TPTP Problem File: SLH0245^1.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_00434_016489__6883782_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1539 ( 699 unt; 256 typ; 0 def)
% Number of atoms : 3648 (1296 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 11735 ( 407 ~; 73 |; 295 &;9530 @)
% ( 0 <=>;1430 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 783 ( 783 >; 0 *; 0 +; 0 <<)
% Number of symbols : 224 ( 221 usr; 39 con; 0-5 aty)
% Number of variables : 3579 ( 377 ^;3103 !; 99 ?;3579 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:55.230
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Stalnaker__Logic_OAx__2_001tf__i,type,
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thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__i,type,
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thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
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thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_v_G,type,
g: set_Epistemic_fm_i ).
thf(sy_v_p,type,
p: epistemic_fm_i ).
% Relevant facts (1277)
thf(fact_0_canonical__model_I2_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ) ).
% canonical_model(2)
thf(fact_1_canonical__model_I3_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P @ S )
=> ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ) ).
% canonical_model(3)
thf(fact_2_canonical__model_I1_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P @ S )
=> ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S )
@ P ) ) ) ).
% canonical_model(1)
thf(fact_3__092_060open_062consistent_AAx__2_A_I_123_092_060_094bold_062_092_060not_062_Ap_125_A_092_060union_062_AG_J_092_060close_062,axiom,
episte2285483198712856234tent_i @ stalnaker_Ax_2_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ p @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ g ) ).
% \<open>consistent Ax_2 ({\<^bold>\<not> p} \<union> G)\<close>
thf(fact_4_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ A2 )
=> ~ ! [I2: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_5_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_i
= ( ^ [A3: epistemic_fm_i] :
? [I: i,P2: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_6_Ax__2_Ointros,axiom,
! [I3: i,P: epistemic_fm_i] : ( stalnaker_Ax_2_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax_2.intros
thf(fact_7_dual__reach,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V2 )
=> ? [W2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) )
& ( member6642669606046002379c_fm_i @ P @ W2 ) ) ) ) ) ).
% dual_reach
thf(fact_8_reach__dualK,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,W3: set_Epistemic_fm_i,I3: i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ W3 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W3 )
=> ( ( member1104366573291651755c_fm_i @ W3
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) )
=> ! [P4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P4 @ W3 )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V2 ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_9_truth__lemma,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P @ V2 )
= ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ V2
@ P ) ) ) ) ).
% truth_lemma
thf(fact_10_mcs_092_060_094sub_062___092_060_094sub_0622__weakly__directed,axiom,
! [A: epistemic_fm_i > $o] :
( ! [P3: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( A @ P3 ) )
=> ( stalna4517306826702924506c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% mcs\<^sub>_\<^sub>2_weakly_directed
thf(fact_11_singleton__insert__inj__eq,axiom,
! [B: product_unit,A2: product_unit,A: set_Product_unit] :
( ( ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_12_singleton__insert__inj__eq,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_13_singleton__insert__inj__eq_H,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ A )
= ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_14_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A2 = B )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_15_valid,axiom,
! [M: episte5489734138418521880t_unit,X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ( stalna4517306826702924506c_fm_i @ M )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ g )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ p ) ) ) ) ).
% valid
thf(fact_16_maximal__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P2 @ S ) ) ) ) ).
% maximal_def
thf(fact_17_maximal__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] : ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal_Extend
thf(fact_18_Extend__subset,axiom,
! [S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] : ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% Extend_subset
thf(fact_19_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_20_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_21_singleton__conv,axiom,
! [A2: product_unit] :
( ( collect_Product_unit
@ ^ [X: product_unit] : ( X = A2 ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singleton_conv
thf(fact_22_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_23_singleton__conv2,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_24_singleton__conv2,axiom,
! [A2: product_unit] :
( ( collect_Product_unit
@ ( ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ A2 ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singleton_conv2
thf(fact_25_Collect__const,axiom,
! [P5: $o] :
( ( P5
=> ( ( collec3087743281813070829c_fm_i
@ ^ [S2: set_Epistemic_fm_i] : P5 )
= top_to3675301018284359218c_fm_i ) )
& ( ~ P5
=> ( ( collec3087743281813070829c_fm_i
@ ^ [S2: set_Epistemic_fm_i] : P5 )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_const
thf(fact_26_Collect__const,axiom,
! [P5: $o] :
( ( P5
=> ( ( collec4904205187116291597c_fm_i
@ ^ [S2: epistemic_fm_i] : P5 )
= top_to8364301187473967186c_fm_i ) )
& ( ~ P5
=> ( ( collec4904205187116291597c_fm_i
@ ^ [S2: epistemic_fm_i] : P5 )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_const
thf(fact_27_Collect__const,axiom,
! [P5: $o] :
( ( P5
=> ( ( collect_Product_unit
@ ^ [S2: product_unit] : P5 )
= top_to1996260823553986621t_unit ) )
& ( ~ P5
=> ( ( collect_Product_unit
@ ^ [S2: product_unit] : P5 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_const
thf(fact_28_Imp__intro,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ Q ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_29_UNIV__I,axiom,
! [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ top_to3675301018284359218c_fm_i ) ).
% UNIV_I
thf(fact_30_UNIV__I,axiom,
! [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ top_to8364301187473967186c_fm_i ) ).
% UNIV_I
thf(fact_31_UNIV__I,axiom,
! [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_32_empty__Collect__eq,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P5 ) )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P5 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_33_empty__Collect__eq,axiom,
! [P5: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P5 ) )
= ( ! [X: epistemic_fm_i] :
~ ( P5 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_34_empty__Collect__eq,axiom,
! [P5: product_unit > $o] :
( ( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ P5 ) )
= ( ! [X: product_unit] :
~ ( P5 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_35_Collect__empty__eq,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P5 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P5 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_36_Collect__empty__eq,axiom,
! [P5: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P5 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
~ ( P5 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_37_Collect__empty__eq,axiom,
! [P5: product_unit > $o] :
( ( ( collect_Product_unit @ P5 )
= bot_bo3957492148770167129t_unit )
= ( ! [X: product_unit] :
~ ( P5 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_38_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_39_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_40_all__not__in__conv,axiom,
! [A: set_Product_unit] :
( ( ! [X: product_unit] :
~ ( member_Product_unit @ X @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% all_not_in_conv
thf(fact_41_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_42_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_43_empty__iff,axiom,
! [C: product_unit] :
~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% empty_iff
thf(fact_44_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_45_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( member1104366573291651755c_fm_i @ X4 @ B2 ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ).
% subsetI
thf(fact_46_subsetI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ! [X4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( member6642669606046002379c_fm_i @ X4 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% subsetI
thf(fact_47_insert__absorb2,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X3 @ ( insert7817948997695205106c_fm_i @ X3 @ A ) )
= ( insert7817948997695205106c_fm_i @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_48_insert__absorb2,axiom,
! [X3: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ X3 @ ( insert_Product_unit2 @ X3 @ A ) )
= ( insert_Product_unit2 @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_49_insert__iff,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ A ) )
= ( ( A2 = B )
| ( member_Product_unit @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_50_insert__iff,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ A ) )
= ( ( A2 = B )
| ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_51_insert__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ A ) )
= ( ( A2 = B )
| ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_52_insertCI,axiom,
! [A2: product_unit,B2: set_Product_unit,B: product_unit] :
( ( ~ ( member_Product_unit @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_53_insertCI,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% insertCI
thf(fact_54_insertCI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ B2 ) ) ) ).
% insertCI
thf(fact_55_Un__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
| ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_56_Un__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
| ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_57_UnCI,axiom,
! [C: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ A ) )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnCI
thf(fact_58_UnCI,axiom,
! [C: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ A ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnCI
thf(fact_59_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( ( epistemic_Imp_i @ X51 @ X52 )
= ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_60_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_61_fm_Oinject_I1_J,axiom,
! [X22: list_char,Y2: list_char] :
( ( ( epistemic_Pro_i @ X22 )
= ( epistemic_Pro_i @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fm.inject(1)
thf(fact_62_frame_Oext__inject,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit,W6: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More2: episte7133455005732253587t_unit] :
( ( ( episte8397520288747125024t_unit @ W5 @ K @ More )
= ( episte8397520288747125024t_unit @ W6 @ K2 @ More2 ) )
= ( ( W5 = W6 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_63_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_i > list_char > $o,More2: product_unit] :
( ( ( episte4955834246032687171t_unit @ Pi @ More )
= ( episte4955834246032687171t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_64_empty__subsetI,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% empty_subsetI
thf(fact_65_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_66_subset__empty,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% subset_empty
thf(fact_67_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_68_singletonI,axiom,
! [A2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singletonI
thf(fact_69_singletonI,axiom,
! [A2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_70_singletonI,axiom,
! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singletonI
thf(fact_71_insert__subset,axiom,
! [X3: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ X3 @ A ) @ B2 )
= ( ( member_Product_unit @ X3 @ B2 )
& ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_72_insert__subset,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B2 )
= ( ( member1104366573291651755c_fm_i @ X3 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_73_insert__subset,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B2 )
= ( ( member6642669606046002379c_fm_i @ X3 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_74_Un__empty,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Un_empty
thf(fact_75_Un__empty,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ( A = bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Un_empty
thf(fact_76_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_77_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A2: epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_79_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_80_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_81_Collect__cong,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ! [X4: set_Epistemic_fm_i] :
( ( P5 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collec3087743281813070829c_fm_i @ P5 )
= ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_cong
thf(fact_82_Collect__cong,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X4: epistemic_fm_i] :
( ( P5 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collec4904205187116291597c_fm_i @ P5 )
= ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_cong
thf(fact_83_Un__insert__right,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( insert_Product_unit2 @ A2 @ ( sup_su793286257634532545t_unit @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_84_Un__insert__right,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_85_Un__insert__left,axiom,
! [A2: product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( insert_Product_unit2 @ A2 @ ( sup_su793286257634532545t_unit @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_86_Un__insert__left,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_87_frame_Oselect__convs_I1_J,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte2675268102452679275t_unit @ ( episte8397520288747125024t_unit @ W5 @ K @ More ) )
= W5 ) ).
% frame.select_convs(1)
thf(fact_88_generalization,axiom,
! [P: epistemic_fm_i,W4: epistemic_fm_i,M2: episte4846006725945195288t_unit,I3: i] :
( ! [M3: episte4846006725945195288t_unit,X4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( episte5182706529199770816c_fm_i @ M3 @ X4 @ P ) )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ).
% generalization
thf(fact_89_generalization,axiom,
! [P: epistemic_fm_i,W4: set_Epistemic_fm_i,M2: episte5489734138418521880t_unit,I3: i] :
( ! [M3: episte5489734138418521880t_unit,X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ).
% generalization
thf(fact_90_soundness__Ax__2,axiom,
! [P: epistemic_fm_i,M2: episte4846006725945195288t_unit,W4: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P )
=> ( ( stalna1726574297087873274c_fm_i @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_Ax_2
thf(fact_91_soundness__Ax__2,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P )
=> ( ( stalna4517306826702924506c_fm_i @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_Ax_2
thf(fact_92_UNIV__witness,axiom,
? [X4: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X4 @ top_to3675301018284359218c_fm_i ) ).
% UNIV_witness
thf(fact_93_UNIV__witness,axiom,
? [X4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X4 @ top_to8364301187473967186c_fm_i ) ).
% UNIV_witness
thf(fact_94_UNIV__witness,axiom,
? [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_95_UNIV__eq__I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X4: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( top_to3675301018284359218c_fm_i = A ) ) ).
% UNIV_eq_I
thf(fact_96_UNIV__eq__I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( top_to8364301187473967186c_fm_i = A ) ) ).
% UNIV_eq_I
thf(fact_97_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X4: product_unit] : ( member_Product_unit @ X4 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_98_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_99_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_100_ex__in__conv,axiom,
! [A: set_Product_unit] :
( ( ? [X: product_unit] : ( member_Product_unit @ X @ A ) )
= ( A != bot_bo3957492148770167129t_unit ) ) ).
% ex_in_conv
thf(fact_101_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_102_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_103_equals0I,axiom,
! [A: set_Product_unit] :
( ! [Y3: product_unit] :
~ ( member_Product_unit @ Y3 @ A )
=> ( A = bot_bo3957492148770167129t_unit ) ) ).
% equals0I
thf(fact_104_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_105_equals0D,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_106_equals0D,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( A = bot_bo3957492148770167129t_unit )
=> ~ ( member_Product_unit @ A2 @ A ) ) ).
% equals0D
thf(fact_107_emptyE,axiom,
! [A2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_108_emptyE,axiom,
! [A2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_109_emptyE,axiom,
! [A2: product_unit] :
~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% emptyE
thf(fact_110_Collect__mono__iff,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( P5 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_111_Collect__mono__iff,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) )
= ( ! [X: epistemic_fm_i] :
( ( P5 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_112_set__eq__subset,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A4 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_113_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_114_Collect__mono,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ! [X4: set_Epistemic_fm_i] :
( ( P5 @ X4 )
=> ( Q2 @ X4 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_115_Collect__mono,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X4: epistemic_fm_i] :
( ( P5 @ X4 )
=> ( Q2 @ X4 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_116_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_117_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A4 )
=> ( member1104366573291651755c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_118_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A4 )
=> ( member6642669606046002379c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_119_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ).
% equalityD2
thf(fact_120_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% equalityD1
thf(fact_121_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A4 )
=> ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_122_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A4 )
=> ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_123_equalityE,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ) ).
% equalityE
thf(fact_124_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% subsetD
thf(fact_125_subsetD,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% subsetD
thf(fact_126_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_127_in__mono,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_128_mk__disjoint__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ? [B4: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ A2 @ B4 ) )
& ~ ( member_Product_unit @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_129_mk__disjoint__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ? [B4: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A2 @ B4 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_130_mk__disjoint__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [B4: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ A2 @ B4 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_131_insert__commute,axiom,
! [X3: epistemic_fm_i,Y4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X3 @ ( insert7817948997695205106c_fm_i @ Y4 @ A ) )
= ( insert7817948997695205106c_fm_i @ Y4 @ ( insert7817948997695205106c_fm_i @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_132_insert__commute,axiom,
! [X3: product_unit,Y4: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ X3 @ ( insert_Product_unit2 @ Y4 @ A ) )
= ( insert_Product_unit2 @ Y4 @ ( insert_Product_unit2 @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_133_insert__eq__iff,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ B @ B2 )
=> ( ( ( insert_Product_unit2 @ A2 @ A )
= ( insert_Product_unit2 @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ B @ C3 ) )
& ~ ( member_Product_unit @ B @ C3 )
& ( B2
= ( insert_Product_unit2 @ A2 @ C3 ) )
& ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_134_insert__eq__iff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B @ B2 )
=> ( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B @ C3 )
& ( B2
= ( insert7817948997695205106c_fm_i @ A2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_135_insert__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ B @ B2 )
=> ( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ B @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ B @ C3 )
& ( B2
= ( insert7698009978809854162c_fm_i @ A2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_136_insert__absorb,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_137_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_138_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_139_insert__ident,axiom,
! [X3: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X3 @ A )
=> ( ~ ( member_Product_unit @ X3 @ B2 )
=> ( ( ( insert_Product_unit2 @ X3 @ A )
= ( insert_Product_unit2 @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_140_insert__ident,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ B2 )
=> ( ( ( insert7817948997695205106c_fm_i @ X3 @ A )
= ( insert7817948997695205106c_fm_i @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_141_insert__ident,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ B2 )
=> ( ( ( insert7698009978809854162c_fm_i @ X3 @ A )
= ( insert7698009978809854162c_fm_i @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_142_Set_Oset__insert,axiom,
! [X3: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ~ ! [B4: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ X3 @ B4 ) )
=> ( member_Product_unit @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_143_Set_Oset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ! [B4: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ B4 ) )
=> ( member6642669606046002379c_fm_i @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_144_Set_Oset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ~ ! [B4: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ X3 @ B4 ) )
=> ( member1104366573291651755c_fm_i @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_145_insertI2,axiom,
! [A2: product_unit,B2: set_Product_unit,B: product_unit] :
( ( member_Product_unit @ A2 @ B2 )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_146_insertI2,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% insertI2
thf(fact_147_insertI2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ B2 ) ) ) ).
% insertI2
thf(fact_148_insertI1,axiom,
! [A2: product_unit,B2: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ A2 @ B2 ) ) ).
% insertI1
thf(fact_149_insertI1,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) ).
% insertI1
thf(fact_150_insertI1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) ) ).
% insertI1
thf(fact_151_insertE,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ A ) )
=> ( ( A2 != B )
=> ( member_Product_unit @ A2 @ A ) ) ) ).
% insertE
thf(fact_152_insertE,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ A ) )
=> ( ( A2 != B )
=> ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_153_insertE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ A ) )
=> ( ( A2 != B )
=> ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_154_weakly__directed,axiom,
! [M2: episte4846006725945195288t_unit,W4: epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( stalna1726574297087873274c_fm_i @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% weakly_directed
thf(fact_155_weakly__directed,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( stalna4517306826702924506c_fm_i @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% weakly_directed
thf(fact_156_Un__left__commute,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_157_Un__left__absorb,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_158_Un__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_159_Un__absorb,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ A )
= A ) ).
% Un_absorb
thf(fact_160_Un__assoc,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_161_ball__Un,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ( P5 @ X ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( P5 @ X ) )
& ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
=> ( P5 @ X ) ) ) ) ).
% ball_Un
thf(fact_162_bex__Un,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
& ( P5 @ X ) ) )
= ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P5 @ X ) )
| ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
& ( P5 @ X ) ) ) ) ).
