @ F2 @ Z @ A )
= ( finite4898207529229619782c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1154_fold__graph__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G5: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite3073240682113661990c_fm_i @ F2 @ Z @ A )
= ( finite3073240682113661990c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1155_fold__graph__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G5: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite969542861950010470c_fm_i @ F2 @ Z @ A )
= ( finite969542861950010470c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1156_fold__graph__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G5: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite5960466103032163910c_fm_i @ F2 @ Z @ A )
= ( finite5960466103032163910c_fm_i @ G5 @ Z @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1157_fold__graph__closed__lemma,axiom,
! [G5: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i,A: set_Epistemic_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i] :
( ( finite4898207529229619782c_fm_i @ G5 @ Z @ A @ X3 )
=> ( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite4898207529229619782c_fm_i @ F2 @ Z @ A @ X3 )
& ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1158_fold__graph__closed__lemma,axiom,
! [G5: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i,A: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( finite3073240682113661990c_fm_i @ G5 @ Z @ A @ X3 )
=> ( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite3073240682113661990c_fm_i @ F2 @ Z @ A @ X3 )
& ( member1104366573291651755c_fm_i @ X3 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1159_fold__graph__closed__lemma,axiom,
! [G5: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z: epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i] :
( ( finite969542861950010470c_fm_i @ G5 @ Z @ A @ X3 )
=> ( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member6642669606046002379c_fm_i @ B7 @ B )
=> ( member6642669606046002379c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z @ B )
=> ( ( finite969542861950010470c_fm_i @ F2 @ Z @ A @ X3 )
& ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1160_fold__graph__closed__lemma,axiom,
! [G5: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( finite5960466103032163910c_fm_i @ G5 @ Z @ A @ X3 )
=> ( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( ( F2 @ A7 @ B7 )
= ( G5 @ A7 @ B7 ) ) ) )
=> ( ! [A7: set_Epistemic_fm_i,B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ( ( member1104366573291651755c_fm_i @ B7 @ B )
=> ( member1104366573291651755c_fm_i @ ( G5 @ A7 @ B7 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z @ B )
=> ( ( finite5960466103032163910c_fm_i @ F2 @ Z @ A @ X3 )
& ( member1104366573291651755c_fm_i @ X3 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1161_finite__lists__length__le,axiom,
! [A: set_se3485332733965609186c_fm_i,N: nat] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( finite5108596011607087409c_fm_i
@ ( collec2200325868635342451c_fm_i
@ ^ [Xs3: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s2509009656797156212c_fm_i @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1162_finite__lists__length__le,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s3357407047445144980c_fm_i @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1163_finite__lists__length__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,N: nat] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( finite5108596011607087409c_fm_i
@ ( collec2200325868635342451c_fm_i
@ ^ [Xs3: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs3 ) @ A )
& ( ( size_s2509009656797156212c_fm_i @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1164_finite__lists__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A )
& ( ( size_s3357407047445144980c_fm_i @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1165_length__filter__le,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( ord_less_eq_nat @ ( size_s3357407047445144980c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) @ ( size_s3357407047445144980c_fm_i @ Xs ) ) ).
% length_filter_le
thf(fact_1166_length__filter__le,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] : ( ord_less_eq_nat @ ( size_s2509009656797156212c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) @ ( size_s2509009656797156212c_fm_i @ Xs ) ) ).
