TPTP Problem File: SLH0596^1.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_00244_008008__6493318_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1605 ( 517 unt; 318 typ; 0 def)
% Number of atoms : 4278 ( 986 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 13987 ( 345 ~; 63 |; 355 &;11307 @)
% ( 0 <=>;1917 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 1740 (1740 >; 0 *; 0 +; 0 <<)
% Number of symbols : 280 ( 277 usr; 17 con; 0-4 aty)
% Number of variables : 4531 ( 592 ^;3762 !; 177 ?;4531 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:08.381
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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% Explicit typings (277)
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member5360357140639596479c_fm_a: episte5999154429349999510c_fm_a > set_Ep9038437136950510070c_fm_a > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
member6642669571620171971c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member5906877432388582473c_fm_a: list_Epistemic_fm_a > set_li769143395467472256c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member2328277852289759785c_fm_a: list_s580375451141968640c_fm_a > set_li4204741992506657632c_fm_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member5029746173879296720c_fm_a: produc2203834439629243303c_fm_a > set_Pr8948481099588399239c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member7847018790254331967c_fm_a: set_Ep8485816078964734998c_fm_a > set_se8358038196042697334c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
member5191902678313347615c_fm_a: set_Ep9038437136950510070c_fm_a > set_se5598658448297020502c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member536094252920883875c_fm_a: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member3183810400728440963c_fm_a: set_se5208064806568342746c_fm_a > set_se8609989359752948410c_fm_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: epistemic_fm_a > $o ).
thf(sy_v_V,type,
v: set_Epistemic_fm_a ).
thf(sy_v_W,type,
w: set_Epistemic_fm_a ).
thf(sy_v_i,type,
i: a ).
% Relevant facts (1277)
thf(fact_0_fm_Oinject_I5_J,axiom,
! [X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a,Y61: set_Epistemic_fm_a,Y62: episte5999154429349999510c_fm_a] :
( ( ( episte3013695276341575662c_fm_a @ X61 @ X62 )
= ( episte3013695276341575662c_fm_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_1_fm_Oinject_I5_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a,Y61: epistemic_fm_a,Y62: episte740340785640729014c_fm_a] :
( ( ( episte5657488632024175118c_fm_a @ X61 @ X62 )
= ( episte5657488632024175118c_fm_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_2_fm_Oinject_I5_J,axiom,
! [X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( ( epistemic_K_a @ X61 @ X62 )
= ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_3_fm_Oinject_I4_J,axiom,
! [X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a,Y51: episte5999154429349999510c_fm_a,Y52: episte5999154429349999510c_fm_a] :
( ( ( episte5200161025749276397c_fm_a @ X51 @ X52 )
= ( episte5200161025749276397c_fm_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_4_fm_Oinject_I4_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a,Y51: episte740340785640729014c_fm_a,Y52: episte740340785640729014c_fm_a] :
( ( ( episte260752218777527565c_fm_a @ X51 @ X52 )
= ( episte260752218777527565c_fm_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_5_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( ( epistemic_Imp_a @ X51 @ X52 )
= ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_6_fm_Odistinct_I9_J,axiom,
! [X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] :
( episte4474038146033622457c_fm_a
!= ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_7_fm_Odistinct_I9_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( episte5073044243917183961c_fm_a
!= ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_8_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_9_fm_Odistinct_I7_J,axiom,
! [X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a] :
( episte4474038146033622457c_fm_a
!= ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_10_fm_Odistinct_I7_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( episte5073044243917183961c_fm_a
!= ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_11_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_12_fm_Odistinct_I29_J,axiom,
! [X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a,X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] :
( ( episte5200161025749276397c_fm_a @ X51 @ X52 )
!= ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_13_fm_Odistinct_I29_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a,X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( ( episte260752218777527565c_fm_a @ X51 @ X52 )
!= ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_14_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Imp_a @ X51 @ X52 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_15_assms_I5_J,axiom,
( member536094252920883875c_fm_a @ w
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ i @ P ) @ v ) ) ) ) ) ).
% assms(5)
thf(fact_16_assms_I3_J,axiom,
episte2285483198712856226tent_a @ a2 @ w ).
% assms(3)
thf(fact_17_assms_I1_J,axiom,
episte2285483198712856226tent_a @ a2 @ v ).
% assms(1)
thf(fact_18_duality__taut,axiom,
! [I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,G: list_char > $o,H: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G @ H @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ Q @ episte4474038146033622457c_fm_a ) ) ) @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ Q @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ episte4474038146033622457c_fm_a ) ) ) ) ).
% duality_taut
thf(fact_19_duality__taut,axiom,
! [I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,G: list_char > $o,H: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G @ H @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ Q @ episte5073044243917183961c_fm_a ) ) ) @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ Q @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ episte5073044243917183961c_fm_a ) ) ) ) ).
% duality_taut
thf(fact_20_duality__taut,axiom,
! [I: a,P2: epistemic_fm_a,Q: epistemic_fm_a,G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_21_AxB_Ocases,axiom,
! [A: episte5999154429349999510c_fm_a] :
( ( episte4027724377868766071c_fm_a @ A )
=> ~ ! [P3: episte5999154429349999510c_fm_a,I2: set_Epistemic_fm_a] :
( A
!= ( episte5200161025749276397c_fm_a @ P3 @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ P3 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ).
% AxB.cases
thf(fact_22_AxB_Ocases,axiom,
! [A: episte740340785640729014c_fm_a] :
( ( episte89022042721419671c_fm_a @ A )
=> ~ ! [P3: episte740340785640729014c_fm_a,I2: epistemic_fm_a] :
( A
!= ( episte260752218777527565c_fm_a @ P3 @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ P3 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ).
% AxB.cases
thf(fact_23_AxB_Ocases,axiom,
! [A: epistemic_fm_a] :
( ( epistemic_AxB_a @ A )
=> ~ ! [P3: epistemic_fm_a,I2: a] :
( A
!= ( epistemic_Imp_a @ P3 @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% AxB.cases
thf(fact_24_AxB_Osimps,axiom,
( episte4027724377868766071c_fm_a
= ( ^ [A2: episte5999154429349999510c_fm_a] :
? [P: episte5999154429349999510c_fm_a,I3: set_Epistemic_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ P @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ P @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_25_AxB_Osimps,axiom,
( episte89022042721419671c_fm_a
= ( ^ [A2: episte740340785640729014c_fm_a] :
? [P: episte740340785640729014c_fm_a,I3: epistemic_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ P @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ P @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_26_AxB_Osimps,axiom,
( epistemic_AxB_a
= ( ^ [A2: epistemic_fm_a] :
? [P: epistemic_fm_a,I3: a] :
( A2
= ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_27_AxB_Ointros,axiom,
! [P2: episte5999154429349999510c_fm_a,I: set_Epistemic_fm_a] : ( episte4027724377868766071c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ).
% AxB.intros
thf(fact_28_AxB_Ointros,axiom,
! [P2: episte740340785640729014c_fm_a,I: epistemic_fm_a] : ( episte89022042721419671c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ).
% AxB.intros
thf(fact_29_AxB_Ointros,axiom,
! [P2: epistemic_fm_a,I: a] : ( epistemic_AxB_a @ ( epistemic_Imp_a @ P2 @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% AxB.intros
thf(fact_30_assms_I2_J,axiom,
maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ a2 ) @ v ).
% assms(2)
thf(fact_31_assms_I4_J,axiom,
maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ a2 ) @ w ).
% assms(4)
thf(fact_32_consistent__hereditary,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a,S2: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S2 @ S )
=> ( episte2285483198712856226tent_a @ A3 @ S2 ) ) ) ).
% consistent_hereditary
thf(fact_33_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte6853238276675109012c_fm_a @ G2 @ H2 @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) )
= ( ( episte6853238276675109012c_fm_a @ G2 @ H2 @ P2 )
=> ( episte6853238276675109012c_fm_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_34_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte6292435812075825844c_fm_a @ G2 @ H2 @ ( episte260752218777527565c_fm_a @ P2 @ Q ) )
= ( ( episte6292435812075825844c_fm_a @ G2 @ H2 @ P2 )
=> ( episte6292435812075825844c_fm_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_35_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ P2 @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P2 )
=> ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_36_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( episte6853238276675109012c_fm_a @ Ux @ H2 @ ( episte3013695276341575662c_fm_a @ I @ P2 ) )
= ( H2 @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ).
% eval.simps(6)
thf(fact_37_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] :
( ( episte6292435812075825844c_fm_a @ Ux @ H2 @ ( episte5657488632024175118c_fm_a @ I @ P2 ) )
= ( H2 @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ).
% eval.simps(6)
thf(fact_38_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H2 @ ( epistemic_K_a @ I @ P2 ) )
= ( H2 @ ( epistemic_K_a @ I @ P2 ) ) ) ).
% eval.simps(6)
thf(fact_39_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_a > $o] :
~ ( epistemic_eval_a @ Uu @ Uv @ epistemic_FF_a ) ).
% eval.simps(1)
thf(fact_40_subsetI,axiom,
! [A3: set_li4204741992506657632c_fm_a,B: set_li4204741992506657632c_fm_a] :
( ! [X: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X @ A3 )
=> ( member2328277852289759785c_fm_a @ X @ B ) )
=> ( ord_le2512102717711966976c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A3: set_se8609989359752948410c_fm_a,B: set_se8609989359752948410c_fm_a] :
( ! [X: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X @ A3 )
=> ( member3183810400728440963c_fm_a @ X @ B ) )
=> ( ord_le4042760284959298138c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A3: set_Ep6479678258574082331c_fm_a,B: set_Ep6479678258574082331c_fm_a] :
( ! [X: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X @ A3 )
=> ( member2634950385374053500c_fm_a @ X @ B ) )
=> ( ord_le8649530489803071867c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A3: set_se6091493367310676795c_fm_a,B: set_se6091493367310676795c_fm_a] :
( ! [X: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X @ A3 )
=> ( member293847127289681564c_fm_a @ X @ B ) )
=> ( ord_le582293785304634267c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_44_subsetI,axiom,
! [A3: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_45_subsetI,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ! [X: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X @ A3 )
=> ( member5029746173879296720c_fm_a @ X @ B ) )
=> ( ord_le8222437384497874983c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_46_subsetI,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( member536094252920883875c_fm_a @ X @ B ) )
=> ( ord_le7112219575281605754c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_47_subsetI,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( member6642669571620171971c_fm_a @ X @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ).
% subsetI
thf(fact_48_subset__antisym,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( ord_le7112219575281605754c_fm_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_49_subset__antisym,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A3 @ B )
=> ( ( ord_le8222437384497874983c_fm_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_50_subset__antisym,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_51_order__refl,axiom,
! [X2: set_se5208064806568342746c_fm_a] : ( ord_le7112219575281605754c_fm_a @ X2 @ X2 ) ).
% order_refl
thf(fact_52_order__refl,axiom,
! [X2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X2 @ X2 ) ).
% order_refl
thf(fact_53_order__refl,axiom,
! [X2: set_Epistemic_fm_a > $o] : ( ord_le1568123225352354019fm_a_o @ X2 @ X2 ) ).
% order_refl
thf(fact_54_order__refl,axiom,
! [X2: $o > set_Epistemic_fm_a] : ( ord_le1382570192702674727c_fm_a @ X2 @ X2 ) ).
% order_refl
thf(fact_55_order__refl,axiom,
! [X2: set_Pr8948481099588399239c_fm_a] : ( ord_le8222437384497874983c_fm_a @ X2 @ X2 ) ).
% order_refl
thf(fact_56_order__refl,axiom,
! [X2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X2 @ X2 ) ).
% order_refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: set_se5208064806568342746c_fm_a] : ( ord_le7112219575281605754c_fm_a @ A @ A ) ).
% dual_order.refl
thf(fact_58_dual__order_Orefl,axiom,
! [A: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A @ A ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A: set_Epistemic_fm_a > $o] : ( ord_le1568123225352354019fm_a_o @ A @ A ) ).
% dual_order.refl
thf(fact_60_dual__order_Orefl,axiom,
! [A: $o > set_Epistemic_fm_a] : ( ord_le1382570192702674727c_fm_a @ A @ A ) ).
% dual_order.refl
thf(fact_61_dual__order_Orefl,axiom,
! [A: set_Pr8948481099588399239c_fm_a] : ( ord_le8222437384497874983c_fm_a @ A @ A ) ).