% bex_Un
thf(fact_163_UnI2,axiom,
! [C: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnI2
thf(fact_164_UnI2,axiom,
! [C: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnI2
thf(fact_165_UnI1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnI1
thf(fact_166_UnI1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnI1
thf(fact_167_UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% UnE
thf(fact_168_UnE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% UnE
thf(fact_169_frame_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W7: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More3: episte7133455005732253587t_unit] :
( R
!= ( episte8397520288747125024t_unit @ W7 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_170_kripke_Oext__induct,axiom,
! [P5: episte7133455005732253587t_unit > $o,R: episte7133455005732253587t_unit] :
( ! [Pi3: set_Epistemic_fm_i > list_char > $o,More3: product_unit] : ( P5 @ ( episte4955834246032687171t_unit @ Pi3 @ More3 ) )
=> ( P5 @ R ) ) ).
% kripke.ext_induct
thf(fact_171_UNIV__def,axiom,
( top_to3675301018284359218c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $true ) ) ).
% UNIV_def
thf(fact_172_UNIV__def,axiom,
( top_to8364301187473967186c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $true ) ) ).
% UNIV_def
thf(fact_173_UNIV__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit
@ ^ [X: product_unit] : $true ) ) ).
% UNIV_def
thf(fact_174_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_175_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_176_empty__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit
@ ^ [X: product_unit] : $false ) ) ).
% empty_def
thf(fact_177_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P5 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_178_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P5 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_179_insert__Collect,axiom,
! [A2: product_unit,P5: product_unit > $o] :
( ( insert_Product_unit2 @ A2 @ ( collect_Product_unit @ P5 ) )
= ( collect_Product_unit
@ ^ [U: product_unit] :
( ( U != A2 )
=> ( P5 @ U ) ) ) ) ).
% insert_Collect
thf(fact_180_insert__Collect,axiom,
! [A2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P5 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U: set_Epistemic_fm_i] :
( ( U != A2 )
=> ( P5 @ U ) ) ) ) ).
% insert_Collect
thf(fact_181_insert__Collect,axiom,
! [A2: epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P5 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U: epistemic_fm_i] :
( ( U != A2 )
=> ( P5 @ U ) ) ) ) ).
% insert_Collect
thf(fact_182_insert__compr,axiom,
( insert_Product_unit2
= ( ^ [A3: product_unit,B3: set_Product_unit] :
( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A3 )
| ( member_Product_unit @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_183_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A3 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_184_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A3 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_185_Collect__disj__eq,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P5 @ X )
| ( Q2 @ X ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_186_Collect__disj__eq,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P5 @ X )
| ( Q2 @ X ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_187_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A4 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_188_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A4 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_189_empty__not__UNIV,axiom,
bot_bo4194595901900360558c_fm_i != top_to8364301187473967186c_fm_i ).
% empty_not_UNIV
thf(fact_190_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_191_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_192_subset__UNIV,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ top_to8364301187473967186c_fm_i ) ).
% subset_UNIV
thf(fact_193_insert__UNIV,axiom,
! [X3: epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X3 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% insert_UNIV
thf(fact_194_insert__UNIV,axiom,
! [X3: product_unit] :
( ( insert_Product_unit2 @ X3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_195_singleton__inject,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_196_singleton__inject,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_197_insert__not__empty,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ A2 @ A )
!= bot_bo4194595901900360558c_fm_i ) ).
% insert_not_empty
thf(fact_198_insert__not__empty,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ A2 @ A )
!= bot_bo3957492148770167129t_unit ) ).
% insert_not_empty
thf(fact_199_doubleton__eq__iff,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i,D: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
= ( insert7817948997695205106c_fm_i @ C @ ( insert7817948997695205106c_fm_i @ D @ bot_bo4194595901900360558c_fm_i ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_200_doubleton__eq__iff,axiom,
! [A2: product_unit,B: product_unit,C: product_unit,D: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit2 @ C @ ( insert_Product_unit2 @ D @ bot_bo3957492148770167129t_unit ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_201_singleton__iff,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_202_singleton__iff,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_203_singleton__iff,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_204_singletonD,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_205_singletonD,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_206_singletonD,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_207_subset__insertI2,axiom,
! [A: set_Product_unit,B2: set_Product_unit,B: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_208_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_209_subset__insertI,axiom,
! [B2: set_Product_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ B2 @ ( insert_Product_unit2 @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_210_subset__insertI,axiom,
! [B2: set_Epistemic_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_211_subset__insert,axiom,
! [X3: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X3 @ A )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X3 @ B2 ) )
= ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_212_subset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B2 ) )
= ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_213_subset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B2 ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_214_insert__mono,axiom,
! [C2: set_Product_unit,D2: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ C2 @ D2 )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ A2 @ C2 ) @ ( insert_Product_unit2 @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_215_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_216_Un__UNIV__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% Un_UNIV_right
thf(fact_217_Un__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_218_Un__UNIV__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ B2 )
= top_to8364301187473967186c_fm_i ) ).
% Un_UNIV_left
thf(fact_219_Un__UNIV__left,axiom,
! [B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B2 )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_220_Un__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Un_empty_right
thf(fact_221_Un__empty__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Un_empty_right
thf(fact_222_Un__empty__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_223_Un__empty__left,axiom,
! [B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_224_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_225_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ~ ! [A5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ A )
=> ! [B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ B2 )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_226_Un__absorb2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_227_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_228_Un__upper2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_upper2
thf(fact_229_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_upper1
thf(fact_230_Un__least,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_231_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_232_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_233_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_234_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_235_semantics_Osimps_I5_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ P @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_236_semantics_Osimps_I1_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
~ ( episte2498298383968489120c_fm_i @ M2 @ W4 @ epistemic_FF_i ) ).
% semantics.simps(1)
thf(fact_237_fm_Odistinct_I15_J,axiom,
! [X22: list_char,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_238_fm_Odistinct_I17_J,axiom,
! [X22: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_239_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( episte2285483198712856234tent_i @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_240_fm_Odistinct_I1_J,axiom,
! [X22: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X22 ) ) ).
% fm.distinct(1)
thf(fact_241_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( A @ P )
=> ( member6642669606046002379c_fm_i @ P @ V2 ) ) ) ) ).
% ax_in_maximal
thf(fact_242_Collect__conv__if2,axiom,
! [P5: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P5 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P5 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_243_Collect__conv__if2,axiom,
! [P5: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P5 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P5 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_244_Collect__conv__if2,axiom,
! [P5: product_unit > $o,A2: product_unit] :
( ( ( P5 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( A2 = X )
& ( P5 @ X ) ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( A2 = X )
& ( P5 @ X ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_conv_if2
thf(fact_245_Collect__conv__if,axiom,
! [P5: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P5 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P5 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_246_Collect__conv__if,axiom,
! [P5: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P5 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P5 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_247_Collect__conv__if,axiom,
! [P5: product_unit > $o,A2: product_unit] :
( ( ( P5 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A2 )
& ( P5 @ X ) ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A2 )
& ( P5 @ X ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_conv_if
thf(fact_248_insert__def,axiom,
( insert_Product_unit2
= ( ^ [A3: product_unit] :
( sup_su793286257634532545t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_249_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_250_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A3 ) ) ) ) ) ).
% insert_def
thf(fact_251_kripke_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W7: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi3: set_Epistemic_fm_i > list_char > $o,More3: product_unit] :
( R
!= ( episte8397520288747125024t_unit @ W7 @ K3 @ ( episte4955834246032687171t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_252_subset__singleton__iff,axiom,
! [X5: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ X5 @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( ( X5 = bot_bo3957492148770167129t_unit )
| ( X5
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_253_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_254_subset__singletonD,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
=> ( ( A = bot_bo3957492148770167129t_unit )
| ( A
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_255_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_256_singleton__Un__iff,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2 = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_257_singleton__Un__iff,axiom,
! [X3: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit )
= ( sup_su793286257634532545t_unit @ A @ B2 ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B2
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
& ( B2 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
& ( B2
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_258_Un__singleton__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2 = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_259_Un__singleton__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit,X3: product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B2 )
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B2
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
& ( B2 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
& ( B2
= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_260_insert__is__Un,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% insert_is_Un
thf(fact_261_insert__is__Un,axiom,
( insert_Product_unit2
= ( ^ [A3: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A3 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% insert_is_Un
thf(fact_262_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P @ V2 )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P @ Q ) @ V2 )
=> ( member6642669606046002379c_fm_i @ Q @ V2 ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_263_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V2 @ W2 )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 ) ) ) ) ).
% maximal_extension
thf(fact_264_kripke_Ocases,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W7: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi3: set_Epistemic_fm_i > list_char > $o] :
( R
!= ( episte8397520288747125024t_unit @ W7 @ K3 @ ( episte4955834246032687171t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_265_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P @ V2 )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ V2 ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_266_consistent__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% consistent_Extend
thf(fact_267_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% boolean_algebra.disj_one_right
thf(fact_268_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_269_boolean__algebra_Odisj__one__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ top_top_Product_unit )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_270_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ X3 )
= top_to8364301187473967186c_fm_i ) ).
% boolean_algebra.disj_one_left
thf(fact_271_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X3 )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_272_boolean__algebra_Odisj__one__left,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X3 )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_273_sup__top__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% sup_top_right
thf(fact_274_sup__top__right,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X3 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% sup_top_right
thf(fact_275_sup__top__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ top_top_Product_unit )
= top_top_Product_unit ) ).
% sup_top_right
thf(fact_276_sup__top__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ X3 )
= top_to8364301187473967186c_fm_i ) ).
% sup_top_left
thf(fact_277_sup__top__left,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X3 )
= top_to1996260823553986621t_unit ) ).
% sup_top_left
thf(fact_278_sup__top__left,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X3 )
= top_top_Product_unit ) ).
% sup_top_left
thf(fact_279_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_280_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_281_sup__bot_Oright__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ A2 @ bot_bot_Product_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_282_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_283_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ A2 @ B ) )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_284_sup__bot_Oneutr__eq__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A2 @ B ) )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_285_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_286_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_287_sup__bot_Oleft__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_288_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_289_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A2 @ B )
= bot_bo3957492148770167129t_unit )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_290_sup__bot_Oeq__neutr__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( sup_sup_Product_unit @ A2 @ B )
= bot_bot_Product_unit )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_291_sup__eq__bot__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 )
= bot_bo4194595901900360558c_fm_i )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_eq_bot_iff
thf(fact_292_sup__eq__bot__iff,axiom,
! [X3: set_Product_unit,Y4: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ X3 @ Y4 )
= bot_bo3957492148770167129t_unit )
= ( ( X3 = bot_bo3957492148770167129t_unit )
& ( Y4 = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_293_sup__eq__bot__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ( sup_sup_Product_unit @ X3 @ Y4 )
= bot_bot_Product_unit )
= ( ( X3 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_294_bot__eq__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_eq_sup_iff
thf(fact_295_bot__eq__sup__iff,axiom,
! [X3: set_Product_unit,Y4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ X3 @ Y4 ) )
= ( ( X3 = bot_bo3957492148770167129t_unit )
& ( Y4 = bot_bo3957492148770167129t_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_296_bot__eq__sup__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( ( X3 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_297_sup__bot__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= X3 ) ).
% sup_bot_right
thf(fact_298_sup__bot__right,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X3 @ bot_bo3957492148770167129t_unit )
= X3 ) ).
% sup_bot_right
thf(fact_299_sup__bot__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ bot_bot_Product_unit )
= X3 ) ).
% sup_bot_right
thf(fact_300_sup_Oidem,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_301_sup_Oidem,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_302_sup__idem,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_303_sup__idem,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_304_sup_Oleft__idem,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_305_sup_Oleft__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ A2 @ B ) )
= ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_306_sup__left__idem,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_307_sup__left__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_left_idem
thf(fact_308_sup_Oright__idem,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ B )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_309_sup_Oright__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ B )
= ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_310_le__sup__iff,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le3221252021190050221t_unit @ X3 @ Z2 )
& ( ord_le3221252021190050221t_unit @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_311_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ Z2 )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_312_sup_Obounded__iff,axiom,
! [B: product_unit,C: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 )
= ( ( ord_le3221252021190050221t_unit @ B @ A2 )
& ( ord_le3221252021190050221t_unit @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_313_sup_Obounded__iff,axiom,
! [B: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 )
= ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_314_sup__bot__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_315_sup__bot__left,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_316_sup__bot__left,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_317_top__set__def,axiom,
( top_to3675301018284359218c_fm_i
= ( collec3087743281813070829c_fm_i @ top_to393988326734912811fm_i_o ) ) ).
% top_set_def
thf(fact_318_top__set__def,axiom,
( top_to8364301187473967186c_fm_i
= ( collec4904205187116291597c_fm_i @ top_to2858054242046946059fm_i_o ) ) ).
% top_set_def
thf(fact_319_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_320_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_321_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_322_bot__set__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% bot_set_def
thf(fact_323_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A4: 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 @ A4 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_324_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A4 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_325_inf__sup__aci_I8_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_326_inf__sup__aci_I8_J,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_327_inf__sup__aci_I7_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_328_inf__sup__aci_I7_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_329_inf__sup__aci_I6_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ Z2 )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_330_inf__sup__aci_I6_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_331_inf__sup__aci_I5_J,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y5 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_332_inf__sup__aci_I5_J,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( sup_sup_Product_unit @ Y5 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_333_sup_Oassoc,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ C )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C ) ) ) ).
% sup.assoc
thf(fact_334_sup_Oassoc,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ C )
= ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ B @ C ) ) ) ).
% sup.assoc
thf(fact_335_sup__assoc,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ Z2 )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_336_sup__assoc,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_337_sup_Ocommute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B6 @ A3 ) ) ) ).
% sup.commute
thf(fact_338_sup_Ocommute,axiom,
( sup_sup_Product_unit
= ( ^ [A3: product_unit,B6: product_unit] : ( sup_sup_Product_unit @ B6 @ A3 ) ) ) ).
% sup.commute
thf(fact_339_sup__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y5 @ X ) ) ) ).
% sup_commute
thf(fact_340_sup__commute,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( sup_sup_Product_unit @ Y5 @ X ) ) ) ).
% sup_commute
thf(fact_341_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Epistemic_fm_i,K4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A
= ( sup_su1936195050962291414c_fm_i @ K4 @ A2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( sup_su1936195050962291414c_fm_i @ K4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_342_boolean__algebra__cancel_Osup1,axiom,
! [A: product_unit,K4: product_unit,A2: product_unit,B: product_unit] :
( ( A
= ( sup_sup_Product_unit @ K4 @ A2 ) )
=> ( ( sup_sup_Product_unit @ A @ B )
= ( sup_sup_Product_unit @ K4 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_343_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Epistemic_fm_i,K4: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B2
= ( sup_su1936195050962291414c_fm_i @ K4 @ B ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= ( sup_su1936195050962291414c_fm_i @ K4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_344_boolean__algebra__cancel_Osup2,axiom,
! [B2: product_unit,K4: product_unit,B: product_unit,A2: product_unit] :
( ( B2
= ( sup_sup_Product_unit @ K4 @ B ) )
=> ( ( sup_sup_Product_unit @ A2 @ B2 )
= ( sup_sup_Product_unit @ K4 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_345_sup_Oleft__commute,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A2 @ C ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_346_sup_Oleft__commute,axiom,
! [B: product_unit,A2: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ B @ ( sup_sup_Product_unit @ A2 @ C ) )
= ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_347_sup__left__commute,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_348_sup__left__commute,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_349_inf__sup__ord_I4_J,axiom,
! [Y4: product_unit,X3: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_350_inf__sup__ord_I4_J,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_351_inf__sup__ord_I3_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_352_inf__sup__ord_I3_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_353_le__supE,axiom,
! [A2: product_unit,B: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ X3 )
=> ~ ( ( ord_le3221252021190050221t_unit @ A2 @ X3 )
=> ~ ( ord_le3221252021190050221t_unit @ B @ X3 ) ) ) ).
% le_supE
thf(fact_354_le__supE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ X3 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ X3 ) ) ) ).
% le_supE
thf(fact_355_le__supI,axiom,
! [A2: product_unit,X3: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ B @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_356_le__supI,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_357_sup__ge1,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_358_sup__ge1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% sup_ge1
thf(fact_359_sup__ge2,axiom,
! [Y4: product_unit,X3: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_360_sup__ge2,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) ) ).
% sup_ge2
thf(fact_361_le__supI1,axiom,
! [X3: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ A2 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_362_le__supI1,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_363_le__supI2,axiom,
! [X3: product_unit,B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ B )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_364_le__supI2,axiom,
! [X3: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ B )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_365_sup_Omono,axiom,
! [C: product_unit,A2: product_unit,D: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ( ord_le3221252021190050221t_unit @ D @ B )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D ) @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_366_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D ) @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_367_sup__mono,axiom,
! [A2: product_unit,C: product_unit,B: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ ( sup_sup_Product_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_368_sup__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ ( sup_su1936195050962291414c_fm_i @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_369_sup__least,axiom,
! [Y4: product_unit,X3: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ Z2 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_370_sup__least,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_371_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X: product_unit,Y5: product_unit] :
( ( sup_sup_Product_unit @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_372_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_373_sup_OorderE,axiom,
! [B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( A2
= ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_374_sup_OorderE,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_375_sup_OorderI,axiom,
! [A2: product_unit,B: product_unit] :
( ( A2
= ( sup_sup_Product_unit @ A2 @ B ) )
=> ( ord_le3221252021190050221t_unit @ B @ A2 ) ) ).
% sup.orderI
thf(fact_376_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ B @ A2 ) ) ).