% length_filter_le
thf(fact_1167_set__n__lists,axiom,
! [N: nat,Xs: list_s8081015415394010888c_fm_i] :
( ( set_li1699392436468070557c_fm_i @ ( n_list3831374397297245712c_fm_i @ N @ Xs ) )
= ( collec2200325868635342451c_fm_i
@ ^ [Ys2: list_s8081015415394010888c_fm_i] :
( ( ( size_s2509009656797156212c_fm_i @ Ys2 )
= N )
& ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Ys2 ) @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1168_set__n__lists,axiom,
! [N: nat,Xs: list_Epistemic_fm_i] :
( ( set_li9010496130497932989c_fm_i @ ( n_list3185984609264368688c_fm_i @ N @ Xs ) )
= ( collec5759350117362652307c_fm_i
@ ^ [Ys2: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Ys2 )
= N )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ys2 ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1169_finite__lists__distinct__length__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,N: nat] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( finite5108596011607087409c_fm_i
@ ( collec2200325868635342451c_fm_i
@ ^ [Xs3: list_s8081015415394010888c_fm_i] :
( ( ( size_s2509009656797156212c_fm_i @ Xs3 )
= N )
& ( distin8272202008718213115c_fm_i @ Xs3 )
& ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1170_finite__lists__distinct__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Xs3 )
= N )
& ( distin5379319190628067867c_fm_i @ Xs3 )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1171_distinct_Osimps_I2_J,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ X3 @ Xs ) )
= ( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( distin5379319190628067867c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1172_distinct_Osimps_I2_J,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( distin8272202008718213115c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) )
= ( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
& ( distin8272202008718213115c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1173_distinct__filter,axiom,
! [Xs: list_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( distin5379319190628067867c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) ).
% distinct_filter
thf(fact_1174_distinct__filter,axiom,
! [Xs: list_s8081015415394010888c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( distin8272202008718213115c_fm_i @ Xs )
=> ( distin8272202008718213115c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) ).
% distinct_filter
thf(fact_1175_finite__distinct__list,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ? [Xs2: list_s8081015415394010888c_fm_i] :
( ( ( set_se200842218512397079c_fm_i @ Xs2 )
= A )
& ( distin8272202008718213115c_fm_i @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1176_finite__distinct__list,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ? [Xs2: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs2 )
= A )
& ( distin5379319190628067867c_fm_i @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1177_distinct__disjoint__shuffles,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( ( distin5379319190628067867c_fm_i @ Ys )
=> ( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( distin5379319190628067867c_fm_i @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1178_distinct__disjoint__shuffles,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( distin8272202008718213115c_fm_i @ Xs )
=> ( ( distin8272202008718213115c_fm_i @ Ys )
=> ( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( distin8272202008718213115c_fm_i @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1179_set__remove1__eq,axiom,
! [Xs: list_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X3 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% set_remove1_eq
thf(fact_1180_set__remove1__eq,axiom,
! [Xs: list_s8081015415394010888c_fm_i,X3: set_Epistemic_fm_i] :
( ( distin8272202008718213115c_fm_i @ Xs )
=> ( ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ X3 @ Xs ) )
= ( minus_1073301138667672009c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% set_remove1_eq
thf(fact_1181_distinct__concat,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin4780646717405202849c_fm_i @ Xs )
=> ( ! [Ys3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys3 ) )
=> ( ! [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( member6475149752759350353c_fm_i @ Zs2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys3 ) @ ( set_Epistemic_fm_i2 @ Zs2 ) )
= bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1182_distinct__concat,axiom,
! [Xs: list_l8939085672862518926c_fm_i] :
( ( distin3563480452828447617c_fm_i @ Xs )
=> ( ! [Ys3: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys3 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( distin8272202008718213115c_fm_i @ Ys3 ) )
=> ( ! [Ys3: list_s8081015415394010888c_fm_i,Zs2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys3 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( ( member605545779687026225c_fm_i @ Zs2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Ys3 ) @ ( set_se200842218512397079c_fm_i @ Zs2 ) )
= bot_bo145720340923748686c_fm_i ) ) ) )