% dual_order.refl
thf(fact_62_dual__order_Orefl,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% dual_order.refl
thf(fact_63_Collect__subset,axiom,
! [A3: set_se8609989359752948410c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ord_le4042760284959298138c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_64_Collect__subset,axiom,
! [A3: set_Ep6479678258574082331c_fm_a,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ord_le8649530489803071867c_fm_a
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_65_Collect__subset,axiom,
! [A3: set_se6091493367310676795c_fm_a,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ord_le582293785304634267c_fm_a
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_66_Collect__subset,axiom,
! [A3: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_67_Collect__subset,axiom,
! [A3: set_li4204741992506657632c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ord_le2512102717711966976c_fm_a
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_68_Collect__subset,axiom,
! [A3: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_69_Collect__subset,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,P4: produc2203834439629243303c_fm_a > $o] :
( ord_le8222437384497874983c_fm_a
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_70_Collect__subset,axiom,
! [A3: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_71_Collect__subset,axiom,
! [A3: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_72_conj__subset__def,axiom,
! [A3: set_li4204741992506657632c_fm_a,P4: list_s580375451141968640c_fm_a > $o,Q2: list_s580375451141968640c_fm_a > $o] :
( ( ord_le2512102717711966976c_fm_a @ A3
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le2512102717711966976c_fm_a @ A3 @ ( collec3923057941238076011c_fm_a @ P4 ) )
& ( ord_le2512102717711966976c_fm_a @ A3 @ ( collec3923057941238076011c_fm_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_73_conj__subset__def,axiom,
! [A3: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o,Q2: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ A3
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le2931979837777675552c_fm_a @ A3 @ ( collec5191077796991884427c_fm_a @ P4 ) )
& ( ord_le2931979837777675552c_fm_a @ A3 @ ( collec5191077796991884427c_fm_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_74_conj__subset__def,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,P4: produc2203834439629243303c_fm_a > $o,Q2: produc2203834439629243303c_fm_a > $o] :
( ( ord_le8222437384497874983c_fm_a @ A3
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le8222437384497874983c_fm_a @ A3 @ ( collec7391038856303340306c_fm_a @ P4 ) )
& ( ord_le8222437384497874983c_fm_a @ A3 @ ( collec7391038856303340306c_fm_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_75_conj__subset__def,axiom,
! [A3: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A3
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le7112219575281605754c_fm_a @ A3 @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( ord_le7112219575281605754c_fm_a @ A3 @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_76_conj__subset__def,axiom,
! [A3: set_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A3
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) )
= ( ( ord_le3275665582123262618c_fm_a @ A3 @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( ord_le3275665582123262618c_fm_a @ A3 @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_77_prop__restrict,axiom,
! [X2: set_se5208064806568342746c_fm_a,Z: set_se8609989359752948410c_fm_a,X4: set_se8609989359752948410c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( member3183810400728440963c_fm_a @ X2 @ Z )
=> ( ( ord_le4042760284959298138c_fm_a @ Z
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_78_prop__restrict,axiom,
! [X2: epistemic_fm_a > epistemic_fm_a,Z: set_Ep6479678258574082331c_fm_a,X4: set_Ep6479678258574082331c_fm_a,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( member2634950385374053500c_fm_a @ X2 @ Z )
=> ( ( ord_le8649530489803071867c_fm_a @ Z
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_79_prop__restrict,axiom,
! [X2: set_Epistemic_fm_a > epistemic_fm_a,Z: set_se6091493367310676795c_fm_a,X4: set_se6091493367310676795c_fm_a,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( member293847127289681564c_fm_a @ X2 @ Z )
=> ( ( ord_le582293785304634267c_fm_a @ Z
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_80_prop__restrict,axiom,
! [X2: a,Z: set_a,X4: set_a,P4: a > $o] :
( ( member_a @ X2 @ Z )
=> ( ( ord_less_eq_set_a @ Z
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_81_prop__restrict,axiom,
! [X2: list_s580375451141968640c_fm_a,Z: set_li4204741992506657632c_fm_a,X4: set_li4204741992506657632c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ( member2328277852289759785c_fm_a @ X2 @ Z )
=> ( ( ord_le2512102717711966976c_fm_a @ Z
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_82_prop__restrict,axiom,
! [X2: list_Epistemic_fm_a,Z: set_li769143395467472256c_fm_a,X4: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( member5906877432388582473c_fm_a @ X2 @ Z )
=> ( ( ord_le2931979837777675552c_fm_a @ Z
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_83_prop__restrict,axiom,
! [X2: produc2203834439629243303c_fm_a,Z: set_Pr8948481099588399239c_fm_a,X4: set_Pr8948481099588399239c_fm_a,P4: produc2203834439629243303c_fm_a > $o] :
( ( member5029746173879296720c_fm_a @ X2 @ Z )
=> ( ( ord_le8222437384497874983c_fm_a @ Z
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_84_prop__restrict,axiom,
! [X2: set_Epistemic_fm_a,Z: set_se5208064806568342746c_fm_a,X4: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X2 @ Z )
=> ( ( ord_le7112219575281605754c_fm_a @ Z
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_85_prop__restrict,axiom,
! [X2: epistemic_fm_a,Z: set_Epistemic_fm_a,X4: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X2 @ Z )
=> ( ( ord_le3275665582123262618c_fm_a @ Z
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X2 ) ) ) ).
% prop_restrict
thf(fact_86_Collect__restrict,axiom,
! [X4: set_se8609989359752948410c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ord_le4042760284959298138c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_87_Collect__restrict,axiom,
! [X4: set_Ep6479678258574082331c_fm_a,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ord_le8649530489803071867c_fm_a
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_88_Collect__restrict,axiom,
! [X4: set_se6091493367310676795c_fm_a,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ord_le582293785304634267c_fm_a
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_89_Collect__restrict,axiom,
! [X4: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_90_Collect__restrict,axiom,
! [X4: set_li4204741992506657632c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ord_le2512102717711966976c_fm_a
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_91_Collect__restrict,axiom,
! [X4: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_92_Collect__restrict,axiom,
! [X4: set_Pr8948481099588399239c_fm_a,P4: produc2203834439629243303c_fm_a > $o] :
( ord_le8222437384497874983c_fm_a
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_93_Collect__restrict,axiom,
! [X4: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_94_Collect__restrict,axiom,
! [X4: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ X4 )
& ( P4 @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_95_subset__CollectI,axiom,
! [B: set_se8609989359752948410c_fm_a,A3: set_se8609989359752948410c_fm_a,Q2: set_se5208064806568342746c_fm_a > $o,P4: set_se5208064806568342746c_fm_a > $o] :
( ( ord_le4042760284959298138c_fm_a @ B @ A3 )
=> ( ! [X: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le4042760284959298138c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_96_subset__CollectI,axiom,
! [B: set_Ep6479678258574082331c_fm_a,A3: set_Ep6479678258574082331c_fm_a,Q2: ( epistemic_fm_a > epistemic_fm_a ) > $o,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( ord_le8649530489803071867c_fm_a @ B @ A3 )
=> ( ! [X: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le8649530489803071867c_fm_a
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_97_subset__CollectI,axiom,
! [B: set_se6091493367310676795c_fm_a,A3: set_se6091493367310676795c_fm_a,Q2: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( ord_le582293785304634267c_fm_a @ B @ A3 )
=> ( ! [X: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le582293785304634267c_fm_a
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_98_subset__CollectI,axiom,
! [B: set_a,A3: set_a,Q2: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_99_subset__CollectI,axiom,
! [B: set_li4204741992506657632c_fm_a,A3: set_li4204741992506657632c_fm_a,Q2: list_s580375451141968640c_fm_a > $o,P4: list_s580375451141968640c_fm_a > $o] :
( ( ord_le2512102717711966976c_fm_a @ B @ A3 )
=> ( ! [X: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le2512102717711966976c_fm_a
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_100_subset__CollectI,axiom,
! [B: set_li769143395467472256c_fm_a,A3: set_li769143395467472256c_fm_a,Q2: list_Epistemic_fm_a > $o,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B @ A3 )
=> ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_101_subset__CollectI,axiom,
! [B: set_Pr8948481099588399239c_fm_a,A3: set_Pr8948481099588399239c_fm_a,Q2: produc2203834439629243303c_fm_a > $o,P4: produc2203834439629243303c_fm_a > $o] :
( ( ord_le8222437384497874983c_fm_a @ B @ A3 )
=> ( ! [X: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le8222437384497874983c_fm_a
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_102_subset__CollectI,axiom,
! [B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a,Q2: set_Epistemic_fm_a > $o,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B @ A3 )
=> ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_103_subset__CollectI,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,Q2: epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B @ A3 )
=> ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ B )
=> ( ( Q2 @ X )
=> ( P4 @ X ) ) )
=> ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ B )
& ( Q2 @ X3 ) ) )
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_104_subset__Collect__iff,axiom,
! [B: set_se8609989359752948410c_fm_a,A3: set_se8609989359752948410c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( ord_le4042760284959298138c_fm_a @ B @ A3 )
=> ( ( ord_le4042760284959298138c_fm_a @ B
@ ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_105_subset__Collect__iff,axiom,
! [B: set_Ep6479678258574082331c_fm_a,A3: set_Ep6479678258574082331c_fm_a,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( ord_le8649530489803071867c_fm_a @ B @ A3 )
=> ( ( ord_le8649530489803071867c_fm_a @ B
@ ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_106_subset__Collect__iff,axiom,
! [B: set_se6091493367310676795c_fm_a,A3: set_se6091493367310676795c_fm_a,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( ord_le582293785304634267c_fm_a @ B @ A3 )
=> ( ( ord_le582293785304634267c_fm_a @ B
@ ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_107_subset__Collect__iff,axiom,
! [B: set_a,A3: set_a,P4: a > $o] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_108_subset__Collect__iff,axiom,
! [B: set_li4204741992506657632c_fm_a,A3: set_li4204741992506657632c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ( ord_le2512102717711966976c_fm_a @ B @ A3 )
=> ( ( ord_le2512102717711966976c_fm_a @ B
@ ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_109_subset__Collect__iff,axiom,
! [B: set_li769143395467472256c_fm_a,A3: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B @ A3 )
=> ( ( ord_le2931979837777675552c_fm_a @ B
@ ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_110_subset__Collect__iff,axiom,
! [B: set_Pr8948481099588399239c_fm_a,A3: set_Pr8948481099588399239c_fm_a,P4: produc2203834439629243303c_fm_a > $o] :
( ( ord_le8222437384497874983c_fm_a @ B @ A3 )
=> ( ( ord_le8222437384497874983c_fm_a @ B
@ ( collec7391038856303340306c_fm_a
@ ^ [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_111_subset__Collect__iff,axiom,
! [B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B @ A3 )
=> ( ( ord_le7112219575281605754c_fm_a @ B
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_112_subset__Collect__iff,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ B )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_113_Ax5_Ointros,axiom,
! [I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] : ( episte3174827985664174276c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ).
% Ax5.intros
thf(fact_114_Ax5_Ointros,axiom,
! [I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] : ( episte7251237963447243492c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ).