% sup.orderI
thf(fact_377_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X3: product_unit,Y4: product_unit] :
( ! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: product_unit,Y3: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X4 )
=> ( ( ord_le3221252021190050221t_unit @ Z3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y3 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_378_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y3 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_379_sup_Oabsorb1,axiom,
! [B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( ( sup_sup_Product_unit @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_380_sup_Oabsorb1,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_381_sup_Oabsorb2,axiom,
! [A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ( ( sup_sup_Product_unit @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_382_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_383_sup__absorb1,axiom,
! [Y4: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_384_sup__absorb1,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 )
= X3 ) ) ).
% sup_absorb1
thf(fact_385_sup__absorb2,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( sup_sup_Product_unit @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_386_sup__absorb2,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_387_sup_OboundedE,axiom,
! [B: product_unit,C: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 )
=> ~ ( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_388_sup_OboundedE,axiom,
! [B: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_389_sup_OboundedI,axiom,
! [B: product_unit,A2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_390_sup_OboundedI,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_391_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B6: product_unit,A3: product_unit] :
( A3
= ( sup_sup_Product_unit @ A3 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_392_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B6: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( A3
= ( sup_su1936195050962291414c_fm_i @ A3 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_393_sup_Ocobounded1,axiom,
! [A2: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ A2 @ ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_394_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_395_sup_Ocobounded2,axiom,
! [B: product_unit,A2: product_unit] : ( ord_le3221252021190050221t_unit @ B @ ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_396_sup_Ocobounded2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_397_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B6: product_unit,A3: product_unit] :
( ( sup_sup_Product_unit @ A3 @ B6 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_398_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B6: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B6 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_399_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A3: product_unit,B6: product_unit] :
( ( sup_sup_Product_unit @ A3 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_400_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_401_sup_OcoboundedI1,axiom,
! [C: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_402_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_403_sup_OcoboundedI2,axiom,
! [C: product_unit,B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_404_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_405_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_406_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X3 @ bot_bo3957492148770167129t_unit )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_407_boolean__algebra_Odisj__zero__right,axiom,
! [X3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ bot_bot_Product_unit )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_408_completeness,axiom,
! [P5: episte5489734138418521880t_unit > $o,P: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M3: episte5489734138418521880t_unit] :
( ( P5 @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ bot_bo4194595901900360558c_fm_i )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ( ( P5
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) )
=> ( epistemic_AK_i @ A @ P ) ) ) ).
% completeness
thf(fact_409_extendS__def,axiom,
! [A: epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_410_maximal_H__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] : ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal'_Extend
thf(fact_411_maximal__maximal_H,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal_maximal'
thf(fact_412_the__elem__eq,axiom,
! [X3: epistemic_fm_i] :
( ( the_el2173195912186371479c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= X3 ) ).
% the_elem_eq
thf(fact_413_the__elem__eq,axiom,
! [X3: product_unit] :
( ( the_el608902216710161154t_unit @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= X3 ) ).
% the_elem_eq
thf(fact_414_iso__tuple__UNIV__I,axiom,
! [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ top_to3675301018284359218c_fm_i ) ).
% iso_tuple_UNIV_I
thf(fact_415_iso__tuple__UNIV__I,axiom,
! [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ top_to8364301187473967186c_fm_i ) ).
% iso_tuple_UNIV_I
thf(fact_416_iso__tuple__UNIV__I,axiom,
! [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_417_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_418_order__refl,axiom,
! [X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ X3 ) ).
% order_refl
thf(fact_419_Ax,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i] :
( ( A @ P )
=> ( epistemic_AK_i @ A @ P ) ) ).
% Ax
thf(fact_420_K__trans,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i,R: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P @ Q ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ Q @ R ) @ ( epistemic_Imp_i @ P @ R ) ) ) ) ).
% K_trans
thf(fact_421_R1,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ Q ) )
=> ( epistemic_AK_i @ A @ Q ) ) ) ).
% R1
thf(fact_422_K__imp__trans_H,axiom,
! [A: epistemic_fm_i > $o,Q: epistemic_fm_i,R: epistemic_fm_i,P: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P @ Q ) @ ( epistemic_Imp_i @ P @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_423_K__imp__trans,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i,R: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ R ) ) ) ) ).
% K_imp_trans
thf(fact_424_R2,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,I3: i] :
( ( epistemic_AK_i @ A @ P )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I3 @ P ) ) ) ).
% R2
thf(fact_425_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A4 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_426_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A4 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_427_K__map,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i,I3: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_map
thf(fact_428_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ Q ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_A2'
thf(fact_429_soundness,axiom,
! [A: epistemic_fm_i > $o,P5: episte4846006725945195288t_unit > $o,P: epistemic_fm_i,M2: episte4846006725945195288t_unit,W4: epistemic_fm_i] :
( ! [M3: episte4846006725945195288t_unit,W8: epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member6642669606046002379c_fm_i @ W8 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( episte5182706529199770816c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ( epistemic_AK_i @ A @ P )
=> ( ( P5 @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ P ) ) ) ) ) ).
% soundness
thf(fact_430_soundness,axiom,
! [A: epistemic_fm_i > $o,P5: episte5489734138418521880t_unit > $o,P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit,W8: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member1104366573291651755c_fm_i @ W8 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ( epistemic_AK_i @ A @ P )
=> ( ( P5 @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ) ).
% soundness
thf(fact_431_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( epistemic_AK_i @ A @ P )
=> ( member6642669606046002379c_fm_i @ P @ V2 ) ) ) ) ).
% deriv_in_maximal
thf(fact_432_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ P ) ) ) ).
% K_L_dual
thf(fact_433_K__LK,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_434_order__antisym__conv,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_435_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_436_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( X3 = Y4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 ) ) ).
% order_eq_refl
thf(fact_438_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B ) @ C )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_439_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_440_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B6 )
& ( ord_le3843937902494030498c_fm_i @ B6 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_441_antisym,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_442_dual__order_Otrans,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B )
=> ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_443_dual__order_Oantisym,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_444_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B6 @ A3 )
& ( ord_le3843937902494030498c_fm_i @ A3 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_445_order__trans,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y4 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_446_order_Otrans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_447_order__antisym,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% order_antisym
thf(fact_448_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_449_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2 = B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_450_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
& ( ord_le3843937902494030498c_fm_i @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_451_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M3: episte5489734138418521880t_unit,X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ bot_bo4194595901900360558c_fm_i )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) )
=> ( epistemic_AK_i @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_452_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit2 @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_453_old_Ounit_Oexhaust,axiom,
! [Y4: product_unit] : ( Y4 = product_Unity ) ).
% old.unit.exhaust
thf(fact_454_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_455_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu: product_unit,Uv: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_456_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_457_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P @ V2 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_consequent
thf(fact_458_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
=> ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_459_bot_Oextremum__uniqueI,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ bot_bot_Product_unit )
=> ( A2 = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_460_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_461_bot_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_unique
thf(fact_462_bot_Oextremum__unique,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ bot_bot_Product_unit )
= ( A2 = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_463_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_464_bot_Oextremum,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ).
% bot.extremum
thf(fact_465_bot_Oextremum,axiom,
! [A2: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A2 ) ).
% bot.extremum
thf(fact_466_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_467_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_468_top_Oextremum__uniqueI,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A2 )
=> ( A2 = top_top_Product_unit ) ) ).
% top.extremum_uniqueI
thf(fact_469_top_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A2 )
=> ( A2 = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_uniqueI
thf(fact_470_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_471_top_Oextremum__unique,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A2 )
= ( A2 = top_top_Product_unit ) ) ).
% top.extremum_unique
thf(fact_472_top_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A2 )
= ( A2 = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_unique
thf(fact_473_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_474_top__greatest,axiom,
! [A2: product_unit] : ( ord_le3221252021190050221t_unit @ A2 @ top_top_Product_unit ) ).
% top_greatest
thf(fact_475_top__greatest,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ top_to8364301187473967186c_fm_i ) ).
% top_greatest
thf(fact_476__092_060open_062_092_060not_062_AAx__2_059_AG_A_092_060turnstile_062_Ap_092_060close_062,axiom,
~ ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ g )
& ( epistemic_AK_i @ stalnaker_Ax_2_i @ ( epistemic_imply_i @ Qs @ p ) ) ) ).
% \<open>\<not> Ax_2; G \<turnstile> p\<close>
thf(fact_477_sup__Un__eq,axiom,
! [R2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R2 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( sup_su2582925890723967158c_fm_i @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_478_sup__Un__eq,axiom,
! [R2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R2 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_479_maximal_H__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ X @ S ) ) ) ) ) ).
% maximal'_def
thf(fact_480_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_481_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,V2: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ W2 )
=> ( ( finite3304564979551393739c_fm_i @ W2 )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_482_top__empty__eq,axiom,
( top_to393988326734912811fm_i_o
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ top_to3675301018284359218c_fm_i ) ) ) ).
% top_empty_eq
thf(fact_483_top__empty__eq,axiom,
( top_to2858054242046946059fm_i_o
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ top_to8364301187473967186c_fm_i ) ) ) ).
% top_empty_eq
thf(fact_484_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_485_Field__empty,axiom,
( ( field_Epistemic_fm_i @ bot_bo3668964140382854387c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% Field_empty
thf(fact_486_Field__empty,axiom,
( ( field_Product_unit @ bot_bo1654022762305692595t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Field_empty
thf(fact_487_Field__Un,axiom,
! [R: set_Pr4658907567593863815c_fm_i,S4: set_Pr4658907567593863815c_fm_i] :
( ( field_Epistemic_fm_i @ ( sup_su2687297166538467419c_fm_i @ R @ S4 ) )
= ( sup_su1936195050962291414c_fm_i @ ( field_Epistemic_fm_i @ R ) @ ( field_Epistemic_fm_i @ S4 ) ) ) ).
% Field_Un
thf(fact_488_mono__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,S4: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ S4 )
=> ( ord_le3843937902494030498c_fm_i @ ( field_Epistemic_fm_i @ R ) @ ( field_Epistemic_fm_i @ S4 ) ) ) ).
% mono_Field
thf(fact_489_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_490_infinite__UNIV__fm,axiom,
~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ).
% infinite_UNIV_fm
thf(fact_491_soundness__imply,axiom,
! [A: epistemic_fm_i > $o,P5: episte4846006725945195288t_unit > $o,Ps: list_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte4846006725945195288t_unit,W8: epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member6642669606046002379c_fm_i @ W8 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( episte5182706529199770816c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P ) )
=> ! [M: episte4846006725945195288t_unit] :
( ( P5 @ M )
=> ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Ps ) )
=> ( episte5182706529199770816c_fm_i @ M @ X2 @ Xa ) )
=> ( episte5182706529199770816c_fm_i @ M @ X2 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_492_soundness__imply,axiom,
! [A: epistemic_fm_i > $o,P5: episte5489734138418521880t_unit > $o,Ps: list_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit,W8: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member1104366573291651755c_fm_i @ W8 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( P5 @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Ps ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_493_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A4: epistemic_fm_i > $o,S5: set_Epistemic_fm_i] :
~ ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ S5 )
& ( epistemic_AK_i @ A4 @ ( epistemic_imply_i @ Qs2 @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_494_strong__soundness,axiom,
! [A: epistemic_fm_i > $o,P5: episte4846006725945195288t_unit > $o,G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte4846006725945195288t_unit,W8: epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member6642669606046002379c_fm_i @ W8 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( episte5182706529199770816c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte4846006725945195288t_unit] :
( ( P5 @ M )
=> ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte5182706529199770816c_fm_i @ M @ X2 @ Xa ) )
=> ( episte5182706529199770816c_fm_i @ M @ X2 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_495_strong__soundness,axiom,
! [A: epistemic_fm_i > $o,P5: episte5489734138418521880t_unit > $o,G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit,W8: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( ( P5 @ M3 )
=> ( ( member1104366573291651755c_fm_i @ W8 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ W8 @ P3 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( P5 @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_496_K__right__mp,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ ( epistemic_Imp_i @ P @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_497_strong__soundness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i,P5: episte5489734138418521880t_unit > $o] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [Uu2: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( P5 @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_498_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S )
=> ? [S6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S6 @ S )
& ( finite3304564979551393739c_fm_i @ S6 )
& ~ ( episte2285483198712856234tent_i @ A @ S6 ) ) ) ).
% inconsistent_finite
thf(fact_499_main_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [Uu2: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_500_strong__completeness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit,X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i
@ ^ [Uu2: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_501_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P ) ) ) ).
% inconsistent_imply
thf(fact_502_pred__subset__eq,axiom,
! [R2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R2 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ord_le5389487502678872194c_fm_i @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_503_pred__subset__eq,axiom,
! [R2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R2 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ord_le3843937902494030498c_fm_i @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_504_strong__completeness,axiom,
! [P5: episte5489734138418521880t_unit > $o,G: set_Epistemic_fm_i,P: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M3: episte5489734138418521880t_unit] :
( ( P5 @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ( ( P5
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_505_Extend__subset_H,axiom,
! [S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] :
( ( ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) )
!= bot_bo4194595901900360558c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% Extend_subset'
thf(fact_506_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_507_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_508_bot__empty__eq,axiom,
( bot_bo4642748612307482820unit_o
= ( ^ [X: product_unit] : ( member_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ).
% bot_empty_eq
thf(fact_509__C_K_C,axiom,
! [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ g )
=> ~ ( epistemic_AK_i @ stalnaker_Ax_2_i @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ p @ epistemic_FF_i ) @ Qs ) @ epistemic_FF_i ) ) ) ).
% "*"
thf(fact_510_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_511_finite__Collect__not,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
=> ( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( P5 @ X ) ) )
= ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i ) ) ) ).
% finite_Collect_not
thf(fact_512_finite__Collect__not,axiom,
! [P5: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
=> ( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( P5 @ X ) ) )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Collect_not
thf(fact_513_finite__Collect__not,axiom,
! [P5: product_unit > $o] :
( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P5 ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
~ ( P5 @ X ) ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Collect_not
thf(fact_514_finite__Un,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) )
= ( ( finite3304564979551393739c_fm_i @ F2 )
& ( finite3304564979551393739c_fm_i @ G ) ) ) ).
% finite_Un
thf(fact_515_Field__card__of,axiom,
! [A: set_Epistemic_fm_i] :
( ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ A ) )
= A ) ).
% Field_card_of
thf(fact_516_Field__card__of,axiom,
! [A: set_Product_unit] :
( ( field_Product_unit @ ( bNF_Ca7083678892733797993t_unit @ A ) )
= A ) ).
% Field_card_of
thf(fact_517_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_i] : ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% List.finite_set
thf(fact_518_list_Oinject,axiom,
! [X21: epistemic_fm_i,X222: list_Epistemic_fm_i,Y21: epistemic_fm_i,Y22: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X21 @ X222 )
= ( cons_Epistemic_fm_i @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_519_finite__Collect__conjI,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q2 ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P5 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_520_finite__Collect__conjI,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q2 ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P5 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_521_finite__Collect__disjI,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P5 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_522_finite__Collect__disjI,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P5 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_523_finite__Plus__UNIV__iff,axiom,
( ( finite7028649318107984188c_fm_i @ top_to795506832365754251c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_524_finite__Plus__UNIV__iff,axiom,
( ( finite6693373646013997607t_unit @ top_to3751750602860535030t_unit )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_525_finite__Plus__UNIV__iff,axiom,
( ( finite5258787827736110737c_fm_i @ top_to6207264864741117024c_fm_i )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_526_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_527_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_528_finite__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( insert_Product_unit2 @ A2 @ A ) )
= ( finite4290736615968046902t_unit @ A ) ) ).
% finite_insert
thf(fact_529_list_Osimps_I15_J,axiom,
! [X21: product_unit,X222: list_Product_unit] :
( ( set_Product_unit2 @ ( cons_Product_unit @ X21 @ X222 ) )
= ( insert_Product_unit2 @ X21 @ ( set_Product_unit2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_530_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X222: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_531_set__ConsD,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) )
=> ( ( Y4 = X3 )
| ( member1104366573291651755c_fm_i @ Y4 @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_532_set__ConsD,axiom,
! [Y4: epistemic_fm_i,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) )
=> ( ( Y4 = X3 )
| ( member6642669606046002379c_fm_i @ Y4 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_533_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_534_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_535_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_i,X222: list_s8081015415394010888c_fm_i] : ( member1104366573291651755c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_536_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X222: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_537_list_Oset__intros_I2_J,axiom,
! [Y4: set_Epistemic_fm_i,X222: list_s8081015415394010888c_fm_i,X21: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ ( set_se200842218512397079c_fm_i @ X222 ) )
=> ( member1104366573291651755c_fm_i @ Y4 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_538_list_Oset__intros_I2_J,axiom,
! [Y4: epistemic_fm_i,X222: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ ( set_Epistemic_fm_i2 @ X222 ) )
=> ( member6642669606046002379c_fm_i @ Y4 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_539_not__Cons__self2,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( cons_Epistemic_fm_i @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_540_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_541_imply_Osimps_I2_J,axiom,
! [P: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ Ps ) @ Q )
= ( epistemic_Imp_i @ P @ ( epistemic_imply_i @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_542_K__swap,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i,G: list_Epistemic_fm_i,R: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ ( cons_Epistemic_fm_i @ Q @ G ) ) @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ ( cons_Epistemic_fm_i @ P @ G ) ) @ R ) ) ) ).
% K_swap
thf(fact_543_K__imply__Cons,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_544_K__imply__head,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Ps: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ Ps ) @ P ) ) ).