=> ( distin8272202008718213115c_fm_i @ ( concat3510228398265320643c_fm_i @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1183_in__set__remove1,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( A2 != B2 )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ B2 @ Xs ) ) )
= ( member6642669606046002379c_fm_i @ A2 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1184_in__set__remove1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( A2 != B2 )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ B2 @ Xs ) ) )
= ( member1104366573291651755c_fm_i @ A2 @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1185_remove1__filter__not,axiom,
! [P3: epistemic_fm_i > $o,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( P3 @ X3 )
=> ( ( remove8001893566138066622c_fm_i @ X3 @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P3 @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_1186_remove1__filter__not,axiom,
! [P3: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( P3 @ X3 )
=> ( ( remove2021789187851727006c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) )
= ( filter3188398074982218495c_fm_i @ P3 @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_1187_notin__set__remove1,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1188_notin__set__remove1,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i,Y: set_Epistemic_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1189_remove1__idem,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1190_remove1__idem,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remove2021789187851727006c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1191_filter__remove1,axiom,
! [Q: epistemic_fm_i > $o,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( filter7636273843821131039c_fm_i @ Q @ ( remove8001893566138066622c_fm_i @ X3 @ Xs ) )
= ( remove8001893566138066622c_fm_i @ X3 @ ( filter7636273843821131039c_fm_i @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_1192_filter__remove1,axiom,
! [Q: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( filter3188398074982218495c_fm_i @ Q @ ( remove2021789187851727006c_fm_i @ X3 @ Xs ) )
= ( remove2021789187851727006c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_1193_set__remove1__subset,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ X3 @ Xs ) ) @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% set_remove1_subset
thf(fact_1194_set__remove1__subset,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X3 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1195_distinct__concat__iff,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) )
= ( ( distin4780646717405202849c_fm_i @ ( remove5890047714521709364c_fm_i @ nil_Epistemic_fm_i @ Xs ) )
& ! [Ys2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys2 ) )
& ! [Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( member6475149752759350353c_fm_i @ Zs3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys2 ) @ ( set_Epistemic_fm_i2 @ Zs3 ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% distinct_concat_iff
thf(fact_1196_distinct__concat__iff,axiom,
! [Xs: list_l8939085672862518926c_fm_i] :
( ( distin8272202008718213115c_fm_i @ ( concat3510228398265320643c_fm_i @ Xs ) )
= ( ( distin3563480452828447617c_fm_i @ ( remove1440612933533133076c_fm_i @ nil_se670848885773549064c_fm_i @ Xs ) )
& ! [Ys2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( distin8272202008718213115c_fm_i @ Ys2 ) )
& ! [Ys2: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( ( member605545779687026225c_fm_i @ Ys2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
& ( member605545779687026225c_fm_i @ Zs3 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Ys2 ) @ ( set_se200842218512397079c_fm_i @ Zs3 ) )
= bot_bo145720340923748686c_fm_i ) ) ) ) ).
% distinct_concat_iff
thf(fact_1197_set__empty,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs )
= bot_bo4194595901900360558c_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty
thf(fact_1198_set__empty,axiom,
! [Xs: list_s8081015415394010888c_fm_i] :
( ( ( set_se200842218512397079c_fm_i @ Xs )
= bot_bo145720340923748686c_fm_i )
= ( Xs = nil_se670848885773549064c_fm_i ) ) ).
% set_empty
thf(fact_1199_set__empty2,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ Xs ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty2
thf(fact_1200_set__empty2,axiom,
! [Xs: list_s8081015415394010888c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( set_se200842218512397079c_fm_i @ Xs ) )
= ( Xs = nil_se670848885773549064c_fm_i ) ) ).
% set_empty2
thf(fact_1201_filter__False,axiom,
! [Xs: list_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P3 @ X2 ) )
=> ( ( filter7636273843821131039c_fm_i @ P3 @ Xs )
= nil_Epistemic_fm_i ) ) ).
% filter_False
thf(fact_1202_filter__False,axiom,
! [Xs: list_s8081015415394010888c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ~ ( P3 @ X2 ) )
=> ( ( filter3188398074982218495c_fm_i @ P3 @ Xs )
= nil_se670848885773549064c_fm_i ) ) ).
% filter_False
thf(fact_1203_imply_Osimps_I1_J,axiom,
! [Q2: epistemic_fm_i] :
( ( epistemic_imply_i @ nil_Epistemic_fm_i @ Q2 )
= Q2 ) ).