% Ax5.intros
thf(fact_115_Ax5_Ointros,axiom,
! [I: a,P2: epistemic_fm_a] : ( epistemic_Ax5_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% Ax5.intros
thf(fact_116_less__eq__set__def,axiom,
( ord_le2512102717711966976c_fm_a
= ( ^ [A4: set_li4204741992506657632c_fm_a,B2: set_li4204741992506657632c_fm_a] :
( ord_le8346386785241846749fm_a_o
@ ^ [X3: list_s580375451141968640c_fm_a] : ( member2328277852289759785c_fm_a @ X3 @ A4 )
@ ^ [X3: list_s580375451141968640c_fm_a] : ( member2328277852289759785c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_117_less__eq__set__def,axiom,
( ord_le4042760284959298138c_fm_a
= ( ^ [A4: set_se8609989359752948410c_fm_a,B2: set_se8609989359752948410c_fm_a] :
( ord_le8279579953234471683fm_a_o
@ ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ A4 )
@ ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_118_less__eq__set__def,axiom,
( ord_le8649530489803071867c_fm_a
= ( ^ [A4: set_Ep6479678258574082331c_fm_a,B2: set_Ep6479678258574082331c_fm_a] :
( ord_le914026578752367818fm_a_o
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] : ( member2634950385374053500c_fm_a @ X3 @ A4 )
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] : ( member2634950385374053500c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_119_less__eq__set__def,axiom,
( ord_le582293785304634267c_fm_a
= ( ^ [A4: set_se6091493367310676795c_fm_a,B2: set_se6091493367310676795c_fm_a] :
( ord_le8669121985476793002fm_a_o
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] : ( member293847127289681564c_fm_a @ X3 @ A4 )
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] : ( member293847127289681564c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_120_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A4 )
@ ^ [X3: a] : ( member_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_121_less__eq__set__def,axiom,
( ord_le8222437384497874983c_fm_a
= ( ^ [A4: set_Pr8948481099588399239c_fm_a,B2: set_Pr8948481099588399239c_fm_a] :
( ord_le4942933042592423798fm_a_o
@ ^ [X3: produc2203834439629243303c_fm_a] : ( member5029746173879296720c_fm_a @ X3 @ A4 )
@ ^ [X3: produc2203834439629243303c_fm_a] : ( member5029746173879296720c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_122_less__eq__set__def,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( ord_le1568123225352354019fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A4 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_123_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A4 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_124_Ax5__Euclidean,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,U: set_Ep9038437136950510070c_fm_a,V: set_Ep9038437136950510070c_fm_a,W: set_Ep9038437136950510070c_fm_a,I: set_Epistemic_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte3174827985664174276c_fm_a @ A3 )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ U )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ U )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ W )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ W )
=> ( ( member5191902678313347615c_fm_a @ V
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ U ) ) ) ) )
=> ( ( member5191902678313347615c_fm_a @ W
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ U ) ) ) ) )
=> ( member5191902678313347615c_fm_a @ W
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_125_Ax5__Euclidean,axiom,
! [A3: episte740340785640729014c_fm_a > $o,U: set_Ep8485816078964734998c_fm_a,V: set_Ep8485816078964734998c_fm_a,W: set_Ep8485816078964734998c_fm_a,I: epistemic_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte7251237963447243492c_fm_a @ A3 )
=> ( ( episte704884124137261758c_fm_a @ A3 @ U )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ U )
=> ( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( episte704884124137261758c_fm_a @ A3 @ W )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ W )
=> ( ( member7847018790254331967c_fm_a @ V
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ U ) ) ) ) )
=> ( ( member7847018790254331967c_fm_a @ W
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ U ) ) ) ) )
=> ( member7847018790254331967c_fm_a @ W
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_126_Ax5__Euclidean,axiom,
! [A3: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V: set_Epistemic_fm_a,W: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A3 )
=> ( ( episte2285483198712856226tent_a @ A3 @ U )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ U )
=> ( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( episte2285483198712856226tent_a @ A3 @ W )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W )
=> ( ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ U ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ W
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ U ) ) ) ) )
=> ( member536094252920883875c_fm_a @ W
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_127_ax__in__maximal,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( A3 @ P2 )
=> ( member6642669571620171971c_fm_a @ P2 @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_128_AxB__symmetric_H,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,W: set_Ep9038437136950510070c_fm_a,I: set_Epistemic_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte4027724377868766071c_fm_a @ A3 )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ W )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ W )
=> ( ( member5191902678313347615c_fm_a @ W
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ V ) ) ) ) )
=> ( member5191902678313347615c_fm_a @ V
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ W ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_129_AxB__symmetric_H,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,W: set_Ep8485816078964734998c_fm_a,I: epistemic_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte89022042721419671c_fm_a @ A3 )
=> ( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( episte704884124137261758c_fm_a @ A3 @ W )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ W )
=> ( ( member7847018790254331967c_fm_a @ W
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ V ) ) ) ) )
=> ( member7847018790254331967c_fm_a @ V
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ W ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_130_AxB__symmetric_H,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A3 )
=> ( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( episte2285483198712856226tent_a @ A3 @ W )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W )
=> ( ( member536094252920883875c_fm_a @ W
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ V ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ W ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_131_consequent__in__maximal,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( member5360357140639596479c_fm_a @ P2 @ V )
=> ( ( member5360357140639596479c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) @ V )
=> ( member5360357140639596479c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_132_consequent__in__maximal,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( member1118567890201975263c_fm_a @ P2 @ V )
=> ( ( member1118567890201975263c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ Q ) @ V )
=> ( member1118567890201975263c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_133_consequent__in__maximal,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P2 @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P2 @ Q ) @ V )
=> ( member6642669571620171971c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_134_maximal__extension,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ~ ! [W2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V @ W2 )
=> ( ( episte2285483198712856226tent_a @ A3 @ W2 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W2 ) ) ) ) ).
% maximal_extension
thf(fact_135_order__antisym__conv,axiom,
! [Y: set_se5208064806568342746c_fm_a,X2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ Y @ X2 )
=> ( ( ord_le7112219575281605754c_fm_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_136_order__antisym__conv,axiom,
! [Y: epistemic_fm_a > $o,X2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_137_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ Y @ X2 )
=> ( ( ord_le1568123225352354019fm_a_o @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_138_order__antisym__conv,axiom,
! [Y: $o > set_Epistemic_fm_a,X2: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ Y @ X2 )
=> ( ( ord_le1382570192702674727c_fm_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_139_order__antisym__conv,axiom,
! [Y: set_Pr8948481099588399239c_fm_a,X2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ Y @ X2 )
=> ( ( ord_le8222437384497874983c_fm_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_140_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_141_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_143_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__le__eq__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_ord__le__eq__subst,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1568123225352354019fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1568123225352354019fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1382570192702674727c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1382570192702674727c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_ord__le__eq__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_149_ord__le__eq__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_150_ord__le__eq__subst,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_a,F: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_155_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_156_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1568123225352354019fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1568123225352354019fm_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_157_ord__eq__le__subst,axiom,
! [A: $o > set_Epistemic_fm_a,F: set_Epistemic_fm_a > $o > set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1382570192702674727c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1382570192702674727c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_158_ord__eq__le__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_ord__eq__le__subst,axiom,
! [A: epistemic_fm_a > $o,F: set_se5208064806568342746c_fm_a > epistemic_fm_a > $o,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_160_ord__eq__le__subst,axiom,
! [A: set_se5208064806568342746c_fm_a,F: ( epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_161_order__eq__refl,axiom,
! [X2: set_se5208064806568342746c_fm_a,Y: set_se5208064806568342746c_fm_a] :
( ( X2 = Y )
=> ( ord_le7112219575281605754c_fm_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_162_order__eq__refl,axiom,
! [X2: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( X2 = Y )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_163_order__eq__refl,axiom,
! [X2: set_Epistemic_fm_a > $o,Y: set_Epistemic_fm_a > $o] :
( ( X2 = Y )
=> ( ord_le1568123225352354019fm_a_o @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_164_order__eq__refl,axiom,
! [X2: $o > set_Epistemic_fm_a,Y: $o > set_Epistemic_fm_a] :
( ( X2 = Y )
=> ( ord_le1382570192702674727c_fm_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_165_order__eq__refl,axiom,
! [X2: set_Pr8948481099588399239c_fm_a,Y: set_Pr8948481099588399239c_fm_a] :
( ( X2 = Y )
=> ( ord_le8222437384497874983c_fm_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_166_order__eq__refl,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( X2 = Y )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_167_order__subst2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_168_order__subst2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B3 ) @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst2,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_172_order__subst2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le1568123225352354019fm_a_o @ ( F @ B3 ) @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1568123225352354019fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1568123225352354019fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_173_order__subst2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le1382570192702674727c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le1382570192702674727c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le1382570192702674727c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_174_order__subst2,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_175_order__subst2,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B3 ) @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_176_order__subst2,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ ( F @ B3 ) @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_177_order__subst1,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A: set_Epistemic_fm_a,F: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst1,axiom,
! [A: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ ( F @ B3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_182_order__subst1,axiom,
! [A: set_Epistemic_fm_a,F: ( set_Epistemic_fm_a > $o ) > set_Epistemic_fm_a,B3: set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le1568123225352354019fm_a_o @ B3 @ C )
=> ( ! [X: set_Epistemic_fm_a > $o,Y2: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A: set_Epistemic_fm_a,F: ( $o > set_Epistemic_fm_a ) > set_Epistemic_fm_a,B3: $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le1382570192702674727c_fm_a @ B3 @ C )
=> ( ! [X: $o > set_Epistemic_fm_a,Y2: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ X @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst1,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_185_order__subst1,axiom,
! [A: set_se5208064806568342746c_fm_a,F: ( epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A @ ( F @ B3 ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_186_order__subst1,axiom,
! [A: epistemic_fm_a > $o,F: set_se5208064806568342746c_fm_a > epistemic_fm_a > $o,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A @ ( F @ B3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ! [X: set_se5208064806568342746c_fm_a,Y2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_187_mem__Collect__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( member3183810400728440963c_fm_a @ A @ ( collec4484617205158951365c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_188_mem__Collect__eq,axiom,
! [A: epistemic_fm_a > epistemic_fm_a,P4: ( epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( member2634950385374053500c_fm_a @ A @ ( collec8583107250894946746c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_189_mem__Collect__eq,axiom,
! [A: set_Epistemic_fm_a > epistemic_fm_a,P4: ( set_Epistemic_fm_a > epistemic_fm_a ) > $o] :
( ( member293847127289681564c_fm_a @ A @ ( collec8907789384498502618c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_190_mem__Collect__eq,axiom,
! [A: a,P4: a > $o] :
( ( member_a @ A @ ( collect_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_191_mem__Collect__eq,axiom,
! [A: list_s580375451141968640c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ( member2328277852289759785c_fm_a @ A @ ( collec3923057941238076011c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_192_mem__Collect__eq,axiom,
! [A: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( member5906877432388582473c_fm_a @ A @ ( collec5191077796991884427c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
! [A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
! [A: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_195_Collect__mem__eq,axiom,
! [A3: set_se8609989359752948410c_fm_a] :
( ( collec4484617205158951365c_fm_a
@ ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A3: set_Ep6479678258574082331c_fm_a] :
( ( collec8583107250894946746c_fm_a
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] : ( member2634950385374053500c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A3: set_se6091493367310676795c_fm_a] :
( ( collec8907789384498502618c_fm_a
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] : ( member293847127289681564c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_198_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A3: set_li4204741992506657632c_fm_a] :
( ( collec3923057941238076011c_fm_a
@ ^ [X3: list_s580375451141968640c_fm_a] : ( member2328277852289759785c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A3: set_li769143395467472256c_fm_a] :
( ( collec5191077796991884427c_fm_a
@ ^ [X3: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A3: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_203_Collect__cong,axiom,
! [P4: list_s580375451141968640c_fm_a > $o,Q2: list_s580375451141968640c_fm_a > $o] :
( ! [X: list_s580375451141968640c_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec3923057941238076011c_fm_a @ P4 )
= ( collec3923057941238076011c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_204_Collect__cong,axiom,
! [P4: list_Epistemic_fm_a > $o,Q2: list_Epistemic_fm_a > $o] :
( ! [X: list_Epistemic_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec5191077796991884427c_fm_a @ P4 )
= ( collec5191077796991884427c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_205_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_206_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_207_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se5208064806568342746c_fm_a,Z2: set_se5208064806568342746c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A2 @ B4 )
& ( ord_le7112219575281605754c_fm_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_208_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A2: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_209_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a > $o,Z2: set_Epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Epistemic_fm_a > $o,B4: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ A2 @ B4 )
& ( ord_le1568123225352354019fm_a_o @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_210_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > set_Epistemic_fm_a,Z2: $o > set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: $o > set_Epistemic_fm_a,B4: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ A2 @ B4 )
& ( ord_le1382570192702674727c_fm_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_211_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr8948481099588399239c_fm_a,Z2: set_Pr8948481099588399239c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Pr8948481099588399239c_fm_a,B4: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A2 @ B4 )
& ( ord_le8222437384497874983c_fm_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_212_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ B4 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_213_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G3: epistemic_fm_a > $o] :
! [X3: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_214_le__fun__def,axiom,
( ord_le1568123225352354019fm_a_o
= ( ^ [F2: set_Epistemic_fm_a > $o,G3: set_Epistemic_fm_a > $o] :
! [X3: set_Epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_215_le__fun__def,axiom,
( ord_le1382570192702674727c_fm_a
= ( ^ [F2: $o > set_Epistemic_fm_a,G3: $o > set_Epistemic_fm_a] :
! [X3: $o] : ( ord_le3275665582123262618c_fm_a @ ( F2 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_216_le__funI,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G2 ) ) ).
% le_funI
thf(fact_217_le__funI,axiom,
! [F: set_Epistemic_fm_a > $o,G2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le1568123225352354019fm_a_o @ F @ G2 ) ) ).
% le_funI
thf(fact_218_le__funI,axiom,
! [F: $o > set_Epistemic_fm_a,G2: $o > set_Epistemic_fm_a] :
( ! [X: $o] : ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le1382570192702674727c_fm_a @ F @ G2 ) ) ).