% K_imply_head
thf(fact_545_K__ImpI,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,G: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ G ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Imp_i @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_546_K__mp,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P @ Q ) @ G ) ) @ Q ) ) ).
% K_mp
thf(fact_547_K__Boole,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) @ G ) @ epistemic_FF_i ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P ) ) ) ).
% K_Boole
thf(fact_548_not__finite__existsD,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P5 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_549_not__finite__existsD,axiom,
! [P5: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
=> ? [X_1: epistemic_fm_i] : ( P5 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_550_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( ord_le3843937902494030498c_fm_i @ X4 @ A2 )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_551_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( ord_le3843937902494030498c_fm_i @ A2 @ X4 )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_552_ex__new__if__finite,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ A )
=> ? [A6: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_553_ex__new__if__finite,axiom,
! [A: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ A )
=> ? [A6: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_554_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A6: product_unit] :
~ ( member_Product_unit @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_555_finite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_UNIV
thf(fact_556_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_557_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5137985727275662011t_unit @ top_to7296822901894720322t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_558_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite3703399908997775141c_fm_i @ top_to528965126920526508c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_559_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_560_finite__prod,axiom,
( ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_561_finite__prod,axiom,
( ( finite5137985727275662011t_unit @ top_to7296822901894720322t_unit )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_562_finite__prod,axiom,
( ( finite3703399908997775141c_fm_i @ top_to528965126920526508c_fm_i )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_563_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_564_Finite__Set_Ofinite__set,axiom,
( ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% Finite_Set.finite_set
thf(fact_565_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_566_infinite__imp__nonempty,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ( S != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_567_infinite__imp__nonempty,axiom,
! [S: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ( S != bot_bo3957492148770167129t_unit ) ) ).
% infinite_imp_nonempty
thf(fact_568_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_569_finite_OemptyI,axiom,
finite4290736615968046902t_unit @ bot_bo3957492148770167129t_unit ).
% finite.emptyI
thf(fact_570_rev__finite__subset,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_571_infinite__super,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_572_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( finite3304564979551393739c_fm_i @ B2 )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_573_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_574_finite_OinsertI,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( finite4290736615968046902t_unit @ ( insert_Product_unit2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_575_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_576_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B2 )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_577_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B2 )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_578_infinite__Un,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) )
= ( ~ ( finite3304564979551393739c_fm_i @ S )
| ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_Un
thf(fact_579_Un__infinite,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_580_finite__UnI,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( finite3304564979551393739c_fm_i @ G )
=> ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_581_finite__has__minimal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X4: product_unit] :
( ( member_Product_unit @ X4 @ A )
& ! [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ A )
=> ( ( ord_le3221252021190050221t_unit @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_582_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_583_finite__has__maximal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X4: product_unit] :
( ( member_Product_unit @ X4 @ A )
& ! [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ A )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_584_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_585_infinite__finite__induct,axiom,
! [P5: set_se3485332733965609186c_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ! [A7: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X4: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X4 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_586_infinite__finite__induct,axiom,
! [P5: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X4: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X4 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_587_infinite__finite__induct,axiom,
! [P5: set_Product_unit > $o,A: set_Product_unit] :
( ! [A7: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_Product_unit2 @ X4 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_588_finite__ne__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( F2 != bot_bo145720340923748686c_fm_i )
=> ( ! [X4: set_Epistemic_fm_i] : ( P5 @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
=> ( ! [X4: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( F3 != bot_bo145720340923748686c_fm_i )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X4 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_589_finite__ne__induct,axiom,
! [F2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( F2 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X4: epistemic_fm_i] : ( P5 @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X4: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X4 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_590_finite__ne__induct,axiom,
! [F2: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( F2 != bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit] : ( P5 @ ( insert_Product_unit2 @ X4 @ bot_bo3957492148770167129t_unit ) )
=> ( ! [X4: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( F3 != bot_bo3957492148770167129t_unit )
=> ( ~ ( member_Product_unit @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_Product_unit2 @ X4 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_591_finite__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X4: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X4 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_592_finite__induct,axiom,
! [F2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X4: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X4 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_593_finite__induct,axiom,
! [F2: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X4 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_Product_unit2 @ X4 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_594_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( ( A3 = bot_bo4194595901900360558c_fm_i )
| ? [A4: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( A3
= ( insert7817948997695205106c_fm_i @ B6 @ A4 ) )
& ( finite3304564979551393739c_fm_i @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_595_finite_Osimps,axiom,
( finite4290736615968046902t_unit
= ( ^ [A3: set_Product_unit] :
( ( A3 = bot_bo3957492148770167129t_unit )
| ? [A4: set_Product_unit,B6: product_unit] :
( ( A3
= ( insert_Product_unit2 @ B6 @ A4 ) )
& ( finite4290736615968046902t_unit @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_596_finite_Ocases,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( A2 != bot_bo4194595901900360558c_fm_i )
=> ~ ! [A7: set_Epistemic_fm_i] :
( ? [A6: epistemic_fm_i] :
( A2
= ( insert7817948997695205106c_fm_i @ A6 @ A7 ) )
=> ~ ( finite3304564979551393739c_fm_i @ A7 ) ) ) ) ).
% finite.cases
thf(fact_597_finite_Ocases,axiom,
! [A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( A2 != bot_bo3957492148770167129t_unit )
=> ~ ! [A7: set_Product_unit] :
( ? [A6: product_unit] :
( A2
= ( insert_Product_unit2 @ A6 @ A7 ) )
=> ~ ( finite4290736615968046902t_unit @ A7 ) ) ) ) ).
% finite.cases
thf(fact_598_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ~ ! [V3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V3 ) @ V2 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ V3 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_599_finite__subset__induct_H,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ F3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_600_finite__subset__induct_H,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A6 @ A )
=> ( ( ord_le3507040750410214029t_unit @ F3 @ A )
=> ( ~ ( member_Product_unit @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_Product_unit2 @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_601_finite__subset__induct_H,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_602_finite__subset__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_603_finite__subset__induct,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A6 @ A )
=> ( ~ ( member_Product_unit @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_Product_unit2 @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_604_finite__subset__induct,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_605_finite__option__UNIV,axiom,
( ( finite4430841290055254033c_fm_i @ top_to7034302889105699736c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% finite_option_UNIV
thf(fact_606_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_607_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_608_mcs__conjunction__mult,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,S: list_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S ) @ V2 )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ S ) @ V2 ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_609_Extend__bound,axiom,
! [N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% Extend_bound
thf(fact_610_Collect__empty__eq__bot,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P5 )
= bot_bo145720340923748686c_fm_i )
= ( P5 = bot_bo6089404950617257231fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_611_Collect__empty__eq__bot,axiom,
! [P5: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P5 )
= bot_bo4194595901900360558c_fm_i )
= ( P5 = bot_bo2580527446789968623fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_612_Collect__empty__eq__bot,axiom,
! [P5: product_unit > $o] :
( ( ( collect_Product_unit @ P5 )
= bot_bo3957492148770167129t_unit )
= ( P5 = bot_bo4642748612307482820unit_o ) ) ).
% Collect_empty_eq_bot
thf(fact_613_consistent__extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,N: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% consistent_extend
thf(fact_614_K__conjunct__imply,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P ) ) ) ).
% K_conjunct_imply
thf(fact_615_K__imply__conjunct,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P ) ) ) ).
% K_imply_conjunct
thf(fact_616_extend__subset,axiom,
! [N: epistemic_fm_i,S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_subset
thf(fact_617_cofinite__bot,axiom,
( ( cofini2965452609477090530c_fm_i = bot_bo3270057444204665492c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% cofinite_bot
thf(fact_618_cofinite__bot,axiom,
( ( cofini76534143805230541t_unit = bot_bo7697929036666283903t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% cofinite_bot
thf(fact_619_is__singleton__the__elem,axiom,
( is_sin4923117429174228502c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
( A4
= ( insert7817948997695205106c_fm_i @ ( the_el2173195912186371479c_fm_i @ A4 ) @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% is_singleton_the_elem
thf(fact_620_is__singleton__the__elem,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A4: set_Product_unit] :
( A4
= ( insert_Product_unit2 @ ( the_el608902216710161154t_unit @ A4 ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_the_elem
thf(fact_621_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( episte6275404236303465413t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_622_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P @ V2 )
=> ( ! [G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ ( epistemic_Imp_i @ P @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_consequent'
thf(fact_623_is__singletonI,axiom,
! [X3: epistemic_fm_i] : ( is_sin4923117429174228502c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ).
% is_singletonI
thf(fact_624_is__singletonI,axiom,
! [X3: product_unit] : ( is_sin2160648248035936513t_unit @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ).
% is_singletonI
thf(fact_625_eval_Osimps_I5_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_i > $o,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G3 @ H2 @ ( epistemic_Imp_i @ P @ Q ) )
= ( ( epistemic_eval_i @ G3 @ H2 @ P )
=> ( epistemic_eval_i @ G3 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_626_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H2 @ ( epistemic_K_i @ I3 @ P ) )
= ( H2 @ ( epistemic_K_i @ I3 @ P ) ) ) ).
% eval.simps(6)
thf(fact_627_eval_Osimps_I1_J,axiom,
! [Uu3: list_char > $o,Uv2: epistemic_fm_i > $o] :
~ ( epistemic_eval_i @ Uu3 @ Uv2 @ epistemic_FF_i ) ).
% eval.simps(1)
thf(fact_628_A1,axiom,
! [P: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ P )
=> ( epistemic_AK_i @ A @ P ) ) ).
% A1
thf(fact_629_tautology,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ! [G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ P )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ).
% tautology
thf(fact_630_eval_Osimps_I2_J,axiom,
! [G3: list_char > $o,Uw: epistemic_fm_i > $o,X3: list_char] :
( ( epistemic_eval_i @ G3 @ Uw @ ( epistemic_Pro_i @ X3 ) )
= ( G3 @ X3 ) ) ).
% eval.simps(2)
thf(fact_631_soundness__Ax5,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( epistemic_Ax5_i @ P )
=> ( ( episte6275404236303465413t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_632_is__singletonI_H,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( member1104366573291651755c_fm_i @ Y3 @ A )
=> ( X4 = Y3 ) ) )
=> ( is_sin2358732406619996150c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_633_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X4: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( X4 = Y3 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_634_is__singletonI_H,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,Y3: product_unit] :
( ( member_Product_unit @ X4 @ A )
=> ( ( member_Product_unit @ Y3 @ A )
=> ( X4 = Y3 ) ) )
=> ( is_sin2160648248035936513t_unit @ A ) ) ) ).
% is_singletonI'
thf(fact_635_duality__taut,axiom,
! [I3: i,P: epistemic_fm_i,Q: epistemic_fm_i,G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_636_Ax5_Ointros,axiom,
! [I3: i,P: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_637_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A3: epistemic_fm_i] :
? [I: i,P2: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_638_Ax5_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A2 )
=> ~ ! [I2: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_639_conjunct__imply,axiom,
! [G: list_Epistemic_fm_i,P: epistemic_fm_i,G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P ) @ ( epistemic_imply_i @ G @ P ) ) ) ).
% conjunct_imply
thf(fact_640_imply__conjunct,axiom,
! [G: list_Epistemic_fm_i,P: epistemic_fm_i,G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ ( epistemic_imply_i @ G @ P ) @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P ) ) ) ).
% imply_conjunct
thf(fact_641_is__singleton__def,axiom,
( is_sin4923117429174228502c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
? [X: epistemic_fm_i] :
( A4
= ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% is_singleton_def
thf(fact_642_is__singleton__def,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A4: set_Product_unit] :
? [X: product_unit] :
( A4
= ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_def
thf(fact_643_is__singletonE,axiom,
! [A: set_Epistemic_fm_i] :
( ( is_sin4923117429174228502c_fm_i @ A )
=> ~ ! [X4: epistemic_fm_i] :
( A
!= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% is_singletonE
thf(fact_644_is__singletonE,axiom,
! [A: set_Product_unit] :
( ( is_sin2160648248035936513t_unit @ A )
=> ~ ! [X4: product_unit] :
( A
!= ( insert_Product_unit2 @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% is_singletonE
thf(fact_645_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M4 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_646_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_647_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs4: list_Epistemic_fm_i,R: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs4 ) )
=> ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R ) @ ( epistemic_imply_i @ Qs4 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_648_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_649_K5__L,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ P ) ) ) ) ).
% K5_L
thf(fact_650_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_i > list_char > $o,W4: set_Epistemic_fm_i,W3: set_se3485332733965609186c_fm_i,R: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,P: epistemic_fm_i] :
( ( epistemic_eval_i @ ( Pi4 @ W4 ) @ ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W3 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W4 ) @ P )
= ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W3 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W4 @ P ) ) ).
% eval_semantics
thf(fact_651_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U2: set_Epistemic_fm_i,V2: set_Epistemic_fm_i,W3: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U2 )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ W3 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W3 )
=> ( ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ U2 ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W3
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ U2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W3
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_652_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,W3: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ W3 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W3 )
=> ( ( member1104366573291651755c_fm_i @ W3
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ W3 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_653_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_654_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,W3: set_Epistemic_fm_i,I3: i,U2: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member1104366573291651755c_fm_i @ W3
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ W3 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_655_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( episte80897895827131294t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_656_S5_H__S5,axiom,
! [P: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ P )
=> ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ P ) ) ).
% S5'_S5
thf(fact_657_S5__S5_H,axiom,
! [P: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ P )
=> ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ P ) ) ).
% S5_S5'
thf(fact_658_soundness__AxB,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( epistemic_AxB_i @ P )
=> ( ( episte80897895827131294t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_AxB
thf(fact_659_S5__S5_H__assms,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_660_AxT_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxT_i @ A2 )
=> ~ ! [I2: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_661_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A3: epistemic_fm_i] :
? [I: i,P2: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P2 ) @ P2 ) ) ) ) ).
% AxT.simps
thf(fact_662_AxT_Ointros,axiom,
! [I3: i,P: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ P ) ) ).
% AxT.intros
thf(fact_663_Ax4_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A2 )
=> ~ ! [I2: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_664_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A3: epistemic_fm_i] :
? [I: i,P2: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P2 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P2 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_665_Ax4_Ointros,axiom,
! [I3: i,P: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ).
% Ax4.intros
thf(fact_666_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: 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 @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_667_AxB_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxB_i @ A2 )
=> ~ ! [P3: epistemic_fm_i,I2: i] :
( A2
!= ( epistemic_Imp_i @ P3 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_668_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A3: epistemic_fm_i] :
? [P2: epistemic_fm_i,I: i] :
( A3
= ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_669_AxB_Ointros,axiom,
! [P: epistemic_fm_i,I3: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_670_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_671_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M4 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_672_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_673_T__L,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_674_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: 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 @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_675_K4__L,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_676_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P2 ) @ V2 ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_677_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( episte251305198166243731t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_678_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( episte2966952217288386430t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_679_strong__soundness_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_680_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_681_main_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M4 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_682_soundness__AxT4,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P )
| ( epistemic_Ax4_i @ P ) )
=> ( ( ( episte251305198166243731t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_683_soundness__AxT5,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
=> ( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_684_soundness__Ax4,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( epistemic_Ax4_i @ P )
=> ( ( episte2966952217288386430t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_685_soundness__AxTB4,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
=> ( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_686_soundness__AxT,axiom,
! [P: epistemic_fm_i,M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
=> ( ( episte251305198166243731t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P ) ) ) ) ).
% soundness_AxT
thf(fact_687_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_688_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_689_pos__introspection,axiom,
! [M2: episte4846006725945195288t_unit,W4: epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte6394096064484582270t_unit @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_690_pos__introspection,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte2966952217288386430t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_691_truth,axiom,
! [M2: episte4846006725945195288t_unit,W4: epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte540185425077706131t_unit @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_692_truth,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte251305198166243731t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_693_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte80897895827131294t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_694_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_695_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_696_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_697_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte80897895827131294t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_698_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
@ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_699_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [P2: epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
@ ( epistemic_imply_i @ Qs @ P ) ) )
=> ! [M: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M @ X2 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_700_neg__introspection,axiom,
! [M2: episte4846006725945195288t_unit,W4: epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte6568605431049406366t_unit @ M2 )
=> ( ( episte6394096064484582270t_unit @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W4 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_701_neg__introspection,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte80897895827131294t_unit @ M2 )
=> ( ( episte2966952217288386430t_unit @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W4 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P ) @ epistemic_FF_i ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_702_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( ! [M4: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M4 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs2 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_703_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_704_strong__completeness_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ! [M3: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X4 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs3 ) @ G )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_705_insert__subsetI,axiom,
! [X3: product_unit,A: set_Product_unit,X5: set_Product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( ord_le3507040750410214029t_unit @ X5 @ A )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_706_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_707_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_708_subset__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X4: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i ) ) ).
% subset_emptyI
thf(fact_709_subset__emptyI,axiom,
! [A: set_Product_unit] :
( ! [X4: product_unit] :
~ ( member_Product_unit @ X4 @ A )
=> ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% subset_emptyI
thf(fact_710_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X4: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_711_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit2 @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% cone_def
thf(fact_712_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte4537596280781771625c_fm_i @ More2 @ ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_713_prop__restrict,axiom,
! [X3: set_Epistemic_fm_i,Z4: set_se3485332733965609186c_fm_i,X5: set_se3485332733965609186c_fm_i,P5: 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 )
& ( P5 @ X ) ) ) )
=> ( P5 @ X3 ) ) ) ).
% prop_restrict
thf(fact_714_prop__restrict,axiom,
! [X3: epistemic_fm_i,Z4: set_Epistemic_fm_i,X5: set_Epistemic_fm_i,P5: 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 )
& ( P5 @ X ) ) ) )
=> ( P5 @ X3 ) ) ) ).