% imply.simps(1)
thf(fact_1204_filter_Osimps_I1_J,axiom,
! [P3: epistemic_fm_i > $o] :
( ( filter7636273843821131039c_fm_i @ P3 @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% filter.simps(1)
thf(fact_1205_filter_Osimps_I1_J,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( filter3188398074982218495c_fm_i @ P3 @ nil_se670848885773549064c_fm_i )
= nil_se670848885773549064c_fm_i ) ).
% filter.simps(1)
thf(fact_1206_empty__set,axiom,
( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% empty_set
thf(fact_1207_empty__set,axiom,
( bot_bo145720340923748686c_fm_i
= ( set_se200842218512397079c_fm_i @ nil_se670848885773549064c_fm_i ) ) ).
% empty_set
thf(fact_1208_empty__filter__conv,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( nil_Epistemic_fm_i
= ( filter7636273843821131039c_fm_i @ P3 @ Xs ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P3 @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_1209_empty__filter__conv,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( nil_se670848885773549064c_fm_i
= ( filter3188398074982218495c_fm_i @ P3 @ Xs ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ~ ( P3 @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_1210_filter__empty__conv,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( ( filter7636273843821131039c_fm_i @ P3 @ Xs )
= nil_Epistemic_fm_i )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P3 @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_1211_filter__empty__conv,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( ( filter3188398074982218495c_fm_i @ P3 @ Xs )
= nil_se670848885773549064c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ~ ( P3 @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_1212_empty__Shift,axiom,
! [Kl: set_li8269783359719514504c_fm_i,K: epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ Kl )
=> ( ( member6642669606046002379c_fm_i @ K @ ( bNF_Gr1093487169986386109c_fm_i @ Kl @ nil_Epistemic_fm_i ) )
=> ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ ( bNF_Gr8437504169225076033c_fm_i @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1213_empty__Shift,axiom,
! [Kl: set_li2934481837907835752c_fm_i,K: set_Epistemic_fm_i] :
( ( member605545779687026225c_fm_i @ nil_se670848885773549064c_fm_i @ Kl )
=> ( ( member1104366573291651755c_fm_i @ K @ ( bNF_Gr2376118584863155869c_fm_i @ Kl @ nil_se670848885773549064c_fm_i ) )
=> ( member605545779687026225c_fm_i @ nil_se670848885773549064c_fm_i @ ( bNF_Gr8275819806307007265c_fm_i @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1214_subset__code_I3_J,axiom,
~ ( ord_le5389487502678872194c_fm_i @ ( coset_1342939534705115381c_fm_i @ nil_se670848885773549064c_fm_i ) @ ( set_se200842218512397079c_fm_i @ nil_se670848885773549064c_fm_i ) ) ).
% subset_code(3)
thf(fact_1215_subset__code_I3_J,axiom,
~ ( ord_le3843937902494030498c_fm_i @ ( coset_Epistemic_fm_i @ nil_Epistemic_fm_i ) @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% subset_code(3)
thf(fact_1216_conjunct_Osimps_I1_J,axiom,
( ( stalnaker_conjunct_i @ nil_Epistemic_fm_i )
= ( epistemic_Imp_i @ epistemic_FF_i @ epistemic_FF_i ) ) ).
% conjunct.simps(1)
thf(fact_1217_the__elem__set,axiom,
! [X3: epistemic_fm_i] :
( ( the_el2173195912186371479c_fm_i @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) ) )
= X3 ) ).
% the_elem_set
thf(fact_1218_the__elem__set,axiom,
! [X3: set_Epistemic_fm_i] :
( ( the_el1853918686291641207c_fm_i @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ nil_se670848885773549064c_fm_i ) ) )
= X3 ) ).
% the_elem_set
thf(fact_1219_Cons__in__lists__iff,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ ( cons_Epistemic_fm_i @ X3 @ Xs ) @ ( lists_Epistemic_fm_i @ A ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1220_Cons__in__lists__iff,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) @ ( lists_3378462843525889270c_fm_i @ A ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( member605545779687026225c_fm_i @ Xs @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1221_in__listsI,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X2 @ A ) )
=> ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) ) ) ).