% le_funI
thf(fact_219_le__funE,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o,X2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_220_le__funE,axiom,
! [F: set_Epistemic_fm_a > $o,G2: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a] :
( ( ord_le1568123225352354019fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_221_le__funE,axiom,
! [F: $o > set_Epistemic_fm_a,G2: $o > set_Epistemic_fm_a,X2: $o] :
( ( ord_le1382570192702674727c_fm_a @ F @ G2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_222_le__funD,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o,X2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_223_le__funD,axiom,
! [F: set_Epistemic_fm_a > $o,G2: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a] :
( ( ord_le1568123225352354019fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_224_le__funD,axiom,
! [F: $o > set_Epistemic_fm_a,G2: $o > set_Epistemic_fm_a,X2: $o] :
( ( ord_le1382570192702674727c_fm_a @ F @ G2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_225_antisym,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_226_antisym,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_227_antisym,axiom,
! [A: set_Epistemic_fm_a > $o,B3: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ A @ B3 )
=> ( ( ord_le1568123225352354019fm_a_o @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_228_antisym,axiom,
! [A: $o > set_Epistemic_fm_a,B3: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ A @ B3 )
=> ( ( ord_le1382570192702674727c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_229_antisym,axiom,
! [A: set_Pr8948481099588399239c_fm_a,B3: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A @ B3 )
=> ( ( ord_le8222437384497874983c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_230_antisym,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_231_dual__order_Otrans,axiom,
! [B3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ B3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ C @ B3 )
=> ( ord_le7112219575281605754c_fm_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_232_dual__order_Otrans,axiom,
! [B3: epistemic_fm_a > $o,A: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ B3 )
=> ( ord_le4043730696559282883fm_a_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_233_dual__order_Otrans,axiom,
! [B3: set_Epistemic_fm_a > $o,A: set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ B3 @ A )
=> ( ( ord_le1568123225352354019fm_a_o @ C @ B3 )
=> ( ord_le1568123225352354019fm_a_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_234_dual__order_Otrans,axiom,
! [B3: $o > set_Epistemic_fm_a,A: $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ B3 @ A )
=> ( ( ord_le1382570192702674727c_fm_a @ C @ B3 )
=> ( ord_le1382570192702674727c_fm_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_235_dual__order_Otrans,axiom,
! [B3: set_Pr8948481099588399239c_fm_a,A: set_Pr8948481099588399239c_fm_a,C: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ B3 @ A )
=> ( ( ord_le8222437384497874983c_fm_a @ C @ B3 )
=> ( ord_le8222437384497874983c_fm_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_236_dual__order_Otrans,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_237_dual__order_Oantisym,axiom,
! [B3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ B3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_238_dual__order_Oantisym,axiom,
! [B3: epistemic_fm_a > $o,A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_239_dual__order_Oantisym,axiom,
! [B3: set_Epistemic_fm_a > $o,A: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ B3 @ A )
=> ( ( ord_le1568123225352354019fm_a_o @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_240_dual__order_Oantisym,axiom,
! [B3: $o > set_Epistemic_fm_a,A: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ B3 @ A )
=> ( ( ord_le1382570192702674727c_fm_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_241_dual__order_Oantisym,axiom,
! [B3: set_Pr8948481099588399239c_fm_a,A: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ B3 @ A )
=> ( ( ord_le8222437384497874983c_fm_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_242_dual__order_Oantisym,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_243_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_se5208064806568342746c_fm_a,Z2: set_se5208064806568342746c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A2 )
& ( ord_le7112219575281605754c_fm_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A2: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B4 @ A2 )
& ( ord_le4043730696559282883fm_a_o @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a > $o,Z2: set_Epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Epistemic_fm_a > $o,B4: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ B4 @ A2 )
& ( ord_le1568123225352354019fm_a_o @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_246_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > set_Epistemic_fm_a,Z2: $o > set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: $o > set_Epistemic_fm_a,B4: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ B4 @ A2 )
& ( ord_le1382570192702674727c_fm_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_247_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Pr8948481099588399239c_fm_a,Z2: set_Pr8948481099588399239c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Pr8948481099588399239c_fm_a,B4: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ B4 @ A2 )
& ( ord_le8222437384497874983c_fm_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_248_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A2 )
& ( ord_le3275665582123262618c_fm_a @ A2 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_249_order__trans,axiom,
! [X2: set_se5208064806568342746c_fm_a,Y: set_se5208064806568342746c_fm_a,Z3: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X2 @ Y )
=> ( ( ord_le7112219575281605754c_fm_a @ Y @ Z3 )
=> ( ord_le7112219575281605754c_fm_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_250_order__trans,axiom,
! [X2: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z3 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_251_order__trans,axiom,
! [X2: set_Epistemic_fm_a > $o,Y: set_Epistemic_fm_a > $o,Z3: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ X2 @ Y )
=> ( ( ord_le1568123225352354019fm_a_o @ Y @ Z3 )
=> ( ord_le1568123225352354019fm_a_o @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_252_order__trans,axiom,
! [X2: $o > set_Epistemic_fm_a,Y: $o > set_Epistemic_fm_a,Z3: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ X2 @ Y )
=> ( ( ord_le1382570192702674727c_fm_a @ Y @ Z3 )
=> ( ord_le1382570192702674727c_fm_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_253_order__trans,axiom,
! [X2: set_Pr8948481099588399239c_fm_a,Y: set_Pr8948481099588399239c_fm_a,Z3: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ X2 @ Y )
=> ( ( ord_le8222437384497874983c_fm_a @ Y @ Z3 )
=> ( ord_le8222437384497874983c_fm_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_254_order__trans,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_255_order_Otrans,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ord_le7112219575281605754c_fm_a @ A @ C ) ) ) ).
% order.trans
thf(fact_256_order_Otrans,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ord_le4043730696559282883fm_a_o @ A @ C ) ) ) ).
% order.trans
thf(fact_257_order_Otrans,axiom,
! [A: set_Epistemic_fm_a > $o,B3: set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ A @ B3 )
=> ( ( ord_le1568123225352354019fm_a_o @ B3 @ C )
=> ( ord_le1568123225352354019fm_a_o @ A @ C ) ) ) ).
% order.trans
thf(fact_258_order_Otrans,axiom,
! [A: $o > set_Epistemic_fm_a,B3: $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ A @ B3 )
=> ( ( ord_le1382570192702674727c_fm_a @ B3 @ C )
=> ( ord_le1382570192702674727c_fm_a @ A @ C ) ) ) ).
% order.trans
thf(fact_259_order_Otrans,axiom,
! [A: set_Pr8948481099588399239c_fm_a,B3: set_Pr8948481099588399239c_fm_a,C: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A @ B3 )
=> ( ( ord_le8222437384497874983c_fm_a @ B3 @ C )
=> ( ord_le8222437384497874983c_fm_a @ A @ C ) ) ) ).
% order.trans
thf(fact_260_order_Otrans,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ord_le3275665582123262618c_fm_a @ A @ C ) ) ) ).
% order.trans
thf(fact_261_order__antisym,axiom,
! [X2: set_se5208064806568342746c_fm_a,Y: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X2 @ Y )
=> ( ( ord_le7112219575281605754c_fm_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_262_order__antisym,axiom,
! [X2: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_263_order__antisym,axiom,
! [X2: set_Epistemic_fm_a > $o,Y: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ X2 @ Y )
=> ( ( ord_le1568123225352354019fm_a_o @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_264_order__antisym,axiom,
! [X2: $o > set_Epistemic_fm_a,Y: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ X2 @ Y )
=> ( ( ord_le1382570192702674727c_fm_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_265_order__antisym,axiom,
! [X2: set_Pr8948481099588399239c_fm_a,Y: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ X2 @ Y )
=> ( ( ord_le8222437384497874983c_fm_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_266_order__antisym,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_267_ord__le__eq__trans,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le7112219575281605754c_fm_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_268_ord__le__eq__trans,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le4043730696559282883fm_a_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_269_ord__le__eq__trans,axiom,
! [A: set_Epistemic_fm_a > $o,B3: set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le1568123225352354019fm_a_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_270_ord__le__eq__trans,axiom,
! [A: $o > set_Epistemic_fm_a,B3: $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le1382570192702674727c_fm_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_271_ord__le__eq__trans,axiom,
! [A: set_Pr8948481099588399239c_fm_a,B3: set_Pr8948481099588399239c_fm_a,C: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le8222437384497874983c_fm_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_272_ord__le__eq__trans,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le3275665582123262618c_fm_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_273_ord__eq__le__trans,axiom,
! [A: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( A = B3 )
=> ( ( ord_le7112219575281605754c_fm_a @ B3 @ C )
=> ( ord_le7112219575281605754c_fm_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_274_ord__eq__le__trans,axiom,
! [A: epistemic_fm_a > $o,B3: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A = B3 )
=> ( ( ord_le4043730696559282883fm_a_o @ B3 @ C )
=> ( ord_le4043730696559282883fm_a_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_275_ord__eq__le__trans,axiom,
! [A: set_Epistemic_fm_a > $o,B3: set_Epistemic_fm_a > $o,C: set_Epistemic_fm_a > $o] :
( ( A = B3 )
=> ( ( ord_le1568123225352354019fm_a_o @ B3 @ C )
=> ( ord_le1568123225352354019fm_a_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_276_ord__eq__le__trans,axiom,
! [A: $o > set_Epistemic_fm_a,B3: $o > set_Epistemic_fm_a,C: $o > set_Epistemic_fm_a] :
( ( A = B3 )
=> ( ( ord_le1382570192702674727c_fm_a @ B3 @ C )
=> ( ord_le1382570192702674727c_fm_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_277_ord__eq__le__trans,axiom,
! [A: set_Pr8948481099588399239c_fm_a,B3: set_Pr8948481099588399239c_fm_a,C: set_Pr8948481099588399239c_fm_a] :
( ( A = B3 )
=> ( ( ord_le8222437384497874983c_fm_a @ B3 @ C )
=> ( ord_le8222437384497874983c_fm_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_278_ord__eq__le__trans,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A = B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ord_le3275665582123262618c_fm_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_279_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se5208064806568342746c_fm_a,Z2: set_se5208064806568342746c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_se5208064806568342746c_fm_a,Y4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ X3 @ Y4 )
& ( ord_le7112219575281605754c_fm_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_280_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [X3: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y4 )
& ( ord_le4043730696559282883fm_a_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_281_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a > $o,Z2: set_Epistemic_fm_a > $o] : ( Y3 = Z2 ) )
= ( ^ [X3: set_Epistemic_fm_a > $o,Y4: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ X3 @ Y4 )
& ( ord_le1568123225352354019fm_a_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_282_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > set_Epistemic_fm_a,Z2: $o > set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [X3: $o > set_Epistemic_fm_a,Y4: $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ X3 @ Y4 )
& ( ord_le1382570192702674727c_fm_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_283_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr8948481099588399239c_fm_a,Z2: set_Pr8948481099588399239c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_Pr8948481099588399239c_fm_a,Y4: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ X3 @ Y4 )
& ( ord_le8222437384497874983c_fm_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_284_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y4 )
& ( ord_le3275665582123262618c_fm_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_285_Collect__mono__iff,axiom,
! [P4: list_s580375451141968640c_fm_a > $o,Q2: list_s580375451141968640c_fm_a > $o] :
( ( ord_le2512102717711966976c_fm_a @ ( collec3923057941238076011c_fm_a @ P4 ) @ ( collec3923057941238076011c_fm_a @ Q2 ) )
= ( ! [X3: list_s580375451141968640c_fm_a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_286_Collect__mono__iff,axiom,
! [P4: list_Epistemic_fm_a > $o,Q2: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q2 ) )
= ( ! [X3: list_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_287_Collect__mono__iff,axiom,
! [P4: produc2203834439629243303c_fm_a > $o,Q2: produc2203834439629243303c_fm_a > $o] :
( ( ord_le8222437384497874983c_fm_a @ ( collec7391038856303340306c_fm_a @ P4 ) @ ( collec7391038856303340306c_fm_a @ Q2 ) )
= ( ! [X3: produc2203834439629243303c_fm_a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_288_Collect__mono__iff,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q2 ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_289_Collect__mono__iff,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q2 ) )
= ( ! [X3: epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_290_set__eq__subset,axiom,
( ( ^ [Y3: set_se5208064806568342746c_fm_a,Z2: set_se5208064806568342746c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A4 @ B2 )
& ( ord_le7112219575281605754c_fm_a @ B2 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_291_set__eq__subset,axiom,
( ( ^ [Y3: set_Pr8948481099588399239c_fm_a,Z2: set_Pr8948481099588399239c_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_Pr8948481099588399239c_fm_a,B2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A4 @ B2 )
& ( ord_le8222437384497874983c_fm_a @ B2 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_292_set__eq__subset,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_293_subset__trans,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a,C2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( ord_le7112219575281605754c_fm_a @ B @ C2 )
=> ( ord_le7112219575281605754c_fm_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_294_subset__trans,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a,C2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A3 @ B )
=> ( ( ord_le8222437384497874983c_fm_a @ B @ C2 )
=> ( ord_le8222437384497874983c_fm_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_295_subset__trans,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_296_Collect__mono,axiom,
! [P4: list_s580375451141968640c_fm_a > $o,Q2: list_s580375451141968640c_fm_a > $o] :
( ! [X: list_s580375451141968640c_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le2512102717711966976c_fm_a @ ( collec3923057941238076011c_fm_a @ P4 ) @ ( collec3923057941238076011c_fm_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_297_Collect__mono,axiom,
! [P4: list_Epistemic_fm_a > $o,Q2: list_Epistemic_fm_a > $o] :
( ! [X: list_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_298_Collect__mono,axiom,
! [P4: produc2203834439629243303c_fm_a > $o,Q2: produc2203834439629243303c_fm_a > $o] :
( ! [X: produc2203834439629243303c_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le8222437384497874983c_fm_a @ ( collec7391038856303340306c_fm_a @ P4 ) @ ( collec7391038856303340306c_fm_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_299_Collect__mono,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_300_Collect__mono,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_301_subset__refl,axiom,