% prop_restrict
thf(fact_715_Collect__restrict,axiom,
! [X5: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X5 )
& ( P5 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_716_Collect__restrict,axiom,
! [X5: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X5 )
& ( P5 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_717_Epistemic__Logic_Oextend__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,N: epistemic_fm_i] :
( ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N )
= ( wellor8883914554786644734c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ S @ ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S ) @ ( maxima7895811110999683971c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) @ N ) ) ).
% Epistemic_Logic.extend_def
thf(fact_718_extend__under,axiom,
! [M5: epistemic_fm_i,N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ M5 @ ( order_1815523504696313095c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_under
thf(fact_719_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_720_set__rec,axiom,
( set_Product_unit2
= ( rec_li8686922504669531926t_unit @ bot_bo3957492148770167129t_unit
@ ^ [X: product_unit,Uu2: list_Product_unit] : ( insert_Product_unit2 @ X ) ) ) ).
% set_rec
thf(fact_721_kripke_Oselect__convs_I2_J,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte484680259488342554t_unit @ ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_722_under__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ A2 ) @ ( order_1815523504696313095c_fm_i @ R3 @ A2 ) ) ) ).
% under_incr2
thf(fact_723_under__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% under_Field
thf(fact_724_under__empty,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_4888947244107180775c_fm_i @ R @ A2 )
= bot_bo145720340923748686c_fm_i ) ) ).
% under_empty
thf(fact_725_under__empty,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1815523504696313095c_fm_i @ R @ A2 )
= bot_bo4194595901900360558c_fm_i ) ) ).
% under_empty
thf(fact_726_under__empty,axiom,
! [A2: product_unit,R: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_6949322460238004850t_unit @ R @ A2 )
= bot_bo3957492148770167129t_unit ) ) ).
% under_empty
thf(fact_727_subset__CollectI,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q2: set_Epistemic_fm_i > $o,P5: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ B2 )
=> ( ( Q2 @ X4 )
=> ( P5 @ X4 ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P5 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_728_subset__CollectI,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q2: epistemic_fm_i > $o,P5: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ! [X4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ B2 )
=> ( ( Q2 @ X4 )
=> ( P5 @ X4 ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P5 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_729_subset__Collect__iff,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B2
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P5 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B2 )
=> ( P5 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_730_subset__Collect__iff,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B2
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P5 @ X ) ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
=> ( P5 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_731_adm__woL__extendL,axiom,
wellor4631791200432093460c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( maxima7895811110999683971c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% adm_woL_extendL
thf(fact_732_kripke_Oselect__convs_I1_J,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte6627496282998226676t_unit @ ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_733_MCS__Lim__Ord_Oconsistent__Extend,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( Consistent @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.consistent_Extend
thf(fact_734_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_735_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S6 @ S )
& ( finite3304564979551393739c_fm_i @ S6 )
& ~ ( Consistent @ S6 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_736_MCS__Lim__Ord_Omaximal_H__Extend,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( maxima2621630817641868953c_fm_i @ R @ Consistent @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ).
% MCS_Lim_Ord.maximal'_Extend
thf(fact_737_MCS__Lim__Ord_Oextend__subset,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ N @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ord_le5389487502678872194c_fm_i @ S @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_738_MCS__Lim__Ord_Oextend__subset,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_739_MCS__Lim__Ord_Oextend__under,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,M5: set_Epistemic_fm_i,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ M5 @ ( order_4888947244107180775c_fm_i @ R @ N ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_740_MCS__Lim__Ord_Oextend__under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,M5: epistemic_fm_i,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ M5 @ ( order_1815523504696313095c_fm_i @ R @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_741_MCS__Lim__Ord_OExtend__subset_H,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( ( field_Product_unit @ R )
!= bot_bo3957492148770167129t_unit )
=> ( ord_le3507040750410214029t_unit @ S @ ( maxima3365502296767225214t_unit @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_subset'
thf(fact_742_MCS__Lim__Ord_OExtend__subset_H,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( ( field_Epistemic_fm_i @ R )
!= bot_bo4194595901900360558c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_subset'
thf(fact_743_MCS__Lim__Ord_OExtend__bound,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ N @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) @ ( maxima2705192052082030579c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_744_MCS__Lim__Ord_OExtend__bound,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_745_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_746_MCS__Lim__Ord_OextendS__def,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_747_MCS__Lim__Ord_OextendS__def,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,N: product_unit,Prev: set_Product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) )
=> ( ( maxima8314523877101631911t_unit @ Consistent @ S @ N @ Prev )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) )
=> ( ( maxima8314523877101631911t_unit @ Consistent @ S @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_748_MCS__Lim__Ord_OExtend_Ocong,axiom,
maxima2580775659384275475c_fm_i = maxima2580775659384275475c_fm_i ).
% MCS_Lim_Ord.Extend.cong
thf(fact_749_MCS__Lim__Ord__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima8116622286610175645c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_Lim_Ord_axioms
thf(fact_750_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( maxima2879964782442171513c_fm_i @ R @ Consistent @ S )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ X @ bot_bo145720340923748686c_fm_i ) @ S ) )
=> ( member1104366573291651755c_fm_i @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_751_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( maxima2621630817641868953c_fm_i @ R @ Consistent @ S )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( field_Epistemic_fm_i @ R ) )
=> ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_752_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( maxima4602283525647497476t_unit @ R @ Consistent @ S )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ ( field_Product_unit @ R ) )
=> ( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) @ S ) )
=> ( member_Product_unit @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_753_semantics_Osimps_I2_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,X3: list_char] :
( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Pro_i @ X3 ) )
= ( episte6627496282998226676t_unit @ M2 @ W4 @ X3 ) ) ).
% semantics.simps(2)
thf(fact_754_kripke_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte4955834246032687171t_unit @ ( episte6627496282998226676t_unit @ R ) @ ( episte484680259488342554t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_755_MCS_Omaximal__maximal_H,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ( maxima1338112684437001276t_unit @ Consistent @ S )
= ( maxima4602283525647497476t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.maximal_maximal'
thf(fact_756_MCS_Omaximal__maximal_H,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.maximal_maximal'
thf(fact_757_MCS_Omaximal__Extend,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( maxima1338112684437001276t_unit @ Consistent @ ( maxima3365502296767225214t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.maximal_Extend
thf(fact_758_MCS_Omaximal__Extend,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( maxima3264069618988350929c_fm_i @ Consistent @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.maximal_Extend
thf(fact_759_MCS_OExtend__subset,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ord_le3507040750410214029t_unit @ S @ ( maxima3365502296767225214t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.Extend_subset
thf(fact_760_MCS_OExtend__subset,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.Extend_subset
thf(fact_761_frame_Oselect__convs_I2_J,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte852950631662840415t_unit @ ( episte8397520288747125024t_unit @ W5 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_762_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_763_weakly__directed__def,axiom,
( stalna4517306826702924506c_fm_i
= ( ^ [M4: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ( ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M4 @ I @ X ) )
& ( member1104366573291651755c_fm_i @ Y5 @ ( episte852950631662840415t_unit @ M4 @ I @ X ) ) )
=> ? [Aa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Aa @ ( episte2675268102452679275t_unit @ M4 ) )
& ( member1104366573291651755c_fm_i @ Aa @ ( episte852950631662840415t_unit @ M4 @ I @ Z5 ) )
& ( member1104366573291651755c_fm_i @ Aa @ ( episte852950631662840415t_unit @ M4 @ I @ Y5 ) ) ) ) ) ) ) ) ) ).
% weakly_directed_def
thf(fact_764_transitive__def,axiom,
( episte2966952217288386430t_unit
= ( ^ [M4: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ( ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M4 @ I @ Y5 ) )
& ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M4 @ I @ Z5 ) ) )
=> ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M4 @ I @ Y5 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_765_reflexive__def,axiom,
( episte251305198166243731t_unit
= ( ^ [M4: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M4 @ I @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_766_symmetric__def,axiom,
( episte80897895827131294t_unit
= ( ^ [M4: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M4 @ I @ Y5 ) )
= ( member1104366573291651755c_fm_i @ Y5 @ ( episte852950631662840415t_unit @ M4 @ I @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_767_Euclidean__def,axiom,
( episte6275404236303465413t_unit
= ( ^ [M4: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ( member1104366573291651755c_fm_i @ Y5 @ ( episte852950631662840415t_unit @ M4 @ I @ X ) )
=> ( ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M4 @ I @ X ) )
=> ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M4 @ I @ Y5 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_768_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% MCS.infinite_UNIV
thf(fact_769_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_Product_unit > $o] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% MCS.infinite_UNIV
thf(fact_770_MCS_Ointro,axiom,
! [Consistent: set_Product_unit > $o] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ! [S7: set_Product_unit,S6: set_Product_unit] :
( ( Consistent @ S7 )
=> ( ( ord_le3507040750410214029t_unit @ S6 @ S7 )
=> ( Consistent @ S6 ) ) )
=> ( ! [S7: set_Product_unit] :
( ~ ( Consistent @ S7 )
=> ? [S8: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S8 @ S7 )
& ( finite4290736615968046902t_unit @ S8 )
& ~ ( Consistent @ S8 ) ) )
=> ( maxima7871471802976955641t_unit @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_771_MCS_Ointro,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ! [S7: set_Epistemic_fm_i,S6: set_Epistemic_fm_i] :
( ( Consistent @ S7 )
=> ( ( ord_le3843937902494030498c_fm_i @ S6 @ S7 )
=> ( Consistent @ S6 ) ) )
=> ( ! [S7: set_Epistemic_fm_i] :
( ~ ( Consistent @ S7 )
=> ? [S8: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S8 @ S7 )
& ( finite3304564979551393739c_fm_i @ S8 )
& ~ ( Consistent @ S8 ) ) )
=> ( maxima1924290493700099598c_fm_i @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_772_MCS__def,axiom,
( maxima7871471802976955641t_unit
= ( ^ [Consistent2: set_Product_unit > $o] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ! [S5: set_Product_unit,S9: set_Product_unit] :
( ( Consistent2 @ S5 )
=> ( ( ord_le3507040750410214029t_unit @ S9 @ S5 )
=> ( Consistent2 @ S9 ) ) )
& ! [S5: set_Product_unit] :
( ~ ( Consistent2 @ S5 )
=> ? [S9: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S9 @ S5 )
& ( finite4290736615968046902t_unit @ S9 )
& ~ ( Consistent2 @ S9 ) ) ) ) ) ) ).
% MCS_def
thf(fact_773_MCS__def,axiom,
( maxima1924290493700099598c_fm_i
= ( ^ [Consistent2: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ! [S5: set_Epistemic_fm_i,S9: set_Epistemic_fm_i] :
( ( Consistent2 @ S5 )
=> ( ( ord_le3843937902494030498c_fm_i @ S9 @ S5 )
=> ( Consistent2 @ S9 ) ) )
& ! [S5: set_Epistemic_fm_i] :
( ~ ( Consistent2 @ S5 )
=> ? [S9: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S9 @ S5 )
& ( finite3304564979551393739c_fm_i @ S9 )
& ~ ( Consistent2 @ S9 ) ) ) ) ) ) ).
% MCS_def
thf(fact_774_kripke_Oequality,axiom,
! [R: episte5489734138418521880t_unit,R3: episte5489734138418521880t_unit] :
( ( ( episte2675268102452679275t_unit @ R )
= ( episte2675268102452679275t_unit @ R3 ) )
=> ( ( ( episte852950631662840415t_unit @ R )
= ( episte852950631662840415t_unit @ R3 ) )
=> ( ( ( episte6627496282998226676t_unit @ R )
= ( episte6627496282998226676t_unit @ R3 ) )
=> ( ( ( episte484680259488342554t_unit @ R )
= ( episte484680259488342554t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ) ).
% kripke.equality
thf(fact_775_MCS_Omaximal__def,axiom,
! [Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima5540402109999912942c_fm_i @ Consistent )
=> ( ( maxima7741914846562630577c_fm_i @ Consistent @ S )
= ( ! [P2: set_Epistemic_fm_i] :
( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ P2 @ bot_bo145720340923748686c_fm_i ) @ S ) )
=> ( member1104366573291651755c_fm_i @ P2 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_776_MCS_Omaximal__def,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ( maxima1338112684437001276t_unit @ Consistent @ S )
= ( ! [P2: product_unit] :
( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ P2 @ bot_bo3957492148770167129t_unit ) @ S ) )
=> ( member_Product_unit @ P2 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_777_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( ! [P2: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P2 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_778_frame_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte7135767042998387833t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_779_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o,W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte8744582347493487171t_unit @ Pi2 @ ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= ( episte8397520288747125024t_unit @ W5 @ K @ ( episte4955834246032687171t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_780_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_781_Sup__fin_Oinsert,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X3 @ A ) )
= ( sup_sup_Product_unit @ X3 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_782_frame_Oequality,axiom,
! [R: episte5489734138418521880t_unit,R3: episte5489734138418521880t_unit] :
( ( ( episte2675268102452679275t_unit @ R )
= ( episte2675268102452679275t_unit @ R3 ) )
=> ( ( ( episte852950631662840415t_unit @ R )
= ( episte852950631662840415t_unit @ R3 ) )
=> ( ( ( episte7135767042998387833t_unit @ R )
= ( episte7135767042998387833t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ).
% frame.equality
thf(fact_783_Sup__fin_Osingleton,axiom,
! [X3: product_unit] :
( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= X3 ) ).
% Sup_fin.singleton
thf(fact_784_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_785_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_786_Sup__fin_Oin__idem,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X3 @ A )
=> ( ( sup_sup_Product_unit @ X3 @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_787_frame_Oselect__convs_I3_J,axiom,
! [W5: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte7135767042998387833t_unit @ ( episte8397520288747125024t_unit @ W5 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_788_Sup__fin_OboundedE,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X3 )
=> ! [A8: product_unit] :
( ( member_Product_unit @ A8 @ A )
=> ( ord_le3221252021190050221t_unit @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_789_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_790_Sup__fin_OboundedI,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit] :
( ( member_Product_unit @ A6 @ A )
=> ( ord_le3221252021190050221t_unit @ A6 @ X3 ) )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_791_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A6 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_792_Sup__fin_Obounded__iff,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X3 )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A )
=> ( ord_le3221252021190050221t_unit @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_793_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_794_Sup__fin_Osubset__imp,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_795_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_796_Sup__fin_Osubset,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ B2 ) @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_797_Sup__fin_Osubset,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B2 ) @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_798_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_799_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X3 @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X3 @ A ) )
= ( sup_sup_Product_unit @ X3 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_800_Sup__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) @ ( insert7698009978809854162c_fm_i @ X4 @ ( insert7698009978809854162c_fm_i @ Y3 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_801_Sup__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,Y3: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X4 @ Y3 ) @ ( insert_Product_unit2 @ X4 @ ( insert_Product_unit2 @ Y3 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_802_Sup__fin_Ounion,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B2 )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_803_Sup__fin_Ounion,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B2 ) )
= ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_804_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P @ Q ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_805_extend__underS,axiom,
! [M5: epistemic_fm_i,N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ M5 @ ( order_1288088345848551912c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_underS
thf(fact_806_subset__singleton__iff__Uniq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [A3: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) )
= ( uniq_s5891897140743797767c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_807_subset__singleton__iff__Uniq,axiom,
! [A: set_Product_unit] :
( ( ? [A3: product_unit] : ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ A3 @ bot_bo3957492148770167129t_unit ) ) )
= ( uniq_Product_unit
@ ^ [X: product_unit] : ( member_Product_unit @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_808_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A3: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_809_semantics_Osimps_I3_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Dis_i @ P @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P )
| ( episte2498298383968489120c_fm_i @ M2 @ W4 @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_810_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_811_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_812_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_813_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I3: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i,J: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ ( order_1288088345848551912c_fm_i @ R2 @ J ) )
=> ( member6642669606046002379c_fm_i @ I3 @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_814_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I3: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i,J: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ I3 @ ( order_8439613377573682120c_fm_i @ R2 @ J ) )
=> ( member1104366573291651755c_fm_i @ I3 @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_815_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_i,X32: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Dis_i @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_816_underS__empty,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_8439613377573682120c_fm_i @ R @ A2 )
= bot_bo145720340923748686c_fm_i ) ) ).
% underS_empty
thf(fact_817_underS__empty,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1288088345848551912c_fm_i @ R @ A2 )
= bot_bo4194595901900360558c_fm_i ) ) ).
% underS_empty
thf(fact_818_underS__empty,axiom,
! [A2: product_unit,R: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_5809128134054296019t_unit @ R @ A2 )
= bot_bo3957492148770167129t_unit ) ) ).
% underS_empty
thf(fact_819_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_820_underS__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( order_1288088345848551912c_fm_i @ R3 @ A2 ) ) ) ).
% underS_incr2
thf(fact_821_underS__subset__under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( order_1815523504696313095c_fm_i @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_822_MCS__Lim__Ord_Oextend__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,M5: set_Epistemic_fm_i,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ M5 @ ( order_8439613377573682120c_fm_i @ R @ N ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_823_MCS__Lim__Ord_Oextend__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,M5: epistemic_fm_i,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ M5 @ ( order_1288088345848551912c_fm_i @ R @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_824_K__DisL,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P6: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ Ps ) @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P6 @ Ps ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Dis_i @ P @ P6 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_825_K__DisE,axiom,
! [A: epistemic_fm_i > $o,P: epistemic_fm_i,G: list_Epistemic_fm_i,R: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P @ G ) @ R ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ G ) @ R ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Dis_i @ P @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ R ) ) ) ) ) ).