% in_listsI
thf(fact_1222_in__listsI,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X2 @ A ) )
=> ( member605545779687026225c_fm_i @ Xs @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ).
% in_listsI
thf(fact_1223_lists__Int__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( lists_3378462843525889270c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( inf_in3319901132148857174c_fm_i @ ( lists_3378462843525889270c_fm_i @ A ) @ ( lists_3378462843525889270c_fm_i @ B ) ) ) ).
% lists_Int_eq
thf(fact_1224_lists_Ocases,axiom,
! [A2: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ A2 @ ( lists_Epistemic_fm_i @ A ) )
=> ( ( A2 != nil_Epistemic_fm_i )
=> ~ ! [A7: epistemic_fm_i,L: list_Epistemic_fm_i] :
( ( A2
= ( cons_Epistemic_fm_i @ A7 @ L ) )
=> ( ( member6642669606046002379c_fm_i @ A7 @ A )
=> ~ ( member6475149752759350353c_fm_i @ L @ ( lists_Epistemic_fm_i @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1225_lists_Ocases,axiom,
! [A2: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ A2 @ ( lists_3378462843525889270c_fm_i @ A ) )
=> ( ( A2 != nil_se670848885773549064c_fm_i )
=> ~ ! [A7: set_Epistemic_fm_i,L: list_s8081015415394010888c_fm_i] :
( ( A2
= ( cons_s4962720389763977656c_fm_i @ A7 @ L ) )
=> ( ( member1104366573291651755c_fm_i @ A7 @ A )
=> ~ ( member605545779687026225c_fm_i @ L @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_1226_lists_Osimps,axiom,
! [A2: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ A2 @ ( lists_Epistemic_fm_i @ A ) )
= ( ( A2 = nil_Epistemic_fm_i )
| ? [A4: epistemic_fm_i,L2: list_Epistemic_fm_i] :
( ( A2
= ( cons_Epistemic_fm_i @ A4 @ L2 ) )
& ( member6642669606046002379c_fm_i @ A4 @ A )
& ( member6475149752759350353c_fm_i @ L2 @ ( lists_Epistemic_fm_i @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1227_lists_Osimps,axiom,
! [A2: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ A2 @ ( lists_3378462843525889270c_fm_i @ A ) )
= ( ( A2 = nil_se670848885773549064c_fm_i )
| ? [A4: set_Epistemic_fm_i,L2: list_s8081015415394010888c_fm_i] :
( ( A2
= ( cons_s4962720389763977656c_fm_i @ A4 @ L2 ) )
& ( member1104366573291651755c_fm_i @ A4 @ A )
& ( member605545779687026225c_fm_i @ L2 @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ) ) ).
% lists.simps
thf(fact_1228_in__lists__conv__set,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1229_in__lists__conv__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ Xs @ ( lists_3378462843525889270c_fm_i @ A ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1230_in__listsD,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) )
=> ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X6 @ A ) ) ) ).
% in_listsD
thf(fact_1231_in__listsD,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ Xs @ ( lists_3378462843525889270c_fm_i @ A ) )
=> ! [X6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X6 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X6 @ A ) ) ) ).
% in_listsD
thf(fact_1232_lists__IntI,axiom,
! [L3: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ L3 @ ( lists_3378462843525889270c_fm_i @ A ) )
=> ( ( member605545779687026225c_fm_i @ L3 @ ( lists_3378462843525889270c_fm_i @ B ) )
=> ( member605545779687026225c_fm_i @ L3 @ ( lists_3378462843525889270c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ) ).
% lists_IntI
thf(fact_1233_lists__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le1209247765174941992c_fm_i @ ( lists_Epistemic_fm_i @ A ) @ ( lists_Epistemic_fm_i @ B ) ) ) ).
% lists_mono
thf(fact_1234_listsE,axiom,
! [X3: epistemic_fm_i,L3: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ ( cons_Epistemic_fm_i @ X3 @ L3 ) @ ( lists_Epistemic_fm_i @ A ) )
=> ~ ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ( member6475149752759350353c_fm_i @ L3 @ ( lists_Epistemic_fm_i @ A ) ) ) ) ).