! [A3: set_se5208064806568342746c_fm_a] : ( ord_le7112219575281605754c_fm_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_302_subset__refl,axiom,
! [A3: set_Pr8948481099588399239c_fm_a] : ( ord_le8222437384497874983c_fm_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_303_subset__refl,axiom,
! [A3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_304_subset__iff,axiom,
( ord_le2512102717711966976c_fm_a
= ( ^ [A4: set_li4204741992506657632c_fm_a,B2: set_li4204741992506657632c_fm_a] :
! [T: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ T @ A4 )
=> ( member2328277852289759785c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_305_subset__iff,axiom,
( ord_le4042760284959298138c_fm_a
= ( ^ [A4: set_se8609989359752948410c_fm_a,B2: set_se8609989359752948410c_fm_a] :
! [T: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ T @ A4 )
=> ( member3183810400728440963c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_306_subset__iff,axiom,
( ord_le8649530489803071867c_fm_a
= ( ^ [A4: set_Ep6479678258574082331c_fm_a,B2: set_Ep6479678258574082331c_fm_a] :
! [T: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ T @ A4 )
=> ( member2634950385374053500c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_307_subset__iff,axiom,
( ord_le582293785304634267c_fm_a
= ( ^ [A4: set_se6091493367310676795c_fm_a,B2: set_se6091493367310676795c_fm_a] :
! [T: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ T @ A4 )
=> ( member293847127289681564c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_308_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_309_subset__iff,axiom,
( ord_le8222437384497874983c_fm_a
= ( ^ [A4: set_Pr8948481099588399239c_fm_a,B2: set_Pr8948481099588399239c_fm_a] :
! [T: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ T @ A4 )
=> ( member5029746173879296720c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_310_subset__iff,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
! [T: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ T @ A4 )
=> ( member536094252920883875c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_311_subset__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
! [T: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ T @ A4 )
=> ( member6642669571620171971c_fm_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_312_equalityD2,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( A3 = B )
=> ( ord_le7112219575281605754c_fm_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_313_equalityD2,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ( A3 = B )
=> ( ord_le8222437384497874983c_fm_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_314_equalityD2,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A3 = B )
=> ( ord_le3275665582123262618c_fm_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_315_equalityD1,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( A3 = B )
=> ( ord_le7112219575281605754c_fm_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_316_equalityD1,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ( A3 = B )
=> ( ord_le8222437384497874983c_fm_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_317_equalityD1,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A3 = B )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_318_subset__eq,axiom,
( ord_le2512102717711966976c_fm_a
= ( ^ [A4: set_li4204741992506657632c_fm_a,B2: set_li4204741992506657632c_fm_a] :
! [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ A4 )
=> ( member2328277852289759785c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_319_subset__eq,axiom,
( ord_le4042760284959298138c_fm_a
= ( ^ [A4: set_se8609989359752948410c_fm_a,B2: set_se8609989359752948410c_fm_a] :
! [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ A4 )
=> ( member3183810400728440963c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_320_subset__eq,axiom,
( ord_le8649530489803071867c_fm_a
= ( ^ [A4: set_Ep6479678258574082331c_fm_a,B2: set_Ep6479678258574082331c_fm_a] :
! [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ A4 )
=> ( member2634950385374053500c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_321_subset__eq,axiom,
( ord_le582293785304634267c_fm_a
= ( ^ [A4: set_se6091493367310676795c_fm_a,B2: set_se6091493367310676795c_fm_a] :
! [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ A4 )
=> ( member293847127289681564c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_322_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_323_subset__eq,axiom,
( ord_le8222437384497874983c_fm_a
= ( ^ [A4: set_Pr8948481099588399239c_fm_a,B2: set_Pr8948481099588399239c_fm_a] :
! [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ A4 )
=> ( member5029746173879296720c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_324_subset__eq,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A4 )
=> ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_325_subset__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A4 )
=> ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_326_equalityE,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( A3 = B )
=> ~ ( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ~ ( ord_le7112219575281605754c_fm_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_327_equalityE,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ( A3 = B )
=> ~ ( ( ord_le8222437384497874983c_fm_a @ A3 @ B )
=> ~ ( ord_le8222437384497874983c_fm_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_328_equalityE,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A3 = B )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ~ ( ord_le3275665582123262618c_fm_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_329_subsetD,axiom,
! [A3: set_li4204741992506657632c_fm_a,B: set_li4204741992506657632c_fm_a,C: list_s580375451141968640c_fm_a] :
( ( ord_le2512102717711966976c_fm_a @ A3 @ B )
=> ( ( member2328277852289759785c_fm_a @ C @ A3 )
=> ( member2328277852289759785c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_330_subsetD,axiom,
! [A3: set_se8609989359752948410c_fm_a,B: set_se8609989359752948410c_fm_a,C: set_se5208064806568342746c_fm_a] :
( ( ord_le4042760284959298138c_fm_a @ A3 @ B )
=> ( ( member3183810400728440963c_fm_a @ C @ A3 )
=> ( member3183810400728440963c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_331_subsetD,axiom,
! [A3: set_Ep6479678258574082331c_fm_a,B: set_Ep6479678258574082331c_fm_a,C: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le8649530489803071867c_fm_a @ A3 @ B )
=> ( ( member2634950385374053500c_fm_a @ C @ A3 )
=> ( member2634950385374053500c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_332_subsetD,axiom,
! [A3: set_se6091493367310676795c_fm_a,B: set_se6091493367310676795c_fm_a,C: set_Epistemic_fm_a > epistemic_fm_a] :
( ( ord_le582293785304634267c_fm_a @ A3 @ B )
=> ( ( member293847127289681564c_fm_a @ C @ A3 )
=> ( member293847127289681564c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_333_subsetD,axiom,
! [A3: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_334_subsetD,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a,C: produc2203834439629243303c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A3 @ B )
=> ( ( member5029746173879296720c_fm_a @ C @ A3 )
=> ( member5029746173879296720c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_335_subsetD,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( member536094252920883875c_fm_a @ C @ A3 )
=> ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_336_subsetD,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% subsetD
thf(fact_337_in__mono,axiom,
! [A3: set_li4204741992506657632c_fm_a,B: set_li4204741992506657632c_fm_a,X2: list_s580375451141968640c_fm_a] :
( ( ord_le2512102717711966976c_fm_a @ A3 @ B )
=> ( ( member2328277852289759785c_fm_a @ X2 @ A3 )
=> ( member2328277852289759785c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_338_in__mono,axiom,
! [A3: set_se8609989359752948410c_fm_a,B: set_se8609989359752948410c_fm_a,X2: set_se5208064806568342746c_fm_a] :
( ( ord_le4042760284959298138c_fm_a @ A3 @ B )
=> ( ( member3183810400728440963c_fm_a @ X2 @ A3 )
=> ( member3183810400728440963c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_339_in__mono,axiom,
! [A3: set_Ep6479678258574082331c_fm_a,B: set_Ep6479678258574082331c_fm_a,X2: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le8649530489803071867c_fm_a @ A3 @ B )
=> ( ( member2634950385374053500c_fm_a @ X2 @ A3 )
=> ( member2634950385374053500c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_340_in__mono,axiom,
! [A3: set_se6091493367310676795c_fm_a,B: set_se6091493367310676795c_fm_a,X2: set_Epistemic_fm_a > epistemic_fm_a] :
( ( ord_le582293785304634267c_fm_a @ A3 @ B )
=> ( ( member293847127289681564c_fm_a @ X2 @ A3 )
=> ( member293847127289681564c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_341_in__mono,axiom,
! [A3: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_342_in__mono,axiom,
! [A3: set_Pr8948481099588399239c_fm_a,B: set_Pr8948481099588399239c_fm_a,X2: produc2203834439629243303c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ A3 @ B )
=> ( ( member5029746173879296720c_fm_a @ X2 @ A3 )
=> ( member5029746173879296720c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_343_in__mono,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_344_in__mono,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_345_exactly__one__in__maximal,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( member5360357140639596479c_fm_a @ P2 @ V )
= ( ~ ( member5360357140639596479c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_346_exactly__one__in__maximal,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,P2: episte740340785640729014c_fm_a] :
( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( member1118567890201975263c_fm_a @ P2 @ V )
= ( ~ ( member1118567890201975263c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_347_exactly__one__in__maximal,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P2 @ V )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_348_Ax5_Ocases,axiom,
! [A: episte5999154429349999510c_fm_a] :
( ( episte3174827985664174276c_fm_a @ A )
=> ~ ! [I2: set_Epistemic_fm_a,P3: episte5999154429349999510c_fm_a] :
( A
!= ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ P3 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ P3 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_349_Ax5_Ocases,axiom,
! [A: episte740340785640729014c_fm_a] :
( ( episte7251237963447243492c_fm_a @ A )
=> ~ ! [I2: epistemic_fm_a,P3: episte740340785640729014c_fm_a] :
( A
!= ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ P3 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ P3 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_350_Ax5_Ocases,axiom,
! [A: epistemic_fm_a] :
( ( epistemic_Ax5_a @ A )
=> ~ ! [I2: a,P3: epistemic_fm_a] :
( A
!= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_351_Ax5_Osimps,axiom,
( episte3174827985664174276c_fm_a
= ( ^ [A2: episte5999154429349999510c_fm_a] :
? [I3: set_Epistemic_fm_a,P: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ P @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ P @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_352_Ax5_Osimps,axiom,
( episte7251237963447243492c_fm_a
= ( ^ [A2: episte740340785640729014c_fm_a] :
? [I3: epistemic_fm_a,P: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ P @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ P @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_353_Ax5_Osimps,axiom,
( epistemic_Ax5_a
= ( ^ [A2: epistemic_fm_a] :
? [I3: a,P: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_354_Ax4__transitive,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,W: set_Ep9038437136950510070c_fm_a,I: set_Epistemic_fm_a,U: set_Ep9038437136950510070c_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte3818710727560342277c_fm_a @ A3 )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( member5191902678313347615c_fm_a @ W
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ V ) ) ) ) )
=> ( ( member5191902678313347615c_fm_a @ U
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ W ) ) ) ) )
=> ( member5191902678313347615c_fm_a @ U
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_355_Ax4__transitive,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,W: set_Ep8485816078964734998c_fm_a,I: epistemic_fm_a,U: set_Ep8485816078964734998c_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4964217022932375845c_fm_a @ A3 )
=> ( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( member7847018790254331967c_fm_a @ W
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ V ) ) ) ) )
=> ( ( member7847018790254331967c_fm_a @ U
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ W ) ) ) ) )
=> ( member7847018790254331967c_fm_a @ U
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_356_Ax4__transitive,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W: set_Epistemic_fm_a,I: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A3 )
=> ( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( member536094252920883875c_fm_a @ W
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ V ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ W ) ) ) ) )
=> ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_357_AxT__reflexive,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,I: set_Epistemic_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte1661207060592517861c_fm_a @ A3 )
=> ( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( member5191902678313347615c_fm_a @ V
@ ( collec1302195411925791841c_fm_a
@ ( ord_le4874308747680037270c_fm_a
@ ( collec3748559304425083393c_fm_a
@ ^ [P: episte5999154429349999510c_fm_a] : ( member5360357140639596479c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_358_AxT__reflexive,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,I: epistemic_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4361910824569934085c_fm_a @ A3 )
=> ( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( member7847018790254331967c_fm_a @ V
@ ( collec6235220954039818881c_fm_a
@ ( ord_le3671305448118956982c_fm_a
@ ( collec7281052159120012321c_fm_a
@ ^ [P: episte740340785640729014c_fm_a] : ( member1118567890201975263c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_359_AxT__reflexive,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A3 )
=> ( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_360_pred__subset__eq,axiom,
! [R: set_li4204741992506657632c_fm_a,S: set_li4204741992506657632c_fm_a] :
( ( ord_le8346386785241846749fm_a_o
@ ^ [X3: list_s580375451141968640c_fm_a] : ( member2328277852289759785c_fm_a @ X3 @ R )
@ ^ [X3: list_s580375451141968640c_fm_a] : ( member2328277852289759785c_fm_a @ X3 @ S ) )
= ( ord_le2512102717711966976c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_361_pred__subset__eq,axiom,
! [R: set_se8609989359752948410c_fm_a,S: set_se8609989359752948410c_fm_a] :
( ( ord_le8279579953234471683fm_a_o
@ ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ R )
@ ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ S ) )
= ( ord_le4042760284959298138c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_362_pred__subset__eq,axiom,
! [R: set_Ep6479678258574082331c_fm_a,S: set_Ep6479678258574082331c_fm_a] :
( ( ord_le914026578752367818fm_a_o
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] : ( member2634950385374053500c_fm_a @ X3 @ R )
@ ^ [X3: epistemic_fm_a > epistemic_fm_a] : ( member2634950385374053500c_fm_a @ X3 @ S ) )
= ( ord_le8649530489803071867c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_363_pred__subset__eq,axiom,
! [R: set_se6091493367310676795c_fm_a,S: set_se6091493367310676795c_fm_a] :
( ( ord_le8669121985476793002fm_a_o
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] : ( member293847127289681564c_fm_a @ X3 @ R )
@ ^ [X3: set_Epistemic_fm_a > epistemic_fm_a] : ( member293847127289681564c_fm_a @ X3 @ S ) )
= ( ord_le582293785304634267c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_364_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_365_pred__subset__eq,axiom,
! [R: set_Pr8948481099588399239c_fm_a,S: set_Pr8948481099588399239c_fm_a] :
( ( ord_le4942933042592423798fm_a_o
@ ^ [X3: produc2203834439629243303c_fm_a] : ( member5029746173879296720c_fm_a @ X3 @ R )
@ ^ [X3: produc2203834439629243303c_fm_a] : ( member5029746173879296720c_fm_a @ X3 @ S ) )
= ( ord_le8222437384497874983c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_366_pred__subset__eq,axiom,
! [R: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( ord_le1568123225352354019fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ R )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ S ) )
= ( ord_le7112219575281605754c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_367_pred__subset__eq,axiom,
! [R: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ R )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ S ) )
= ( ord_le3275665582123262618c_fm_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_368_MCS_Omaximal_Ocong,axiom,
maxima7173642526191862697c_fm_a = maxima7173642526191862697c_fm_a ).