% K_DisE
thf(fact_826_Refl__under__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_4888947244107180775c_fm_i @ R @ A2 )
= ( sup_su2582925890723967158c_fm_i @ ( order_8439613377573682120c_fm_i @ R @ A2 ) @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ).
% Refl_under_underS
thf(fact_827_Refl__under__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1815523504696313095c_fm_i @ R @ A2 )
= ( sup_su1936195050962291414c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Refl_under_underS
thf(fact_828_Refl__under__underS,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_6949322460238004850t_unit @ R @ A2 )
= ( sup_su793286257634532545t_unit @ ( order_5809128134054296019t_unit @ R @ A2 ) @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Refl_under_underS
thf(fact_829_fm_Odistinct_I13_J,axiom,
! [X22: list_char,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_830_semantics_Osimps_I4_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Con_i @ P @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ P )
& ( episte2498298383968489120c_fm_i @ M2 @ W4 @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_831_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_832_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_833_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_834_refl__on__empty,axiom,
refl_o2748477423337579169c_fm_i @ bot_bo4194595901900360558c_fm_i @ bot_bo3668964140382854387c_fm_i ).
% refl_on_empty
thf(fact_835_refl__on__empty,axiom,
refl_on_Product_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% refl_on_empty
thf(fact_836_Refl__under__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_1815523504696313095c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_under_in
thf(fact_837_Refl__under__in,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_4888947244107180775c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_under_in
thf(fact_838_K__imply__multi,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ B ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ C ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Con_i @ B @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_839_K__multi__imply,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Imp_i @ B @ C ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ A2 @ B ) @ C ) ) ) ).
% K_multi_imply
thf(fact_840_refl__on__Un,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,S4: set_Pr4658907567593863815c_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( refl_o2748477423337579169c_fm_i @ B2 @ S4 )
=> ( refl_o2748477423337579169c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su2687297166538467419c_fm_i @ R @ S4 ) ) ) ) ).
% refl_on_Un
thf(fact_841_mcs__conjunction,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( ( member6642669606046002379c_fm_i @ P @ V2 )
& ( member6642669606046002379c_fm_i @ Q @ V2 ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P @ Q ) @ V2 ) ) ) ) ).
% mcs_conjunction
thf(fact_842_conjunct_Osimps_I2_J,axiom,
! [P: epistemic_fm_i,Ps: list_Epistemic_fm_i] :
( ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ P @ Ps ) )
= ( epistemic_Con_i @ P @ ( stalnaker_conjunct_i @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_843_A2,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ Q ) ) ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ).
% A2
thf(fact_844_K__conj__imply__factor,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i,R: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ Q ) ) @ R ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_845_K__conjunction__out,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ Q ) ) @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_846_K__conjunction__in,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P @ Q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_847_distribution,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ Q ) ) ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ).
% distribution
thf(fact_848_K__thm,axiom,
! [A: epistemic_fm_i > $o,I3: i,P: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P @ Q ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_849_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A4: epistemic_fm_i > $o,A3: epistemic_fm_i] :
( ? [P2: epistemic_fm_i] :
( ( A3 = P2 )
& ! [G5: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H4 @ P2 ) )
| ? [I: i,P2: epistemic_fm_i,Q3: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P2 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P2 @ Q3 ) ) ) @ ( epistemic_K_i @ I @ Q3 ) ) )
| ? [P2: epistemic_fm_i] :
( ( A3 = P2 )
& ( A4 @ P2 ) )
| ? [P2: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( A3 = Q3 )
& ( epistemic_AK_i @ A4 @ P2 )
& ( epistemic_AK_i @ A4 @ ( epistemic_Imp_i @ P2 @ Q3 ) ) )
| ? [P2: epistemic_fm_i,I: i] :
( ( A3
= ( epistemic_K_i @ I @ P2 ) )
& ( epistemic_AK_i @ A4 @ P2 ) ) ) ) ) ).
% AK.simps
thf(fact_850_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A2 )
=> ( ~ ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ A2 )
=> ( ! [I2: i,P3: epistemic_fm_i,Q4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ Q4 ) ) ) @ ( epistemic_K_i @ I2 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P3 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ A2 ) ) )
=> ~ ! [P3: epistemic_fm_i] :
( ? [I2: i] :
( A2
= ( epistemic_K_i @ I2 @ P3 ) )
=> ~ ( epistemic_AK_i @ A @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_851_fm_Oexhaust,axiom,
! [Y4: epistemic_fm_i] :
( ( Y4 != epistemic_FF_i )
=> ( ! [X23: list_char] :
( Y4
!= ( epistemic_Pro_i @ X23 ) )
=> ( ! [X312: epistemic_fm_i,X322: epistemic_fm_i] :
( Y4
!= ( epistemic_Dis_i @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y4
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y4
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y4
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_852_fm_Orel__cases,axiom,
! [R2: i > i > $o,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_i )
=> ( B != epistemic_FF_i ) )
=> ( ! [X4: list_char] :
( ( A2
= ( epistemic_Pro_i @ X4 ) )
=> ! [Y3: list_char] :
( ( B
= ( epistemic_Pro_i @ Y3 ) )
=> ( X4 != Y3 ) ) )
=> ( ! [X1: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R2 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_853_fm_Orel__induct,axiom,
! [R2: i > i > $o,X3: epistemic_fm_i,Y4: epistemic_fm_i,Q2: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R2 @ X3 @ Y4 )
=> ( ( Q2 @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q2 @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q2 @ A31 @ B31 )
=> ( ( Q2 @ A32 @ B32 )
=> ( Q2 @ ( epistemic_Dis_i @ A31 @ A32 ) @ ( epistemic_Dis_i @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_i,A42: epistemic_fm_i,B41: epistemic_fm_i,B42: epistemic_fm_i] :
( ( Q2 @ A41 @ B41 )
=> ( ( Q2 @ A42 @ B42 )
=> ( Q2 @ ( epistemic_Con_i @ A41 @ A42 ) @ ( epistemic_Con_i @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_i,A52: epistemic_fm_i,B51: epistemic_fm_i,B52: epistemic_fm_i] :
( ( Q2 @ A51 @ B51 )
=> ( ( Q2 @ A52 @ B52 )
=> ( Q2 @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R2 @ A61 @ B61 )
=> ( ( Q2 @ A62 @ B62 )
=> ( Q2 @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q2 @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_854_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_855_Sup__fin_Oremove,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X3 @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= X3 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= ( sup_sup_Product_unit @ X3 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_856_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_857_Sup__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X3: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X3 @ A ) )
= ( sup_sup_Product_unit @ X3 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_858_DiffI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_859_DiffI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_860_Diff__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ~ ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_861_Diff__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ~ ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_862_Diff__cancel,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ A )
= bot_bo4194595901900360558c_fm_i ) ).
% Diff_cancel
thf(fact_863_Diff__cancel,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ A )
= bot_bo3957492148770167129t_unit ) ).
% Diff_cancel
thf(fact_864_empty__Diff,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ bot_bo4194595901900360558c_fm_i @ A )
= bot_bo4194595901900360558c_fm_i ) ).
% empty_Diff
thf(fact_865_empty__Diff,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A )
= bot_bo3957492148770167129t_unit ) ).
% empty_Diff
thf(fact_866_Diff__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Diff_empty
thf(fact_867_Diff__empty,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Diff_empty
thf(fact_868_insert__Diff1,axiom,
! [X3: product_unit,B2: set_Product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X3 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X3 @ A ) @ B2 )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_869_insert__Diff1,axiom,
! [X3: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B2 )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_870_insert__Diff1,axiom,
! [X3: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B2 )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_871_Diff__insert0,axiom,
! [X3: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X3 @ A )
=> ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ B2 ) )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_872_Diff__insert0,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_873_Diff__insert0,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B2 ) )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_874_Un__Diff__cancel2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ B2 @ A ) @ A )
= ( sup_su1936195050962291414c_fm_i @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_875_Un__Diff__cancel,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_876_Diff__UNIV,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ top_to8364301187473967186c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% Diff_UNIV
thf(fact_877_Diff__UNIV,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_878_Diff__eq__empty__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_879_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_880_insert__Diff__single,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_881_insert__Diff__single,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( insert_Product_unit2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_882_finite__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_883_finite__Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) ) )
= ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_884_DiffE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% DiffE
thf(fact_885_DiffE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% DiffE
thf(fact_886_DiffD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_887_DiffD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_888_DiffD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B2 ) ) ).
% DiffD2
thf(fact_889_DiffD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ~ ( member1104366573291651755c_fm_i @ C @ B2 ) ) ).
% DiffD2
thf(fact_890_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A4 )
& ~ ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_891_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A4 )
& ~ ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_892_fm_Octr__transfer_I1_J,axiom,
! [R2: i > i > $o] : ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ epistemic_FF_i ) ).
% fm.ctr_transfer(1)
thf(fact_893_fm_Orel__intros_I6_J,axiom,
! [R2: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R2 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_894_fm_Orel__inject_I6_J,axiom,
! [R2: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R2 @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R2 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_895_fm_Orel__intros_I5_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,Y51: epistemic_fm_i,X52: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X52 @ Y52 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_896_fm_Orel__inject_I5_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_i_i @ R2 @ X51 @ Y51 )
& ( epistemic_rel_fm_i_i @ R2 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_897_Un__Diff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) @ ( minus_1543973939109951465c_fm_i @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_898_insert__Diff__if,axiom,
! [X3: product_unit,B2: set_Product_unit,A: set_Product_unit] :
( ( ( member_Product_unit @ X3 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X3 @ A ) @ B2 )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) )
& ( ~ ( member_Product_unit @ X3 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X3 @ A ) @ B2 )
= ( insert_Product_unit2 @ X3 @ ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_899_insert__Diff__if,axiom,
! [X3: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X3 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B2 )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B2 )
= ( insert7817948997695205106c_fm_i @ X3 @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_900_insert__Diff__if,axiom,
! [X3: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X3 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B2 )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B2 )
= ( insert7698009978809854162c_fm_i @ X3 @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_901_double__diff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B2 @ ( minus_1543973939109951465c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_902_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_903_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_904_fm_Orel__intros_I2_J,axiom,
! [X22: list_char,Y2: list_char,R2: i > i > $o] :
( ( X22 = Y2 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Pro_i @ Y2 ) ) ) ).
% fm.rel_intros(2)
thf(fact_905_fm_Orel__inject_I2_J,axiom,
! [R2: i > i > $o,X22: list_char,Y2: list_char] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Pro_i @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fm.rel_inject(2)
thf(fact_906_fm_Orel__distinct_I30_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_907_fm_Orel__distinct_I29_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_908_fm_Orel__distinct_I8_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(8)
thf(fact_909_fm_Orel__distinct_I7_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_910_fm_Orel__distinct_I10_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(10)
thf(fact_911_fm_Orel__distinct_I9_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_912_diff__shunt__var,axiom,
! [X3: set_Product_unit,Y4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ X3 @ Y4 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_913_diff__shunt__var,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X3 @ Y4 )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_914_diff__shunt__var,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X3 @ Y4 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_915_fm_Orel__distinct_I16_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i,X22: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ ( epistemic_Pro_i @ X22 ) ) ).
% fm.rel_distinct(16)
thf(fact_916_fm_Orel__distinct_I15_J,axiom,
! [R2: i > i > $o,X22: list_char,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_917_fm_Orel__distinct_I18_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i,X22: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Pro_i @ X22 ) ) ).
% fm.rel_distinct(18)
thf(fact_918_fm_Orel__distinct_I17_J,axiom,
! [R2: i > i > $o,X22: list_char,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_919_fm_Orel__distinct_I2_J,axiom,
! [R2: i > i > $o,Y2: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ Y2 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(2)
thf(fact_920_fm_Orel__distinct_I1_J,axiom,
! [R2: i > i > $o,Y2: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_Pro_i @ Y2 ) ) ).
% fm.rel_distinct(1)
thf(fact_921_Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% Diff_insert
thf(fact_922_Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% Diff_insert
thf(fact_923_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_924_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_925_insert__Diff,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit2 @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= A ) ) ).
% insert_Diff
thf(fact_926_Diff__insert2,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_927_Diff__insert2,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_928_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_929_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_930_Diff__insert__absorb,axiom,
! [X3: product_unit,A: set_Product_unit] :
( ~ ( member_Product_unit @ X3 @ A )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X3 @ A ) @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_931_subset__Diff__insert,axiom,
! [A: set_Product_unit,B2: set_Product_unit,X3: product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ ( insert_Product_unit2 @ X3 @ C2 ) ) )
= ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ C2 ) )
& ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_932_subset__Diff__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B2 @ C2 ) )
& ~ ( member1104366573291651755c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_933_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X3: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_934_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_935_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_936_finite__empty__induct,axiom,
! [A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( P5 @ A )
=> ( ! [A6: set_Epistemic_fm_i,A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ A6 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) )
=> ( P5 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_937_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P5 @ A )
=> ( ! [A6: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_938_finite__empty__induct,axiom,
! [A: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( P5 @ A )
=> ( ! [A6: product_unit,A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( member_Product_unit @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ( P5 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% finite_empty_induct
thf(fact_939_infinite__coinduct,axiom,
! [X5: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( X5 @ A )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( X5 @ A7 )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A7 )
& ( ( X5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% infinite_coinduct
thf(fact_940_infinite__coinduct,axiom,
! [X5: set_Product_unit > $o,A: set_Product_unit] :
( ( X5 @ A )
=> ( ! [A7: set_Product_unit] :
( ( X5 @ A7 )
=> ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A7 )
& ( ( X5 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) )
| ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% infinite_coinduct
thf(fact_941_infinite__remove,axiom,
! [S: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% infinite_remove
thf(fact_942_infinite__remove,axiom,
! [S: set_Product_unit,A2: product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% infinite_remove
thf(fact_943_subset__insert__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B2 ) )
= ( ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) @ B2 ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_944_subset__insert__iff,axiom,
! [A: set_Product_unit,X3: product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X3 @ B2 ) )
= ( ( ( member_Product_unit @ X3 @ A )
=> ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) @ B2 ) )
& ( ~ ( member_Product_unit @ X3 @ A )
=> ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_945_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B2 ) )
= ( ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_946_Diff__single__insert,axiom,
! [A: set_Product_unit,X3: product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) @ B2 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_947_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_948_remove__induct,axiom,
! [P5: set_se3485332733965609186c_fm_i > $o,B2: set_se3485332733965609186c_fm_i] :
( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ( ~ ( finite7933139204641697195c_fm_i @ B2 )
=> ( P5 @ B2 ) )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B2 )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_949_remove__induct,axiom,
! [P5: set_Product_unit > $o,B2: set_Product_unit] :
( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ( ~ ( finite4290736615968046902t_unit @ B2 )
=> ( P5 @ B2 ) )
=> ( ! [A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( A7 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A7 @ B2 )
=> ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A7 )
=> ( P5 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_950_remove__induct,axiom,
! [P5: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i] :
( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B2 )
=> ( P5 @ B2 ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B2 )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_951_finite__remove__induct,axiom,
! [B2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ B2 )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B2 )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_952_finite__remove__induct,axiom,
! [B2: set_Product_unit,P5: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( P5 @ bot_bo3957492148770167129t_unit )
=> ( ! [A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( A7 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A7 @ B2 )
=> ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A7 )
=> ( P5 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_953_finite__remove__induct,axiom,
! [B2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B2 )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B2 )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_954_underS__under,axiom,
! [I3: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ( member1104366573291651755c_fm_i @ I3 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_8439613377573682120c_fm_i @ R @ I3 )
= ( minus_1073301138667672009c_fm_i @ ( order_4888947244107180775c_fm_i @ R @ I3 ) @ ( insert7698009978809854162c_fm_i @ I3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% underS_under
thf(fact_955_underS__under,axiom,
! [I3: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1288088345848551912c_fm_i @ R @ I3 )
= ( minus_1543973939109951465c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ I3 ) @ ( insert7817948997695205106c_fm_i @ I3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% underS_under
thf(fact_956_underS__under,axiom,
! [I3: product_unit,R: set_Pr5094982260447487303t_unit] :
( ( member_Product_unit @ I3 @ ( field_Product_unit @ R ) )
=> ( ( order_5809128134054296019t_unit @ R @ I3 )
= ( minus_6452836326544984404t_unit @ ( order_6949322460238004850t_unit @ R @ I3 ) @ ( insert_Product_unit2 @ I3 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% underS_under
thf(fact_957_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_958_set__removeAll,axiom,
! [X3: product_unit,Xs: list_Product_unit] :
( ( set_Product_unit2 @ ( remove8762936195178819609t_unit @ X3 @ Xs ) )
= ( minus_6452836326544984404t_unit @ ( set_Product_unit2 @ Xs ) @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% set_removeAll
thf(fact_959_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_960_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_961_minus__set__def,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A4: 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 @ A4 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_962_minus__set__def,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( minus_6519238007984220724fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A4 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_963_removeAll_Osimps_I2_J,axiom,
! [X3: epistemic_fm_i,Y4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( X3 = Y4 )
=> ( ( remove817376155714978094c_fm_i @ X3 @ ( cons_Epistemic_fm_i @ Y4 @ Xs ) )
= ( remove817376155714978094c_fm_i @ X3 @ Xs ) ) )
& ( ( X3 != Y4 )
=> ( ( remove817376155714978094c_fm_i @ X3 @ ( cons_Epistemic_fm_i @ Y4 @ Xs ) )
= ( cons_Epistemic_fm_i @ Y4 @ ( remove817376155714978094c_fm_i @ X3 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_964_remove__def,axiom,
( remove5552923401211475591c_fm_i
= ( ^ [X: epistemic_fm_i,A4: set_Epistemic_fm_i] : ( minus_1543973939109951465c_fm_i @ A4 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% remove_def
thf(fact_965_remove__def,axiom,
( remove_Product_unit
= ( ^ [X: product_unit,A4: set_Product_unit] : ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% remove_def
thf(fact_966_member__remove,axiom,
! [X3: epistemic_fm_i,Y4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( remove5552923401211475591c_fm_i @ Y4 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( X3 != Y4 ) ) ) ).