% listsE
thf(fact_1235_listsE,axiom,
! [X3: set_Epistemic_fm_i,L3: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member605545779687026225c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ L3 ) @ ( lists_3378462843525889270c_fm_i @ A ) )
=> ~ ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ~ ( member605545779687026225c_fm_i @ L3 @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ) ).
% listsE
thf(fact_1236_lists_OCons,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,L3: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( member6475149752759350353c_fm_i @ L3 @ ( lists_Epistemic_fm_i @ A ) )
=> ( member6475149752759350353c_fm_i @ ( cons_Epistemic_fm_i @ A2 @ L3 ) @ ( lists_Epistemic_fm_i @ A ) ) ) ) ).
% lists.Cons
thf(fact_1237_lists_OCons,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,L3: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( member605545779687026225c_fm_i @ L3 @ ( lists_3378462843525889270c_fm_i @ A ) )
=> ( member605545779687026225c_fm_i @ ( cons_s4962720389763977656c_fm_i @ A2 @ L3 ) @ ( lists_3378462843525889270c_fm_i @ A ) ) ) ) ).
% lists.Cons
thf(fact_1238_lists__def2,axiom,
( lists_3378462843525889270c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i] :
( collec2200325868635342451c_fm_i
@ ^ [L2: list_s8081015415394010888c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ L2 ) @ A3 ) ) ) ) ).
% lists_def2
thf(fact_1239_lists__def2,axiom,
( lists_Epistemic_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( collec5759350117362652307c_fm_i
@ ^ [L2: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ L2 ) @ A3 ) ) ) ) ).
% lists_def2
thf(fact_1240_sum__length__filter__compl,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( plus_plus_nat @ ( size_s3357407047445144980c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) )
@ ( size_s3357407047445144980c_fm_i
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( P3 @ X )
@ Xs ) ) )
= ( size_s3357407047445144980c_fm_i @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_1241_sum__length__filter__compl,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( plus_plus_nat @ ( size_s2509009656797156212c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) )
@ ( size_s2509009656797156212c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( P3 @ X )
@ Xs ) ) )
= ( size_s2509009656797156212c_fm_i @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_1242_nlists__def,axiom,
( cardin5471715159450223164c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,N2: nat] :
( collec2200325868635342451c_fm_i
@ ^ [L2: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ L2 ) @ A3 )
& ( ( size_s2509009656797156212c_fm_i @ L2 )
= N2 ) ) ) ) ) ).
% nlists_def
thf(fact_1243_nlists__def,axiom,
( cardin549358664437198940c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,N2: nat] :
( collec5759350117362652307c_fm_i
@ ^ [L2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ L2 ) @ A3 )
& ( ( size_s3357407047445144980c_fm_i @ L2 )
= N2 ) ) ) ) ) ).