% MCS.maximal.cong
thf(fact_369_MCS_Omaximal_Ocong,axiom,
maxima3264069584562520521c_fm_a = maxima3264069584562520521c_fm_a ).
% MCS.maximal.cong
thf(fact_370_K5__L,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte3174827985664174276c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ) ).
% K5_L
thf(fact_371_K5__L,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte7251237963447243492c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ) ).
% K5_L
thf(fact_372_K5__L,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P2 ) ) ) ) ).
% K5_L
thf(fact_373_Greatest__equality,axiom,
! [P4: set_se5208064806568342746c_fm_a > $o,X2: set_se5208064806568342746c_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y2: set_se5208064806568342746c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ Y2 @ X2 ) )
=> ( ( order_8284996552903640883c_fm_a @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_374_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X2 ) )
=> ( ( order_253494837916242250fm_a_o @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_375_Greatest__equality,axiom,
! [P4: ( set_Epistemic_fm_a > $o ) > $o,X2: set_Epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le1568123225352354019fm_a_o @ Y2 @ X2 ) )
=> ( ( order_3524904582381003626fm_a_o @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_376_Greatest__equality,axiom,
! [P4: ( $o > set_Epistemic_fm_a ) > $o,X2: $o > set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y2: $o > set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le1382570192702674727c_fm_a @ Y2 @ X2 ) )
=> ( ( order_3339351549731324334c_fm_a @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_377_Greatest__equality,axiom,
! [P4: set_Pr8948481099588399239c_fm_a > $o,X2: set_Pr8948481099588399239c_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Pr8948481099588399239c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le8222437384497874983c_fm_a @ Y2 @ X2 ) )
=> ( ( order_8796109153055160928c_fm_a @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_378_Greatest__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X2 ) )
=> ( ( order_4585748725732241747c_fm_a @ P4 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_379_GreatestI2__order,axiom,
! [P4: set_se5208064806568342746c_fm_a > $o,X2: set_se5208064806568342746c_fm_a,Q2: set_se5208064806568342746c_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: set_se5208064806568342746c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le7112219575281605754c_fm_a @ Y2 @ X2 ) )
=> ( ! [X: set_se5208064806568342746c_fm_a] :
( ( P4 @ X )
=> ( ! [Y5: set_se5208064806568342746c_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le7112219575281605754c_fm_a @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_8284996552903640883c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_380_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X2: epistemic_fm_a > $o,Q2: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X2 ) )
=> ( ! [X: epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_253494837916242250fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_381_GreatestI2__order,axiom,
! [P4: ( set_Epistemic_fm_a > $o ) > $o,X2: set_Epistemic_fm_a > $o,Q2: ( set_Epistemic_fm_a > $o ) > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le1568123225352354019fm_a_o @ Y2 @ X2 ) )
=> ( ! [X: set_Epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ! [Y5: set_Epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le1568123225352354019fm_a_o @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_3524904582381003626fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_382_GreatestI2__order,axiom,
! [P4: ( $o > set_Epistemic_fm_a ) > $o,X2: $o > set_Epistemic_fm_a,Q2: ( $o > set_Epistemic_fm_a ) > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: $o > set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le1382570192702674727c_fm_a @ Y2 @ X2 ) )
=> ( ! [X: $o > set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( ! [Y5: $o > set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le1382570192702674727c_fm_a @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_3339351549731324334c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_383_GreatestI2__order,axiom,
! [P4: set_Pr8948481099588399239c_fm_a > $o,X2: set_Pr8948481099588399239c_fm_a,Q2: set_Pr8948481099588399239c_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Pr8948481099588399239c_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le8222437384497874983c_fm_a @ Y2 @ X2 ) )
=> ( ! [X: set_Pr8948481099588399239c_fm_a] :
( ( P4 @ X )
=> ( ! [Y5: set_Pr8948481099588399239c_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le8222437384497874983c_fm_a @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_8796109153055160928c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_384_GreatestI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a,Q2: set_Epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X2 ) )
=> ( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_4585748725732241747c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_385_le__rel__bool__arg__iff,axiom,
( ord_le3474282128009451783c_fm_a
= ( ^ [X5: $o > set_se5208064806568342746c_fm_a,Y6: $o > set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le7112219575281605754c_fm_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_386_le__rel__bool__arg__iff,axiom,
( ord_le2986116216980733324fm_a_o
= ( ^ [X5: $o > epistemic_fm_a > $o,Y6: $o > epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le4043730696559282883fm_a_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_387_le__rel__bool__arg__iff,axiom,
( ord_le3429829824310240172fm_a_o
= ( ^ [X5: $o > set_Epistemic_fm_a > $o,Y6: $o > set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1568123225352354019fm_a_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_388_le__rel__bool__arg__iff,axiom,
( ord_le7787317817306898416c_fm_a
= ( ^ [X5: $o > $o > set_Epistemic_fm_a,Y6: $o > $o > set_Epistemic_fm_a] :
( ( ord_le1382570192702674727c_fm_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1382570192702674727c_fm_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_389_le__rel__bool__arg__iff,axiom,
( ord_le4963706944741856564c_fm_a
= ( ^ [X5: $o > set_Pr8948481099588399239c_fm_a,Y6: $o > set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8222437384497874983c_fm_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_390_le__rel__bool__arg__iff,axiom,
( ord_le1382570192702674727c_fm_a
= ( ^ [X5: $o > set_Epistemic_fm_a,Y6: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3275665582123262618c_fm_a @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_391_verit__comp__simplify1_I2_J,axiom,
! [A: set_se5208064806568342746c_fm_a] : ( ord_le7112219575281605754c_fm_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_392_verit__comp__simplify1_I2_J,axiom,
! [A: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_393_verit__comp__simplify1_I2_J,axiom,
! [A: set_Epistemic_fm_a > $o] : ( ord_le1568123225352354019fm_a_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_394_verit__comp__simplify1_I2_J,axiom,
! [A: $o > set_Epistemic_fm_a] : ( ord_le1382570192702674727c_fm_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_395_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr8948481099588399239c_fm_a] : ( ord_le8222437384497874983c_fm_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_396_verit__comp__simplify1_I2_J,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_397_Ax4_Ocases,axiom,
! [A: episte5999154429349999510c_fm_a] :
( ( episte3818710727560342277c_fm_a @ A )
=> ~ ! [I2: set_Epistemic_fm_a,P3: episte5999154429349999510c_fm_a] :
( A
!= ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ P3 ) @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte3013695276341575662c_fm_a @ I2 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_398_Ax4_Ocases,axiom,
! [A: episte740340785640729014c_fm_a] :
( ( episte4964217022932375845c_fm_a @ A )
=> ~ ! [I2: epistemic_fm_a,P3: episte740340785640729014c_fm_a] :
( A
!= ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ P3 ) @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte5657488632024175118c_fm_a @ I2 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_399_Ax4_Ocases,axiom,
! [A: epistemic_fm_a] :
( ( epistemic_Ax4_a @ A )
=> ~ ! [I2: a,P3: epistemic_fm_a] :
( A
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P3 ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_400_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q2 ) ) ).
% predicate1I
thf(fact_401_predicate1I,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_le1568123225352354019fm_a_o @ P4 @ Q2 ) ) ).
% predicate1I
thf(fact_402_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o,X2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q2 )
=> ( ( P4 @ X2 )
=> ( Q2 @ X2 ) ) ) ).
% predicate1D
thf(fact_403_predicate1D,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a] :
( ( ord_le1568123225352354019fm_a_o @ P4 @ Q2 )
=> ( ( P4 @ X2 )
=> ( Q2 @ X2 ) ) ) ).
% predicate1D
thf(fact_404_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X2: epistemic_fm_a,Q2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q2 )
=> ( Q2 @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_405_rev__predicate1D,axiom,
! [P4: set_Epistemic_fm_a > $o,X2: set_Epistemic_fm_a,Q2: set_Epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ( ord_le1568123225352354019fm_a_o @ P4 @ Q2 )
=> ( Q2 @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_406_Ax,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a] :
( ( A3 @ P2 )
=> ( epistemic_AK_a @ A3 @ P2 ) ) ).
% Ax
thf(fact_407_S5_H__S5,axiom,
! [P2: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
@ P2 )
=> ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ P2 ) ) ).
% S5'_S5
thf(fact_408_S5__S5_H,axiom,
! [P2: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ P2 )
=> ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
@ P2 ) ) ).
% S5_S5'
thf(fact_409_K__trans,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,R2: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ Q @ R2 ) @ ( episte5200161025749276397c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_trans
thf(fact_410_K__trans,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,R2: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ Q ) @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ Q @ R2 ) @ ( episte260752218777527565c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_trans
thf(fact_411_K__trans,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a,R2: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P2 @ Q ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ Q @ R2 ) @ ( epistemic_Imp_a @ P2 @ R2 ) ) ) ) ).
% K_trans
thf(fact_412_R1,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ P2 )
=> ( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ Q ) ) ) ).
% R1
thf(fact_413_R1,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ P2 )
=> ( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ Q ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ Q ) ) ) ).
% R1
thf(fact_414_R1,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ P2 )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q ) )
=> ( epistemic_AK_a @ A3 @ Q ) ) ) ).
% R1
thf(fact_415_K__imp__trans_H,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,Q: episte5999154429349999510c_fm_a,R2: episte5999154429349999510c_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ Q @ R2 ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) @ ( episte5200161025749276397c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_416_K__imp__trans_H,axiom,
! [A3: episte740340785640729014c_fm_a > $o,Q: episte740340785640729014c_fm_a,R2: episte740340785640729014c_fm_a,P2: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ Q @ R2 ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ Q ) @ ( episte260752218777527565c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_417_K__imp__trans_H,axiom,
! [A3: epistemic_fm_a > $o,Q: epistemic_fm_a,R2: epistemic_fm_a,P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ Q @ R2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P2 @ Q ) @ ( epistemic_Imp_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_418_K__imp__trans,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,R2: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) )
=> ( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ Q @ R2 ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_419_K__imp__trans,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,R2: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ Q ) )
=> ( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ Q @ R2 ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_420_K__imp__trans,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a,R2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ Q @ R2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_421_R2,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,I: set_Epistemic_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ P2 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ).
% R2
thf(fact_422_R2,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,I: epistemic_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ P2 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ).
% R2
thf(fact_423_R2,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A3 @ P2 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_K_a @ I @ P2 ) ) ) ).
% R2
thf(fact_424_A1,axiom,
! [P2: epistemic_fm_a,A3: epistemic_fm_a > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P2 )
=> ( epistemic_AK_a @ A3 @ P2 ) ) ).
% A1
thf(fact_425_T__L,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,I: set_Epistemic_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte1661207060592517861c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ).
% T_L
thf(fact_426_T__L,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,I: epistemic_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4361910824569934085c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ).
% T_L
thf(fact_427_T__L,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% T_L
thf(fact_428_K4__L,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte3818710727560342277c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ).
% K4_L
thf(fact_429_K4__L,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4964217022932375845c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ).
% K4_L
thf(fact_430_K4__L,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% K4_L
thf(fact_431_S5_H__4,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte1661207060592517861c_fm_a @ A3 )
=> ( ( ord_le7632454964716729543fm_a_o @ episte3174827985664174276c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ) ) ) ).
% S5'_4
thf(fact_432_S5_H__4,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4361910824569934085c_fm_a @ A3 )
=> ( ( ord_le7240545311762076327fm_a_o @ episte7251237963447243492c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ) ) ) ).
% S5'_4
thf(fact_433_S5_H__4,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A3 )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P2 ) ) ) ) ) ) ).
% S5'_4
thf(fact_434_K__map,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,I: set_Epistemic_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) ) ) ).
% K_map
thf(fact_435_K__map,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,I: epistemic_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ Q ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) ) ) ).
% K_map
thf(fact_436_K__map,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_map
thf(fact_437_K__A2_H,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) ) ) ).
% K_A2'
thf(fact_438_K__A2_H,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ Q ) ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) ) ) ).