% member_remove
thf(fact_967_member__remove,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( remove1392734035364174951c_fm_i @ Y4 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( X3 != Y4 ) ) ) ).
% member_remove
thf(fact_968_unit__all__impI,axiom,
! [P5: product_unit > $o,Q2: product_unit > $o] :
( ( ( P5 @ product_Unity )
=> ( Q2 @ product_Unity ) )
=> ! [X2: product_unit] :
( ( P5 @ X2 )
=> ( Q2 @ X2 ) ) ) ).
% unit_all_impI
thf(fact_969_eq__sym__Unity__conv,axiom,
! [X3: $o] :
( ( X3
= ( product_Unity = product_Unity ) )
= X3 ) ).
% eq_sym_Unity_conv
thf(fact_970_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_971_Func__is__emp,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( bNF_We489590214134045204c_fm_i @ A @ B2 )
= bot_bo1633126455607734703c_fm_i )
= ( ( A != bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Func_is_emp
thf(fact_972_Func__is__emp,axiom,
! [A: set_Epistemic_fm_i,B2: set_Product_unit] :
( ( ( bNF_We5948249774593435647t_unit @ A @ B2 )
= bot_bo5160044010258911514t_unit )
= ( ( A != bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Func_is_emp
thf(fact_973_Func__is__emp,axiom,
! [A: set_Product_unit,B2: set_Epistemic_fm_i] :
( ( ( bNF_We667984284430983401c_fm_i @ A @ B2 )
= bot_bo7615558272139493508c_fm_i )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Func_is_emp
thf(fact_974_Func__is__emp,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( bNF_We2086683926456529748t_unit @ A @ B2 )
= bot_bo290337425153025135t_unit )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Func_is_emp
thf(fact_975_Func__emp2,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( bNF_We489590214134045204c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= bot_bo1633126455607734703c_fm_i ) ) ).
% Func_emp2
thf(fact_976_Func__emp2,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( bNF_We5948249774593435647t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo5160044010258911514t_unit ) ) ).
% Func_emp2
thf(fact_977_Func__emp2,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( bNF_We667984284430983401c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= bot_bo7615558272139493508c_fm_i ) ) ).
% Func_emp2
thf(fact_978_Func__emp2,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( bNF_We2086683926456529748t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo290337425153025135t_unit ) ) ).
% Func_emp2
thf(fact_979_List_Oset__insert,axiom,
! [X3: product_unit,Xs: list_Product_unit] :
( ( set_Product_unit2 @ ( insert_Product_unit @ X3 @ Xs ) )
= ( insert_Product_unit2 @ X3 @ ( set_Product_unit2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_980_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_981_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_982_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_983_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_984_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_985_empty__in__Func,axiom,
! [B2: set_Epistemic_fm_i] :
( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member8097212860227194492c_fm_i
@ ^ [X: epistemic_fm_i] : undefi6158949293579472778c_fm_i
@ ( bNF_We489590214134045204c_fm_i @ bot_bo4194595901900360558c_fm_i @ B2 ) ) ) ).
% empty_in_Func
thf(fact_986_empty__in__Func,axiom,
! [B2: set_Epistemic_fm_i] :
( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member3099810079864775633c_fm_i
@ ^ [X: product_unit] : undefi6158949293579472778c_fm_i
@ ( bNF_We667984284430983401c_fm_i @ bot_bo3957492148770167129t_unit @ B2 ) ) ) ).
% empty_in_Func
thf(fact_987_empty__in__Func,axiom,
! [B2: set_Product_unit] :
( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member4534395898142662503t_unit
@ ^ [X: epistemic_fm_i] : undefi2974082611360736373t_unit
@ ( bNF_We5948249774593435647t_unit @ bot_bo4194595901900360558c_fm_i @ B2 ) ) ) ).
% empty_in_Func
thf(fact_988_empty__in__Func,axiom,
! [B2: set_Product_unit] :
( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member3685380118729357628t_unit
@ ^ [X: product_unit] : undefi2974082611360736373t_unit
@ ( bNF_We2086683926456529748t_unit @ bot_bo3957492148770167129t_unit @ B2 ) ) ) ).
% empty_in_Func
thf(fact_989_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_990_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_991_Field__insert,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( field_Epistemic_fm_i @ ( insert347525590040402807c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ B ) @ R ) )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) @ ( field_Epistemic_fm_i @ R ) ) ) ).
% Field_insert
thf(fact_992_Field__insert,axiom,
! [A2: product_unit,B: product_unit,R: set_Pr5094982260447487303t_unit] :
( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R ) )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R ) ) ) ).
% Field_insert
thf(fact_993_semantics_Osimps_I6_J,axiom,
! [M2: episte5489734138418521880t_unit,W4: set_Epistemic_fm_i,I3: i,P: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M2 @ W4 @ ( epistemic_K_i @ I3 @ P ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( inf_in161960956874937808c_fm_i @ ( episte2675268102452679275t_unit @ M2 ) @ ( episte852950631662840415t_unit @ M2 @ I3 @ W4 ) ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_994_Set_Ois__empty__def,axiom,
( is_emp7643256682846039796c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] : ( A4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Set.is_empty_def
thf(fact_995_Set_Ois__empty__def,axiom,
( is_emp5529367068699929823t_unit
= ( ^ [A4: set_Product_unit] : ( A4 = bot_bo3957492148770167129t_unit ) ) ) ).
% Set.is_empty_def
thf(fact_996_inf__right__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_right_idem
thf(fact_997_inf_Oright__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ B )
= ( inf_inf_Product_unit @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_998_inf__left__idem,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_left_idem
thf(fact_999_inf_Oleft__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ A2 @ ( inf_inf_Product_unit @ A2 @ B ) )
= ( inf_inf_Product_unit @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_1000_inf__idem,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_1001_inf_Oidem,axiom,
! [A2: product_unit] :
( ( inf_inf_Product_unit @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_1002_Int__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1003_Int__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1004_IntI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1005_IntI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_1006_le__inf__iff,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
& ( ord_le3221252021190050221t_unit @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1007_le__inf__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
& ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1008_inf_Obounded__iff,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ ( inf_inf_Product_unit @ B @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A2 @ B )
& ( ord_le3221252021190050221t_unit @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1009_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C ) )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
& ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1010_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% boolean_algebra.conj_zero_right
thf(fact_1011_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X3 @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1012_boolean__algebra_Oconj__zero__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1013_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ bot_bo4194595901900360558c_fm_i @ X3 )
= bot_bo4194595901900360558c_fm_i ) ).
% boolean_algebra.conj_zero_left
thf(fact_1014_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X3 )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1015_boolean__algebra_Oconj__zero__left,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X3 )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1016_inf__bot__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% inf_bot_right
thf(fact_1017_inf__bot__right,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X3 @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% inf_bot_right
thf(fact_1018_inf__bot__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_1019_inf__bot__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ bot_bo4194595901900360558c_fm_i @ X3 )
= bot_bo4194595901900360558c_fm_i ) ).
% inf_bot_left
thf(fact_1020_inf__bot__left,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X3 )
= bot_bo3957492148770167129t_unit ) ).
% inf_bot_left
thf(fact_1021_inf__bot__left,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X3 )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_1022_inf__top__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ top_to8364301187473967186c_fm_i @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_1023_inf__top__left,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_1024_inf__top__left,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ top_top_Product_unit @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_1025_inf__top__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ top_to8364301187473967186c_fm_i )
= X3 ) ).
% inf_top_right
thf(fact_1026_inf__top__right,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X3 @ top_to1996260823553986621t_unit )
= X3 ) ).
% inf_top_right
thf(fact_1027_inf__top__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ top_top_Product_unit )
= X3 ) ).
% inf_top_right
thf(fact_1028_inf__eq__top__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 )
= top_to8364301187473967186c_fm_i )
= ( ( X3 = top_to8364301187473967186c_fm_i )
& ( Y4 = top_to8364301187473967186c_fm_i ) ) ) ).
% inf_eq_top_iff
thf(fact_1029_inf__eq__top__iff,axiom,
! [X3: set_Product_unit,Y4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X3 @ Y4 )
= top_to1996260823553986621t_unit )
= ( ( X3 = top_to1996260823553986621t_unit )
& ( Y4 = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_1030_inf__eq__top__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ( inf_inf_Product_unit @ X3 @ Y4 )
= top_top_Product_unit )
= ( ( X3 = top_top_Product_unit )
& ( Y4 = top_top_Product_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_1031_top__eq__inf__iff,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( top_to8364301187473967186c_fm_i
= ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) )
= ( ( X3 = top_to8364301187473967186c_fm_i )
& ( Y4 = top_to8364301187473967186c_fm_i ) ) ) ).
% top_eq_inf_iff
thf(fact_1032_top__eq__inf__iff,axiom,
! [X3: set_Product_unit,Y4: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ X3 @ Y4 ) )
= ( ( X3 = top_to1996260823553986621t_unit )
& ( Y4 = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_1033_top__eq__inf__iff,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( top_top_Product_unit
= ( inf_inf_Product_unit @ X3 @ Y4 ) )
= ( ( X3 = top_top_Product_unit )
& ( Y4 = top_top_Product_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_1034_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= top_to8364301187473967186c_fm_i )
= ( ( A2 = top_to8364301187473967186c_fm_i )
& ( B = top_to8364301187473967186c_fm_i ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1035_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ B )
= top_to1996260823553986621t_unit )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1036_inf__top_Oeq__neutr__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( inf_inf_Product_unit @ A2 @ B )
= top_top_Product_unit )
= ( ( A2 = top_top_Product_unit )
& ( B = top_top_Product_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1037_inf__top_Oleft__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ top_to8364301187473967186c_fm_i @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1038_inf__top_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1039_inf__top_Oleft__neutral,axiom,
! [A2: product_unit] :
( ( inf_inf_Product_unit @ top_top_Product_unit @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1040_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( top_to8364301187473967186c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
= ( ( A2 = top_to8364301187473967186c_fm_i )
& ( B = top_to8364301187473967186c_fm_i ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1041_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ A2 @ B ) )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1042_inf__top_Oneutr__eq__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( top_top_Product_unit
= ( inf_inf_Product_unit @ A2 @ B ) )
= ( ( A2 = top_top_Product_unit )
& ( B = top_top_Product_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1043_inf__top_Oright__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A2 @ top_to8364301187473967186c_fm_i )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1044_inf__top_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A2 @ top_to1996260823553986621t_unit )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1045_inf__top_Oright__neutral,axiom,
! [A2: product_unit] :
( ( inf_inf_Product_unit @ A2 @ top_top_Product_unit )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1046_inf__sup__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_1047_inf__sup__absorb,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ X3 @ Y4 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_1048_sup__inf__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_1049_sup__inf__absorb,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_1050_Int__UNIV,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= top_to8364301187473967186c_fm_i )
= ( ( A = top_to8364301187473967186c_fm_i )
& ( B2 = top_to8364301187473967186c_fm_i ) ) ) ).
% Int_UNIV
thf(fact_1051_Int__UNIV,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_1052_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
= ( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
& ( ord_le3843937902494030498c_fm_i @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1053_Int__insert__right__if1,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( insert_Product_unit2 @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1054_Int__insert__right__if1,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1055_Int__insert__right__if1,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1056_Int__insert__right__if0,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1057_Int__insert__right__if0,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1058_Int__insert__right__if0,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) )
= ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1059_insert__inter__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1060_insert__inter__insert,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ A ) @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( insert_Product_unit2 @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1061_Int__insert__left__if1,axiom,
! [A2: product_unit,C2: set_Product_unit,B2: set_Product_unit] :
( ( member_Product_unit @ A2 @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( insert_Product_unit2 @ A2 @ ( inf_in4660618365625256667t_unit @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1062_Int__insert__left__if1,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1063_Int__insert__left__if1,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1064_Int__insert__left__if0,axiom,
! [A2: product_unit,C2: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( inf_in4660618365625256667t_unit @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1065_Int__insert__left__if0,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1066_Int__insert__left__if0,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1067_Int__Un__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ T2 @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1068_Int__Un__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ S @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1069_Int__Un__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1070_Int__Un__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1071_Un__Int__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ T2 @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1072_Un__Int__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ S @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1073_Un__Int__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1074_Un__Int__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1075_insert__disjoint_I1_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B2 )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B2 )
& ( ( inf_in161960956874937808c_fm_i @ A @ B2 )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_1076_insert__disjoint_I1_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B2 )
& ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_1077_insert__disjoint_I1_J,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ A ) @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A2 @ B2 )
& ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_1078_insert__disjoint_I2_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B2 ) )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B2 )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1079_insert__disjoint_I2_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B2 ) )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B2 )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1080_insert__disjoint_I2_J,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ A ) @ B2 ) )
= ( ~ ( member_Product_unit @ A2 @ B2 )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1081_disjoint__insert_I1_J,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ A2 @ A ) )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B2 )
& ( ( inf_in161960956874937808c_fm_i @ B2 @ A )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_1082_disjoint__insert_I1_J,axiom,
! [B2: set_Epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B2 )
& ( ( inf_in3450601097109690352c_fm_i @ B2 @ A )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_1083_disjoint__insert_I1_J,axiom,
! [B2: set_Product_unit,A2: product_unit,A: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ B2 @ ( insert_Product_unit2 @ A2 @ A ) )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A2 @ B2 )
& ( ( inf_in4660618365625256667t_unit @ B2 @ A )
= bot_bo3957492148770167129t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_1084_disjoint__insert_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B @ B2 ) ) )
= ( ~ ( member1104366573291651755c_fm_i @ B @ A )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1085_disjoint__insert_I2_J,axiom,
! [A: set_Epistemic_fm_i,B: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B @ A )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1086_disjoint__insert_I2_J,axiom,
! [A: set_Product_unit,B: product_unit,B2: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit2 @ B @ B2 ) ) )
= ( ~ ( member_Product_unit @ B @ A )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1087_Diff__disjoint,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ A ) )
= bot_bo4194595901900360558c_fm_i ) ).
% Diff_disjoint
thf(fact_1088_Diff__disjoint,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ A ) )
= bot_bo3957492148770167129t_unit ) ).
% Diff_disjoint
thf(fact_1089_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_1090_inf__Sup__absorb,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ A2 @ A )
=> ( ( inf_inf_Product_unit @ A2 @ ( lattic5294303975357428420t_unit @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1091_Int__insert__right,axiom,
! [A2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( insert_Product_unit2 @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( inf_in4660618365625256667t_unit @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1092_Int__insert__right,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1093_Int__insert__right,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) )
= ( inf_in161960956874937808c_fm_i @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1094_Int__insert__left,axiom,
! [A2: product_unit,C2: set_Product_unit,B2: set_Product_unit] :
( ( ( member_Product_unit @ A2 @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( insert_Product_unit2 @ A2 @ ( inf_in4660618365625256667t_unit @ B2 @ C2 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( inf_in4660618365625256667t_unit @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1095_Int__insert__left,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1096_Int__insert__left,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C2 ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1097_underS__def,axiom,
( order_8439613377573682120c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A3: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B6: set_Epistemic_fm_i] :
( ( B6 != A3 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B6 @ A3 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_1098_underS__def,axiom,
( order_1288088345848551912c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A3: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B6: epistemic_fm_i] :
( ( B6 != A3 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B6 @ A3 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_1099_under__def,axiom,
( order_4888947244107180775c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A3: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B6: set_Epistemic_fm_i] : ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B6 @ A3 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_1100_under__def,axiom,
( order_1815523504696313095c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A3: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B6: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B6 @ A3 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_1101_FieldI2,axiom,
! [I3: epistemic_fm_i,J: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i] :
( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I3 @ J ) @ R2 )
=> ( member6642669606046002379c_fm_i @ J @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% FieldI2
thf(fact_1102_FieldI2,axiom,
! [I3: set_Epistemic_fm_i,J: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i] :
( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I3 @ J ) @ R2 )
=> ( member1104366573291651755c_fm_i @ J @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% FieldI2
thf(fact_1103_FieldI1,axiom,
! [I3: epistemic_fm_i,J: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i] :
( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I3 @ J ) @ R2 )
=> ( member6642669606046002379c_fm_i @ I3 @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% FieldI1
thf(fact_1104_FieldI1,axiom,
! [I3: set_Epistemic_fm_i,J: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i] :
( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I3 @ J ) @ R2 )
=> ( member1104366573291651755c_fm_i @ I3 @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% FieldI1
thf(fact_1105_Un__Int__crazy,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1106_Int__Un__distrib,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1107_Un__Int__distrib,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1108_Int__Un__distrib2,axiom,
! [B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) @ A )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B2 @ A ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1109_Un__Int__distrib2,axiom,
! [B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) @ A )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ A ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1110_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ top_to8364301187473967186c_fm_i )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_1111_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X3 @ top_to1996260823553986621t_unit )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_1112_boolean__algebra_Oconj__one__right,axiom,
! [X3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ top_top_Product_unit )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_1113_distrib__imp1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y3 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_1114_distrib__imp1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ! [X4: product_unit,Y3: product_unit,Z3: product_unit] :
( ( inf_inf_Product_unit @ X4 @ ( sup_sup_Product_unit @ Y3 @ Z3 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ ( inf_inf_Product_unit @ X4 @ Z3 ) ) )
=> ( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_1115_distrib__imp2,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y3 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_1116_distrib__imp2,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ! [X4: product_unit,Y3: product_unit,Z3: product_unit] :
( ( sup_sup_Product_unit @ X4 @ ( inf_inf_Product_unit @ Y3 @ Z3 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X4 @ Y3 ) @ ( sup_sup_Product_unit @ X4 @ Z3 ) ) )
=> ( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_1117_inf__sup__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1118_inf__sup__distrib1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_1119_inf__sup__distrib2,axiom,
! [Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y4 @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_1120_inf__sup__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ X3 ) @ ( inf_inf_Product_unit @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_1121_sup__inf__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1122_sup__inf__distrib1,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_1123_sup__inf__distrib2,axiom,
! [Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_1124_sup__inf__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ X3 ) @ ( sup_sup_Product_unit @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_1125_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1126_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1127_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1128_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1129_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y4 @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1130_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ X3 ) @ ( inf_inf_Product_unit @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1131_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1132_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y4: product_unit,Z2: product_unit,X3: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y4 @ Z2 ) @ X3 )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y4 @ X3 ) @ ( sup_sup_Product_unit @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1133_inf__left__commute,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ Y4 @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_1134_inf_Oleft__commute,axiom,
! [B: product_unit,A2: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ B @ ( inf_inf_Product_unit @ A2 @ C ) )
= ( inf_inf_Product_unit @ A2 @ ( inf_inf_Product_unit @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1135_boolean__algebra__cancel_Oinf2,axiom,
! [B2: product_unit,K4: product_unit,B: product_unit,A2: product_unit] :
( ( B2
= ( inf_inf_Product_unit @ K4 @ B ) )
=> ( ( inf_inf_Product_unit @ A2 @ B2 )
= ( inf_inf_Product_unit @ K4 @ ( inf_inf_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1136_boolean__algebra__cancel_Oinf1,axiom,
! [A: product_unit,K4: product_unit,A2: product_unit,B: product_unit] :
( ( A
= ( inf_inf_Product_unit @ K4 @ A2 ) )
=> ( ( inf_inf_Product_unit @ A @ B )
= ( inf_inf_Product_unit @ K4 @ ( inf_inf_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1137_inf__commute,axiom,
( inf_inf_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( inf_inf_Product_unit @ Y5 @ X ) ) ) ).