% nlists_def
thf(fact_1244_image__eqI,axiom,
! [B2: epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i,X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1245_image__eqI,axiom,
! [B2: set_Epistemic_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i,X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ B2 @ ( image_2913337242412145381c_fm_i @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1246_image__eqI,axiom,
! [B2: epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i,X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_809639422248493861c_fm_i @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1247_image__eqI,axiom,
! [B2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ B2 @ ( image_2698589095351095557c_fm_i @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1248_finite__imageI,axiom,
! [F3: set_Epistemic_fm_i,H2: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( finite3304564979551393739c_fm_i @ ( image_120052958407486725c_fm_i @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_1249_if__image__distrib,axiom,
! [P3: epistemic_fm_i > $o,F2: epistemic_fm_i > epistemic_fm_i,G5: epistemic_fm_i > epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( image_120052958407486725c_fm_i
@ ^ [X: epistemic_fm_i] : ( if_Epistemic_fm_i @ ( P3 @ X ) @ ( F2 @ X ) @ ( G5 @ X ) )
@ S3 )
= ( sup_su1936195050962291414c_fm_i @ ( image_120052958407486725c_fm_i @ F2 @ ( inf_in3450601097109690352c_fm_i @ S3 @ ( collec4904205187116291597c_fm_i @ P3 ) ) )
@ ( image_120052958407486725c_fm_i @ G5
@ ( inf_in3450601097109690352c_fm_i @ S3
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( P3 @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1250_if__image__distrib,axiom,
! [P3: set_Epistemic_fm_i > $o,F2: set_Epistemic_fm_i > epistemic_fm_i,G5: set_Epistemic_fm_i > epistemic_fm_i,S3: set_se3485332733965609186c_fm_i] :
( ( image_809639422248493861c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( if_Epistemic_fm_i @ ( P3 @ X ) @ ( F2 @ X ) @ ( G5 @ X ) )
@ S3 )
= ( sup_su1936195050962291414c_fm_i @ ( image_809639422248493861c_fm_i @ F2 @ ( inf_in161960956874937808c_fm_i @ S3 @ ( collec3087743281813070829c_fm_i @ P3 ) ) )
@ ( image_809639422248493861c_fm_i @ G5
@ ( inf_in161960956874937808c_fm_i @ S3
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1251_filter__shuffles,axiom,
! [P3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( image_8661220427522574853c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 ) @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
= ( shuffl511019056595460925c_fm_i @ ( filter7636273843821131039c_fm_i @ P3 @ Xs ) @ ( filter7636273843821131039c_fm_i @ P3 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_1252_filter__shuffles,axiom,
! [P3: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( image_7977712360212772357c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 ) @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
= ( shuffl5790413406345785629c_fm_i @ ( filter3188398074982218495c_fm_i @ P3 @ Xs ) @ ( filter3188398074982218495c_fm_i @ P3 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_1253_image__Un,axiom,
! [F2: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( image_120052958407486725c_fm_i @ F2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ ( image_120052958407486725c_fm_i @ F2 @ A ) @ ( image_120052958407486725c_fm_i @ F2 @ B ) ) ) ).
% image_Un
thf(fact_1254_imageI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ ( F2 @ X3 ) @ ( image_120052958407486725c_fm_i @ F2 @ A ) ) ) ).
% imageI
thf(fact_1255_imageI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ ( F2 @ X3 ) @ ( image_2913337242412145381c_fm_i @ F2 @ A ) ) ) ).
% imageI
thf(fact_1256_imageI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ ( F2 @ X3 ) @ ( image_809639422248493861c_fm_i @ F2 @ A ) ) ) ).
% imageI
thf(fact_1257_imageI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ ( F2 @ X3 ) @ ( image_2698589095351095557c_fm_i @ F2 @ A ) ) ) ).
% imageI
thf(fact_1258_rev__image__eqI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1259_rev__image__eqI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member1104366573291651755c_fm_i @ B2 @ ( image_2913337242412145381c_fm_i @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1260_rev__image__eqI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_809639422248493861c_fm_i @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1261_rev__image__eqI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member1104366573291651755c_fm_i @ B2 @ ( image_2698589095351095557c_fm_i @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1262_imageE,axiom,
! [B2: epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F2 @ A ) )
=> ~ ! [X2: epistemic_fm_i] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ).
% imageE
thf(fact_1263_imageE,axiom,
! [B2: epistemic_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( image_809639422248493861c_fm_i @ F2 @ A ) )
=> ~ ! [X2: set_Epistemic_fm_i] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ).
% imageE
thf(fact_1264_imageE,axiom,
! [B2: set_Epistemic_fm_i,F2: epistemic_fm_i > set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( image_2913337242412145381c_fm_i @ F2 @ A ) )
=> ~ ! [X2: epistemic_fm_i] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ).
% imageE
thf(fact_1265_imageE,axiom,
! [B2: set_Epistemic_fm_i,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( image_2698589095351095557c_fm_i @ F2 @ A ) )
=> ~ ! [X2: set_Epistemic_fm_i] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ).