% K_A2'
thf(fact_439_K__A2_H,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ Q ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_A2'
thf(fact_440_deriv__in__maximal,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( epistemic_AK_a @ A3 @ P2 )
=> ( member6642669571620171971c_fm_a @ P2 @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_441_S5_H__B,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,I: set_Epistemic_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte1661207060592517861c_fm_a @ A3 )
=> ( ( ord_le7632454964716729543fm_a_o @ episte3174827985664174276c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P2 @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_442_S5_H__B,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,I: epistemic_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte4361910824569934085c_fm_a @ A3 )
=> ( ( ord_le7240545311762076327fm_a_o @ episte7251237963447243492c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P2 @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_443_S5_H__B,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A3 )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_444_KB4__5,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( ord_le7632454964716729543fm_a_o @ episte4027724377868766071c_fm_a @ A3 )
=> ( ( ord_le7632454964716729543fm_a_o @ episte3818710727560342277c_fm_a @ A3 )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_445_KB4__5,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] :
( ( ord_le7240545311762076327fm_a_o @ episte89022042721419671c_fm_a @ A3 )
=> ( ( ord_le7240545311762076327fm_a_o @ episte4964217022932375845c_fm_a @ A3 )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_446_KB4__5,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A3 )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A3 )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_447_AxT_Ointros,axiom,
! [I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] : ( episte1661207060592517861c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ P2 ) ) ).
% AxT.intros
thf(fact_448_AxT_Ointros,axiom,
! [I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] : ( episte4361910824569934085c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ P2 ) ) ).
% AxT.intros
thf(fact_449_AxT_Ointros,axiom,
! [I: a,P2: epistemic_fm_a] : ( epistemic_AxT_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ P2 ) ) ).
% AxT.intros
thf(fact_450_AxT_Osimps,axiom,
( episte1661207060592517861c_fm_a
= ( ^ [A2: episte5999154429349999510c_fm_a] :
? [I3: set_Epistemic_fm_a,P: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ P ) @ P ) ) ) ) ).
% AxT.simps
thf(fact_451_AxT_Osimps,axiom,
( episte4361910824569934085c_fm_a
= ( ^ [A2: episte740340785640729014c_fm_a] :
? [I3: epistemic_fm_a,P: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ P ) @ P ) ) ) ) ).
% AxT.simps
thf(fact_452_AxT_Osimps,axiom,
( epistemic_AxT_a
= ( ^ [A2: epistemic_fm_a] :
? [I3: a,P: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ P ) ) ) ) ).
% AxT.simps
thf(fact_453_AxT_Ocases,axiom,
! [A: episte5999154429349999510c_fm_a] :
( ( episte1661207060592517861c_fm_a @ A )
=> ~ ! [I2: set_Epistemic_fm_a,P3: episte5999154429349999510c_fm_a] :
( A
!= ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_454_AxT_Ocases,axiom,
! [A: episte740340785640729014c_fm_a] :
( ( episte4361910824569934085c_fm_a @ A )
=> ~ ! [I2: epistemic_fm_a,P3: episte740340785640729014c_fm_a] :
( A
!= ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_455_AxT_Ocases,axiom,
! [A: epistemic_fm_a] :
( ( epistemic_AxT_a @ A )
=> ~ ! [I2: a,P3: epistemic_fm_a] :
( A
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_456_K__L__dual,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ episte4474038146033622457c_fm_a ) @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ).
% K_L_dual
thf(fact_457_K__L__dual,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ episte5073044243917183961c_fm_a ) @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ).
% K_L_dual
thf(fact_458_K__L__dual,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P2 ) ) ) ).
% K_L_dual
thf(fact_459_K__LK,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ episte4474038146033622457c_fm_a ) ) ) ).
% K_LK
thf(fact_460_K__LK,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ episte5073044243917183961c_fm_a ) ) ) ).
% K_LK
thf(fact_461_K__LK,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) ) ) ).
% K_LK
thf(fact_462_Ax4_Ointros,axiom,
! [I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a] : ( episte3818710727560342277c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte3013695276341575662c_fm_a @ I @ P2 ) ) ) ) ).
% Ax4.intros
thf(fact_463_Ax4_Ointros,axiom,
! [I: epistemic_fm_a,P2: episte740340785640729014c_fm_a] : ( episte4964217022932375845c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte5657488632024175118c_fm_a @ I @ P2 ) ) ) ) ).
% Ax4.intros
thf(fact_464_Ax4_Ointros,axiom,
! [I: a,P2: epistemic_fm_a] : ( epistemic_Ax4_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P2 ) ) ) ) ).
% Ax4.intros
thf(fact_465_Ax4_Osimps,axiom,
( episte3818710727560342277c_fm_a
= ( ^ [A2: episte5999154429349999510c_fm_a] :
? [I3: set_Epistemic_fm_a,P: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ P ) @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte3013695276341575662c_fm_a @ I3 @ P ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_466_Ax4_Osimps,axiom,
( episte4964217022932375845c_fm_a
= ( ^ [A2: episte740340785640729014c_fm_a] :
? [I3: epistemic_fm_a,P: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ P ) @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte5657488632024175118c_fm_a @ I3 @ P ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_467_Ax4_Osimps,axiom,
( epistemic_Ax4_a
= ( ^ [A2: epistemic_fm_a] :
? [I3: a,P: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_468_AK_Osimps,axiom,
( episte4141561593244286310c_fm_a
= ( ^ [A4: episte5999154429349999510c_fm_a > $o,A2: episte5999154429349999510c_fm_a] :
( ? [P: episte5999154429349999510c_fm_a] :
( ( A2 = P )
& ! [G3: list_char > $o,H4: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G3 @ H4 @ P ) )
| ? [I3: set_Epistemic_fm_a,P: episte5999154429349999510c_fm_a,Q3: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I3 @ P ) @ ( episte3013695276341575662c_fm_a @ I3 @ ( episte5200161025749276397c_fm_a @ P @ Q3 ) ) ) @ ( episte3013695276341575662c_fm_a @ I3 @ Q3 ) ) )
| ? [P: episte5999154429349999510c_fm_a] :
( ( A2 = P )
& ( A4 @ P ) )
| ? [P: episte5999154429349999510c_fm_a,Q3: episte5999154429349999510c_fm_a] :
( ( A2 = Q3 )
& ( episte4141561593244286310c_fm_a @ A4 @ P )
& ( episte4141561593244286310c_fm_a @ A4 @ ( episte5200161025749276397c_fm_a @ P @ Q3 ) ) )
| ? [P: episte5999154429349999510c_fm_a,I3: set_Epistemic_fm_a] :
( ( A2
= ( episte3013695276341575662c_fm_a @ I3 @ P ) )
& ( episte4141561593244286310c_fm_a @ A4 @ P ) ) ) ) ) ).
% AK.simps
thf(fact_469_AK_Osimps,axiom,
( episte3601224840237526918c_fm_a
= ( ^ [A4: episte740340785640729014c_fm_a > $o,A2: episte740340785640729014c_fm_a] :
( ? [P: episte740340785640729014c_fm_a] :
( ( A2 = P )
& ! [G3: list_char > $o,H4: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G3 @ H4 @ P ) )
| ? [I3: epistemic_fm_a,P: episte740340785640729014c_fm_a,Q3: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I3 @ P ) @ ( episte5657488632024175118c_fm_a @ I3 @ ( episte260752218777527565c_fm_a @ P @ Q3 ) ) ) @ ( episte5657488632024175118c_fm_a @ I3 @ Q3 ) ) )
| ? [P: episte740340785640729014c_fm_a] :
( ( A2 = P )
& ( A4 @ P ) )
| ? [P: episte740340785640729014c_fm_a,Q3: episte740340785640729014c_fm_a] :
( ( A2 = Q3 )
& ( episte3601224840237526918c_fm_a @ A4 @ P )
& ( episte3601224840237526918c_fm_a @ A4 @ ( episte260752218777527565c_fm_a @ P @ Q3 ) ) )
| ? [P: episte740340785640729014c_fm_a,I3: epistemic_fm_a] :
( ( A2
= ( episte5657488632024175118c_fm_a @ I3 @ P ) )
& ( episte3601224840237526918c_fm_a @ A4 @ P ) ) ) ) ) ).
% AK.simps
thf(fact_470_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A4: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ? [P: epistemic_fm_a] :
( ( A2 = P )
& ! [G3: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H4 @ P ) )
| ? [I3: a,P: epistemic_fm_a,Q3: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ Q3 ) ) ) @ ( epistemic_K_a @ I3 @ Q3 ) ) )
| ? [P: epistemic_fm_a] :
( ( A2 = P )
& ( A4 @ P ) )
| ? [P: epistemic_fm_a,Q3: epistemic_fm_a] :
( ( A2 = Q3 )
& ( epistemic_AK_a @ A4 @ P )
& ( epistemic_AK_a @ A4 @ ( epistemic_Imp_a @ P @ Q3 ) ) )
| ? [P: epistemic_fm_a,I3: a] :
( ( A2
= ( epistemic_K_a @ I3 @ P ) )
& ( epistemic_AK_a @ A4 @ P ) ) ) ) ) ).
% AK.simps
thf(fact_471_AK_Ocases,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,A: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ A )
=> ( ~ ! [G: list_char > $o,H: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G @ H @ A )
=> ( ! [I2: set_Epistemic_fm_a,P3: episte5999154429349999510c_fm_a,Q4: episte5999154429349999510c_fm_a] :
( A
!= ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I2 @ P3 ) @ ( episte3013695276341575662c_fm_a @ I2 @ ( episte5200161025749276397c_fm_a @ P3 @ Q4 ) ) ) @ ( episte3013695276341575662c_fm_a @ I2 @ Q4 ) ) )
=> ( ~ ( A3 @ A )
=> ( ! [P3: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ P3 )
=> ~ ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ P3 @ A ) ) )
=> ~ ! [P3: episte5999154429349999510c_fm_a] :
( ? [I2: set_Epistemic_fm_a] :
( A
= ( episte3013695276341575662c_fm_a @ I2 @ P3 ) )
=> ~ ( episte4141561593244286310c_fm_a @ A3 @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_472_AK_Ocases,axiom,
! [A3: episte740340785640729014c_fm_a > $o,A: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ A )
=> ( ~ ! [G: list_char > $o,H: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G @ H @ A )
=> ( ! [I2: epistemic_fm_a,P3: episte740340785640729014c_fm_a,Q4: episte740340785640729014c_fm_a] :
( A
!= ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I2 @ P3 ) @ ( episte5657488632024175118c_fm_a @ I2 @ ( episte260752218777527565c_fm_a @ P3 @ Q4 ) ) ) @ ( episte5657488632024175118c_fm_a @ I2 @ Q4 ) ) )
=> ( ~ ( A3 @ A )
=> ( ! [P3: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ P3 )
=> ~ ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ P3 @ A ) ) )
=> ~ ! [P3: episte740340785640729014c_fm_a] :
( ? [I2: epistemic_fm_a] :
( A
= ( episte5657488632024175118c_fm_a @ I2 @ P3 ) )
=> ~ ( episte3601224840237526918c_fm_a @ A3 @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_473_AK_Ocases,axiom,
! [A3: epistemic_fm_a > $o,A: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ A )
=> ( ~ ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ A )
=> ( ! [I2: a,P3: epistemic_fm_a,Q4: epistemic_fm_a] :
( A
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P3 ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P3 @ Q4 ) ) ) @ ( epistemic_K_a @ I2 @ Q4 ) ) )
=> ( ~ ( A3 @ A )
=> ( ! [P3: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ P3 )
=> ~ ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P3 @ A ) ) )
=> ~ ! [P3: epistemic_fm_a] :
( ? [I2: a] :
( A
= ( epistemic_K_a @ I2 @ P3 ) )
=> ~ ( epistemic_AK_a @ A3 @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_474_K__thm,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ Q @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) @ episte4474038146033622457c_fm_a ) ) @ episte4474038146033622457c_fm_a ) ) ) ).
% K_thm
thf(fact_475_K__thm,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ Q @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) @ episte5073044243917183961c_fm_a ) ) @ episte5073044243917183961c_fm_a ) ) ) ).
% K_thm
thf(fact_476_K__thm,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Con_a @ P2 @ Q ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ).
% K_thm
thf(fact_477_S5__S5_H__assms,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_478_Powp__mono,axiom,
! [A3: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A3 @ B )
=> ( ord_le1568123225352354019fm_a_o @ ( powp_Epistemic_fm_a @ A3 ) @ ( powp_Epistemic_fm_a @ B ) ) ) ).
% Powp_mono
thf(fact_479_Powp__mono,axiom,
! [A3: set_Epistemic_fm_a > $o,B: set_Epistemic_fm_a > $o] :
( ( ord_le1568123225352354019fm_a_o @ A3 @ B )
=> ( ord_le8279579953234471683fm_a_o @ ( powp_s8172341239314103577c_fm_a @ A3 ) @ ( powp_s8172341239314103577c_fm_a @ B ) ) ) ).
% Powp_mono
thf(fact_480_A2,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) ) ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) ) ).
% A2
thf(fact_481_A2,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte260752218777527565c_fm_a @ P2 @ Q ) ) ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) ) ).