% inf_commute
thf(fact_1138_inf_Ocommute,axiom,
( inf_inf_Product_unit
= ( ^ [A3: product_unit,B6: product_unit] : ( inf_inf_Product_unit @ B6 @ A3 ) ) ) ).
% inf.commute
thf(fact_1139_inf__assoc,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_1140_inf_Oassoc,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ C )
= ( inf_inf_Product_unit @ A2 @ ( inf_inf_Product_unit @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1141_inf__sup__aci_I1_J,axiom,
( inf_inf_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( inf_inf_Product_unit @ Y5 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_1142_inf__sup__aci_I2_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Z2 )
= ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_1143_inf__sup__aci_I3_J,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) )
= ( inf_inf_Product_unit @ Y4 @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_1144_inf__sup__aci_I4_J,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ ( inf_inf_Product_unit @ X3 @ Y4 ) )
= ( inf_inf_Product_unit @ X3 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_1145_Collect__conj__eq,axiom,
! [P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P5 @ X )
& ( Q2 @ X ) ) )
= ( inf_in161960956874937808c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1146_Collect__conj__eq,axiom,
! [P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P5 @ X )
& ( Q2 @ X ) ) )
= ( inf_in3450601097109690352c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1147_Int__Collect,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P5 ) ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P5 @ X3 ) ) ) ).
% Int_Collect
thf(fact_1148_Int__Collect,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P5 ) ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P5 @ X3 ) ) ) ).
% Int_Collect
thf(fact_1149_Int__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A4 )
& ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1150_Int__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A4 )
& ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1151_IntD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ).
% IntD2
thf(fact_1152_IntD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ).
% IntD2
thf(fact_1153_IntD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_1154_IntD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_1155_IntE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ~ ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% IntE
thf(fact_1156_IntE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B2 ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ~ ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% IntE
thf(fact_1157_Int__UNIV__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ top_to8364301187473967186c_fm_i @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1158_Int__UNIV__left,axiom,
! [B2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1159_Int__UNIV__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ top_to8364301187473967186c_fm_i )
= A ) ).
% Int_UNIV_right
thf(fact_1160_Int__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% Int_UNIV_right
thf(fact_1161_refl__onD,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1162_refl__onD,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1163_refl__onD1,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,X3: epistemic_fm_i,Y4: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X3 @ Y4 ) @ R )
=> ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1164_refl__onD1,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ X3 @ Y4 ) @ R )
=> ( member1104366573291651755c_fm_i @ X3 @ A ) ) ) ).
% refl_onD1
thf(fact_1165_refl__onD2,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,X3: epistemic_fm_i,Y4: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X3 @ Y4 ) @ R )
=> ( member6642669606046002379c_fm_i @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1166_refl__onD2,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ X3 @ Y4 ) @ R )
=> ( member1104366573291651755c_fm_i @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1167_Un__Int__assoc__eq,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1168_Diff__triv,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
=> ( ( minus_1543973939109951465c_fm_i @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_1169_Diff__triv,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
=> ( ( minus_6452836326544984404t_unit @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_1170_Int__Diff__disjoint,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
= bot_bo4194595901900360558c_fm_i ) ).
% Int_Diff_disjoint
thf(fact_1171_Int__Diff__disjoint,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ A @ B2 ) @ ( minus_6452836326544984404t_unit @ A @ B2 ) )
= bot_bo3957492148770167129t_unit ) ).
% Int_Diff_disjoint
thf(fact_1172_Int__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1173_Int__lower1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_1174_Int__lower2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1175_Int__absorb1,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1176_Int__absorb2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_1177_Int__greatest,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1178_Int__Collect__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( P5 @ X4 )
=> ( Q2 @ X4 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P5 ) ) @ ( inf_in161960956874937808c_fm_i @ B2 @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1179_Int__Collect__mono,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,P5: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ! [X4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( P5 @ X4 )
=> ( Q2 @ X4 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P5 ) ) @ ( inf_in3450601097109690352c_fm_i @ B2 @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1180_inf_OcoboundedI2,axiom,
! [B: product_unit,C: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1181_inf_OcoboundedI2,axiom,
! [B: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1182_inf_OcoboundedI1,axiom,
! [A2: product_unit,C: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1183_inf_OcoboundedI1,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1184_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B6: product_unit,A3: product_unit] :
( ( inf_inf_Product_unit @ A3 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1185_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B6: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1186_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A3: product_unit,B6: product_unit] :
( ( inf_inf_Product_unit @ A3 @ B6 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_1187_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B6 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_1188_inf_Ocobounded2,axiom,
! [A2: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1189_inf_Ocobounded2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1190_inf_Ocobounded1,axiom,
! [A2: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1191_inf_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1192_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A3: product_unit,B6: product_unit] :
( A3
= ( inf_inf_Product_unit @ A3 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1193_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( A3
= ( inf_in3450601097109690352c_fm_i @ A3 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1194_inf__greatest,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Z2 )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1195_inf__greatest,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1196_inf_OboundedI,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ( ( ord_le3221252021190050221t_unit @ A2 @ C )
=> ( ord_le3221252021190050221t_unit @ A2 @ ( inf_inf_Product_unit @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1197_inf_OboundedI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1198_inf_OboundedE,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ ( inf_inf_Product_unit @ B @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ~ ( ord_le3221252021190050221t_unit @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1199_inf_OboundedE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1200_inf__absorb2,axiom,
! [Y4: product_unit,X3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X3 )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1201_inf__absorb2,axiom,
! [Y4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1202_inf__absorb1,axiom,
! [X3: product_unit,Y4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ Y4 )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1203_inf__absorb1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1204_inf_Oabsorb2,axiom,
! [B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( ( inf_inf_Product_unit @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1205_inf_Oabsorb2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1206_inf_Oabsorb1,axiom,
! [A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ( ( inf_inf_Product_unit @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1207_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1208_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X: product_unit,Y5: product_unit] :
( ( inf_inf_Product_unit @ X @ Y5 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1209_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X @ Y5 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1210_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X3: product_unit,Y4: product_unit] :
( ! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: product_unit,Y3: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y3 )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ Z3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_Product_unit @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1211_inf__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 )
= ( F @ X3 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1212_inf_OorderI,axiom,
! [A2: product_unit,B: product_unit] :
( ( A2
= ( inf_inf_Product_unit @ A2 @ B ) )
=> ( ord_le3221252021190050221t_unit @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1213_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1214_inf_OorderE,axiom,
! [A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ( A2
= ( inf_inf_Product_unit @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1215_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1216_le__infI2,axiom,
! [B: product_unit,X3: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1217_le__infI2,axiom,
! [B: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1218_le__infI1,axiom,
! [A2: product_unit,X3: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1219_le__infI1,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1220_inf__mono,axiom,
! [A2: product_unit,C: product_unit,B: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A2 @ B ) @ ( inf_inf_Product_unit @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1221_inf__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ ( inf_in3450601097109690352c_fm_i @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1222_le__infI,axiom,
! [X3: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ A2 )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ B )
=> ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1223_le__infI,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ B )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1224_le__infE,axiom,
! [X3: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X3 @ ( inf_inf_Product_unit @ A2 @ B ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X3 @ A2 )
=> ~ ( ord_le3221252021190050221t_unit @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1225_le__infE,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1226_inf__le2,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1227_inf__le2,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1228_inf__le1,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ X3 ) ).
% inf_le1
thf(fact_1229_inf__le1,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ X3 ) ).
% inf_le1
thf(fact_1230_inf__sup__ord_I1_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1231_inf__sup__ord_I1_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1232_inf__sup__ord_I2_J,axiom,
! [X3: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1233_inf__sup__ord_I2_J,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1234_distrib__sup__le,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X3 @ ( inf_inf_Product_unit @ Y4 @ Z2 ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y4 ) @ ( sup_sup_Product_unit @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1235_distrib__sup__le,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y4 @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1236_distrib__inf__le,axiom,
! [X3: product_unit,Y4: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y4 ) @ ( inf_inf_Product_unit @ X3 @ Z2 ) ) @ ( inf_inf_Product_unit @ X3 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1237_distrib__inf__le,axiom,
! [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1238_disjoint__iff__not__equal,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ B2 )
=> ( X != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1239_disjoint__iff__not__equal,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A )
=> ! [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ B2 )
=> ( X != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1240_Int__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% Int_empty_right
thf(fact_1241_Int__empty__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_right
thf(fact_1242_Int__empty__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ bot_bo4194595901900360558c_fm_i @ B2 )
= bot_bo4194595901900360558c_fm_i ) ).
% Int_empty_left
thf(fact_1243_Int__empty__left,axiom,
! [B2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ B2 )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_left
thf(fact_1244_disjoint__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B2 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ~ ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1245_disjoint__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ~ ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1246_disjoint__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A )
=> ~ ( member_Product_unit @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1247_Int__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ! [X4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ~ ( member1104366573291651755c_fm_i @ X4 @ B2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B2 )
= bot_bo145720340923748686c_fm_i ) ) ).
% Int_emptyI
thf(fact_1248_Int__emptyI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ! [X4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X4 @ B2 ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_1249_Int__emptyI,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A )
=> ~ ( member_Product_unit @ X4 @ B2 ) )
=> ( ( inf_in4660618365625256667t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit ) ) ).
% Int_emptyI
thf(fact_1250_Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1251_Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B2 @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1252_Int__Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
= A ) ).
% Int_Diff_Un
thf(fact_1253_Un__Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
= A ) ).
% Un_Diff_Int
thf(fact_1254_reflI,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ! [X4: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X4 @ X4 ) @ R )
=> ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R ) ) ).
% reflI
thf(fact_1255_reflI,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ! [X4: product_unit] : ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X4 @ X4 ) @ R )
=> ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R ) ) ).
% reflI
thf(fact_1256_reflD,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1257_reflD,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1258_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A2 @ X3 )
= bot_bo4194595901900360558c_fm_i )
=> ( ( ( sup_su1936195050962291414c_fm_i @ A2 @ X3 )
= top_to8364301187473967186c_fm_i )
=> ( ( ( inf_in3450601097109690352c_fm_i @ A2 @ Y4 )
= bot_bo4194595901900360558c_fm_i )
=> ( ( ( sup_su1936195050962291414c_fm_i @ A2 @ Y4 )
= top_to8364301187473967186c_fm_i )
=> ( X3 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1259_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Product_unit,X3: set_Product_unit,Y4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ X3 )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ X3 )
= top_to1996260823553986621t_unit )
=> ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y4 )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ Y4 )
= top_to1996260823553986621t_unit )
=> ( X3 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1260_boolean__algebra_Ocomplement__unique,axiom,
! [A2: product_unit,X3: product_unit,Y4: product_unit] :
( ( ( inf_inf_Product_unit @ A2 @ X3 )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ A2 @ X3 )
= top_top_Product_unit )
=> ( ( ( inf_inf_Product_unit @ A2 @ Y4 )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ A2 @ Y4 )
= top_top_Product_unit )
=> ( X3 = Y4 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1261_refl__on__singleton,axiom,
! [X3: epistemic_fm_i] : ( refl_o2748477423337579169c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) @ ( insert347525590040402807c_fm_i @ ( produc4253568889897832087c_fm_i @ X3 @ X3 ) @ bot_bo3668964140382854387c_fm_i ) ) ).
% refl_on_singleton
thf(fact_1262_refl__on__singleton,axiom,
! [X3: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X3 @ X3 ) @ bot_bo1654022762305692595t_unit ) ) ).
% refl_on_singleton
thf(fact_1263_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu: product_unit,Uv: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1264_inf__Int__eq,axiom,
! [R2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R2 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1265_inf__Int__eq,axiom,
! [R2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( inf_in853329969422372621fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R2 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( inf_in161960956874937808c_fm_i @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1266_inf__set__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: 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 @ A4 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1267_inf__set__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A4 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_1268_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_1269_null__rec_I1_J,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
~ ( null_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) ).
% null_rec(1)
thf(fact_1270_insert__Chain,axiom,
! [R: set_Pr5094982260447487303t_unit,C2: set_Product_unit,I3: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( member5877623283571906838t_unit @ C2 @ ( chains_Product_unit @ R ) )
=> ( ( member_Product_unit @ I3 @ ( field_Product_unit @ R ) )
=> ( ! [J2: product_unit] :
( ( member_Product_unit @ J2 @ C2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ J2 @ I3 ) @ R )
| ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I3 @ J2 ) @ R ) ) )
=> ( member5877623283571906838t_unit @ ( insert_Product_unit2 @ I3 @ C2 ) @ ( chains_Product_unit @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1271_insert__Chain,axiom,
! [R: set_Pr3972831103112126087c_fm_i,C2: set_se3485332733965609186c_fm_i,I3: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1461078328125707403c_fm_i @ C2 @ ( chains7915783162359121842c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ I3 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ! [J2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ J2 @ C2 )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ J2 @ I3 ) @ R )
| ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I3 @ J2 ) @ R ) ) )
=> ( member1461078328125707403c_fm_i @ ( insert7698009978809854162c_fm_i @ I3 @ C2 ) @ ( chains7915783162359121842c_fm_i @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1272_insert__Chain,axiom,
! [R: set_Pr4658907567593863815c_fm_i,C2: set_Epistemic_fm_i,I3: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ C2 @ ( chains2242491043985051090c_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ I3 @ ( field_Epistemic_fm_i @ R ) )
=> ( ! [J2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ J2 @ C2 )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ J2 @ I3 ) @ R )
| ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I3 @ J2 ) @ R ) ) )
=> ( member1104366573291651755c_fm_i @ ( insert7817948997695205106c_fm_i @ I3 @ C2 ) @ ( chains2242491043985051090c_fm_i @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1273_linear__order__on__singleton,axiom,
! [X3: epistemic_fm_i] : ( order_5410817280966914263c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) @ ( insert347525590040402807c_fm_i @ ( produc4253568889897832087c_fm_i @ X3 @ X3 ) @ bot_bo3668964140382854387c_fm_i ) ) ).
% linear_order_on_singleton
thf(fact_1274_linear__order__on__singleton,axiom,
! [X3: product_unit] : ( order_888282469287196226t_unit @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X3 @ X3 ) @ bot_bo1654022762305692595t_unit ) ) ).
% linear_order_on_singleton
thf(fact_1275_lnear__order__on__empty,axiom,
order_5410817280966914263c_fm_i @ bot_bo4194595901900360558c_fm_i @ bot_bo3668964140382854387c_fm_i ).
% lnear_order_on_empty
thf(fact_1276_lnear__order__on__empty,axiom,
order_888282469287196226t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% lnear_order_on_empty
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: list_Epistemic_fm_i,Y4: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X3: list_Epistemic_fm_i,Y4: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [P5: $o] :
( ( P5 = $true )
| ( P5 = $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,Y4: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $false @ X3 @ Y4 )
= Y4 ) ).
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,Y4: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ stalnaker_Ax_2_i @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ stalnaker_Ax_2_i ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P2 ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ stalnaker_Ax_2_i ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ p @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ g ) )
@ ( epistemic_Imp_i @ p @ epistemic_FF_i ) ) ).
%------------------------------------------------------------------------------