% imageE
thf(fact_1266_Compr__image__eq,axiom,
! [F2: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( image_120052958407486725c_fm_i @ F2 @ A ) )
& ( P3 @ X ) ) )
= ( image_120052958407486725c_fm_i @ F2
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1267_Compr__image__eq,axiom,
! [F2: set_Epistemic_fm_i > epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P3: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( image_809639422248493861c_fm_i @ F2 @ A ) )
& ( P3 @ X ) ) )
= ( image_809639422248493861c_fm_i @ F2
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1268_Compr__image__eq,axiom,
! [F2: epistemic_fm_i > set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( image_2913337242412145381c_fm_i @ F2 @ A ) )
& ( P3 @ X ) ) )
= ( image_2913337242412145381c_fm_i @ F2
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1269_Compr__image__eq,axiom,
! [F2: set_Epistemic_fm_i > set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( image_2698589095351095557c_fm_i @ F2 @ A ) )
& ( P3 @ X ) ) )
= ( image_2698589095351095557c_fm_i @ F2
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1270_pigeonhole__infinite,axiom,
! [A: set_se3485332733965609186c_fm_i,F2: set_Epistemic_fm_i > epistemic_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ ( image_809639422248493861c_fm_i @ F2 @ A ) )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1271_pigeonhole__infinite,axiom,
! [A: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ ( image_120052958407486725c_fm_i @ F2 @ A ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A4 @ A )
& ( ( F2 @ A4 )
= ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1272_image__Collect__subsetI,axiom,
! [P3: epistemic_fm_i > $o,F2: epistemic_fm_i > set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( P3 @ X2 )
=> ( member1104366573291651755c_fm_i @ ( F2 @ X2 ) @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ ( image_2913337242412145381c_fm_i @ F2 @ ( collec4904205187116291597c_fm_i @ P3 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1273_image__Collect__subsetI,axiom,
! [P3: set_Epistemic_fm_i > $o,F2: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( P3 @ X2 )
=> ( member1104366573291651755c_fm_i @ ( F2 @ X2 ) @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ ( image_2698589095351095557c_fm_i @ F2 @ ( collec3087743281813070829c_fm_i @ P3 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1274_image__Collect__subsetI,axiom,
! [P3: epistemic_fm_i > $o,F2: epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( P3 @ X2 )
=> ( member6642669606046002379c_fm_i @ ( F2 @ X2 ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F2 @ ( collec4904205187116291597c_fm_i @ P3 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1275_image__Collect__subsetI,axiom,
! [P3: set_Epistemic_fm_i > $o,F2: set_Epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( P3 @ X2 )
=> ( member6642669606046002379c_fm_i @ ( F2 @ X2 ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_809639422248493861c_fm_i @ F2 @ ( collec3087743281813070829c_fm_i @ P3 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1276_all__subset__image,axiom,
! [F2: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( ! [B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ ( image_120052958407486725c_fm_i @ F2 @ A ) )
=> ( P3 @ B3 ) ) )
= ( ! [B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A )
=> ( P3 @ ( image_120052958407486725c_fm_i @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1277_subset__image__iff,axiom,
! [B: set_Epistemic_fm_i,F2: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F2 @ A ) )
= ( ? [AA: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ AA @ A )
& ( B
= ( image_120052958407486725c_fm_i @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
% Helper facts (9)
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__i_J_T,axiom,
! [X3: epistemic_fm_i,Y: epistemic_fm_i] :
( ( if_Epistemic_fm_i @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__i_J_T,axiom,
! [X3: epistemic_fm_i,Y: epistemic_fm_i] :
( ( if_Epistemic_fm_i @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X3: list_s8081015415394010888c_fm_i,Y: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X3: list_s8081015415394010888c_fm_i,Y: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( sup_su1936195050962291414c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P2: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P2
@ ( collec4904205187116291597c_fm_i
@ ^ [Q3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q3 ) @ u ) ) )
@ s ) )
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P2: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P2
@ ( collec4904205187116291597c_fm_i
@ ^ [Q3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q3 ) @ w ) ) )
@ s ) ) )
= ( set_Epistemic_fm_i2 @ s ) ) ).
%------------------------------------------------------------------------------