% A2
thf(fact_482_A2,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% A2
thf(fact_483_K__conj__imply__factor,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,R2: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) @ R2 ) @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_484_K__conj__imply__factor,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,R2: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) @ R2 ) @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_485_K__conj__imply__factor,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a,R2: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ Q ) ) @ R2 ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P2 @ Q ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_486_K__conjunction__out,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) @ ( episte3013695276341575662c_fm_a @ I @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_487_K__conjunction__out,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) @ ( episte5657488632024175118c_fm_a @ I @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_488_K__conjunction__out,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ Q ) ) @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P2 @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_489_K__conjunction__in,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,I: set_Epistemic_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) ) @ ( episte8719284489194433015c_fm_a @ ( episte3013695276341575662c_fm_a @ I @ P2 ) @ ( episte3013695276341575662c_fm_a @ I @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_490_K__conjunction__in,axiom,
! [A3: episte740340785640729014c_fm_a > $o,I: epistemic_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) ) @ ( episte3685526487207141399c_fm_a @ ( episte5657488632024175118c_fm_a @ I @ P2 ) @ ( episte5657488632024175118c_fm_a @ I @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_491_K__conjunction__in,axiom,
! [A3: epistemic_fm_a > $o,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P2 @ Q ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_492_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( ( epistemic_Con_a @ X41 @ X42 )
= ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_493_fm_Oinject_I3_J,axiom,
! [X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a,Y41: episte5999154429349999510c_fm_a,Y42: episte5999154429349999510c_fm_a] :
( ( ( episte8719284489194433015c_fm_a @ X41 @ X42 )
= ( episte8719284489194433015c_fm_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_494_fm_Oinject_I3_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a,Y41: episte740340785640729014c_fm_a,Y42: episte740340785640729014c_fm_a] :
( ( ( episte3685526487207141399c_fm_a @ X41 @ X42 )
= ( episte3685526487207141399c_fm_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_495_K__imply__weaken,axiom,
! [A3: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,Ps2: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Ps2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_496_fm_Odistinct_I25_J,axiom,
! [X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a,X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a] :
( ( episte8719284489194433015c_fm_a @ X41 @ X42 )
!= ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_497_fm_Odistinct_I25_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a,X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( episte3685526487207141399c_fm_a @ X41 @ X42 )
!= ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_498_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_499_fm_Odistinct_I27_J,axiom,
! [X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a,X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] :
( ( episte8719284489194433015c_fm_a @ X41 @ X42 )
!= ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_500_fm_Odistinct_I27_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a,X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( ( episte3685526487207141399c_fm_a @ X41 @ X42 )
!= ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_501_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_502_fm_Odistinct_I5_J,axiom,
! [X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a] :
( episte4474038146033622457c_fm_a
!= ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_503_fm_Odistinct_I5_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( episte5073044243917183961c_fm_a
!= ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_504_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_505_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte6853238276675109012c_fm_a @ G2 @ H2 @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) )
= ( ( episte6853238276675109012c_fm_a @ G2 @ H2 @ P2 )
& ( episte6853238276675109012c_fm_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_506_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte6292435812075825844c_fm_a @ G2 @ H2 @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) )
= ( ( episte6292435812075825844c_fm_a @ G2 @ H2 @ P2 )
& ( episte6292435812075825844c_fm_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_507_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Con_a @ P2 @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P2 )
& ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_508_tautology__imply__superset,axiom,
! [Ps: list_E2761013248121217820c_fm_a,Qs2: list_E2761013248121217820c_fm_a,R2: episte5999154429349999510c_fm_a] :
( ( ord_le4874308747680037270c_fm_a @ ( set_Ep4008781527546218027c_fm_a @ Ps ) @ ( set_Ep4008781527546218027c_fm_a @ Qs2 ) )
=> ! [G: list_char > $o,H: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G @ H @ ( episte5200161025749276397c_fm_a @ ( episte7011297269853982921c_fm_a @ Ps @ R2 ) @ ( episte7011297269853982921c_fm_a @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_509_tautology__imply__superset,axiom,
! [Ps: list_E5775909407618282812c_fm_a,Qs2: list_E5775909407618282812c_fm_a,R2: episte740340785640729014c_fm_a] :
( ( ord_le3671305448118956982c_fm_a @ ( set_Ep2297100812554489931c_fm_a @ Ps ) @ ( set_Ep2297100812554489931c_fm_a @ Qs2 ) )
=> ! [G: list_char > $o,H: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G @ H @ ( episte260752218777527565c_fm_a @ ( episte1272216372155829993c_fm_a @ Ps @ R2 ) @ ( episte1272216372155829993c_fm_a @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_510_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs2: list_Epistemic_fm_a,R2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs2 ) )
=> ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R2 ) @ ( epistemic_imply_a @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_511_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A4: epistemic_fm_a > $o,S3: set_Epistemic_fm_a] :
~ ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ S3 )
& ( epistemic_AK_a @ A4 @ ( epistemic_imply_a @ Qs @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_512_K__right__mp,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,Ps: list_E2761013248121217820c_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ Ps @ P2 ) )
=> ( ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ Ps @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_513_K__right__mp,axiom,
! [A3: episte740340785640729014c_fm_a > $o,Ps: list_E5775909407618282812c_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ Ps @ P2 ) )
=> ( ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ Ps @ ( episte260752218777527565c_fm_a @ P2 @ Q ) ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_514_K__right__mp,axiom,
! [A3: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ ( epistemic_Imp_a @ P2 @ Q ) ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_515_K__imply__multi,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,A: episte5999154429349999510c_fm_a,B3: episte5999154429349999510c_fm_a,C: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ A @ B3 ) )
=> ( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ A @ C ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ A @ ( episte8719284489194433015c_fm_a @ B3 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_516_K__imply__multi,axiom,
! [A3: episte740340785640729014c_fm_a > $o,A: episte740340785640729014c_fm_a,B3: episte740340785640729014c_fm_a,C: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ A @ B3 ) )
=> ( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ A @ C ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ A @ ( episte3685526487207141399c_fm_a @ B3 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_517_K__imply__multi,axiom,
! [A3: epistemic_fm_a > $o,A: epistemic_fm_a,B3: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ A @ B3 ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ A @ C ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ A @ ( epistemic_Con_a @ B3 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_518_K__multi__imply,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,A: episte5999154429349999510c_fm_a,B3: episte5999154429349999510c_fm_a,C: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ A @ ( episte5200161025749276397c_fm_a @ B3 @ C ) ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( episte8719284489194433015c_fm_a @ A @ B3 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_519_K__multi__imply,axiom,
! [A3: episte740340785640729014c_fm_a > $o,A: episte740340785640729014c_fm_a,B3: episte740340785640729014c_fm_a,C: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ A @ ( episte260752218777527565c_fm_a @ B3 @ C ) ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( episte3685526487207141399c_fm_a @ A @ B3 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_520_K__multi__imply,axiom,
! [A3: epistemic_fm_a > $o,A: epistemic_fm_a,B3: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ A @ ( epistemic_Imp_a @ B3 @ C ) ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ A @ B3 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_521_mcs__conjunction,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,V: set_Ep9038437136950510070c_fm_a,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte7220869009624332446c_fm_a @ A3 @ V )
=> ( ( maxima9990746573876165c_fm_a @ ( episte7220869009624332446c_fm_a @ A3 ) @ V )
=> ( ( ( member5360357140639596479c_fm_a @ P2 @ V )
& ( member5360357140639596479c_fm_a @ Q @ V ) )
=> ( member5360357140639596479c_fm_a @ ( episte8719284489194433015c_fm_a @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_522_mcs__conjunction,axiom,
! [A3: episte740340785640729014c_fm_a > $o,V: set_Ep8485816078964734998c_fm_a,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte704884124137261758c_fm_a @ A3 @ V )
=> ( ( maxima1560535778301255141c_fm_a @ ( episte704884124137261758c_fm_a @ A3 ) @ V )
=> ( ( ( member1118567890201975263c_fm_a @ P2 @ V )
& ( member1118567890201975263c_fm_a @ Q @ V ) )
=> ( member1118567890201975263c_fm_a @ ( episte3685526487207141399c_fm_a @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_523_mcs__conjunction,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( ( member6642669571620171971c_fm_a @ P2 @ V )
& ( member6642669571620171971c_fm_a @ Q @ V ) )
=> ( member6642669571620171971c_fm_a @ ( epistemic_Con_a @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_524_subset__code_I1_J,axiom,
! [Xs: list_l985973790606564998c_fm_a,B: set_li4204741992506657632c_fm_a] :
( ( ord_le2512102717711966976c_fm_a @ ( set_li3422124509070804117c_fm_a @ Xs ) @ B )
= ( ! [X3: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X3 @ ( set_li3422124509070804117c_fm_a @ Xs ) )
=> ( member2328277852289759785c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_525_subset__code_I1_J,axiom,
! [Xs: list_s682303562179440864c_fm_a,B: set_se8609989359752948410c_fm_a] :
( ( ord_le4042760284959298138c_fm_a @ ( set_se6798755039406792943c_fm_a @ Xs ) @ B )
= ( ! [X3: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X3 @ ( set_se6798755039406792943c_fm_a @ Xs ) )
=> ( member3183810400728440963c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_526_subset__code_I1_J,axiom,
! [Xs: list_E1147942331785229941c_fm_a,B: set_Ep6479678258574082331c_fm_a] :
( ( ord_le8649530489803071867c_fm_a @ ( set_Ep5814651402734953232c_fm_a @ Xs ) @ B )
= ( ! [X3: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X3 @ ( set_Ep5814651402734953232c_fm_a @ Xs ) )
=> ( member2634950385374053500c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_527_subset__code_I1_J,axiom,
! [Xs: list_s3694379548216983701c_fm_a,B: set_se6091493367310676795c_fm_a] :
( ( ord_le582293785304634267c_fm_a @ ( set_se3662515899849057584c_fm_a @ Xs ) @ B )
= ( ! [X3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X3 @ ( set_se3662515899849057584c_fm_a @ Xs ) )
=> ( member293847127289681564c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_528_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_529_subset__code_I1_J,axiom,
! [Xs: list_P3099032918374241197c_fm_a,B: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ ( set_Pr6083076219314900284c_fm_a @ Xs ) @ B )
= ( ! [X3: produc2203834439629243303c_fm_a] :
( ( member5029746173879296720c_fm_a @ X3 @ ( set_Pr6083076219314900284c_fm_a @ Xs ) )
=> ( member5029746173879296720c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_530_subset__code_I1_J,axiom,
! [Xs: list_s580375451141968640c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) @ B )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( member536094252920883875c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_531_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ B )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_532_conjunct__imply,axiom,
! [G5: list_E2761013248121217820c_fm_a,P2: episte5999154429349999510c_fm_a,G: list_char > $o,H: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G @ H @ ( episte5200161025749276397c_fm_a @ ( episte5200161025749276397c_fm_a @ ( stalna1247017718033373878c_fm_a @ G5 ) @ P2 ) @ ( episte7011297269853982921c_fm_a @ G5 @ P2 ) ) ) ).
% conjunct_imply
thf(fact_533_conjunct__imply,axiom,
! [G5: list_E5775909407618282812c_fm_a,P2: episte740340785640729014c_fm_a,G: list_char > $o,H: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G @ H @ ( episte260752218777527565c_fm_a @ ( episte260752218777527565c_fm_a @ ( stalna5968238273478151382c_fm_a @ G5 ) @ P2 ) @ ( episte1272216372155829993c_fm_a @ G5 @ P2 ) ) ) ).
% conjunct_imply
thf(fact_534_conjunct__imply,axiom,
! [G5: list_Epistemic_fm_a,P2: epistemic_fm_a,G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P2 ) @ ( epistemic_imply_a @ G5 @ P2 ) ) ) ).
% conjunct_imply
thf(fact_535_imply__conjunct,axiom,
! [G5: list_E2761013248121217820c_fm_a,P2: episte5999154429349999510c_fm_a,G: list_char > $o,H: episte5999154429349999510c_fm_a > $o] : ( episte6853238276675109012c_fm_a @ G @ H @ ( episte5200161025749276397c_fm_a @ ( episte7011297269853982921c_fm_a @ G5 @ P2 ) @ ( episte5200161025749276397c_fm_a @ ( stalna1247017718033373878c_fm_a @ G5 ) @ P2 ) ) ) ).
% imply_conjunct
thf(fact_536_imply__conjunct,axiom,
! [G5: list_E5775909407618282812c_fm_a,P2: episte740340785640729014c_fm_a,G: list_char > $o,H: episte740340785640729014c_fm_a > $o] : ( episte6292435812075825844c_fm_a @ G @ H @ ( episte260752218777527565c_fm_a @ ( episte1272216372155829993c_fm_a @ G5 @ P2 ) @ ( episte260752218777527565c_fm_a @ ( stalna5968238273478151382c_fm_a @ G5 ) @ P2 ) ) ) ).
% imply_conjunct
thf(fact_537_imply__conjunct,axiom,
! [G5: list_Epistemic_fm_a,P2: epistemic_fm_a,G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G5 @ P2 ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P2 ) ) ) ).
% imply_conjunct
thf(fact_538_K__conjunct__imply,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,G5: list_E2761013248121217820c_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( stalna1247017718033373878c_fm_a @ G5 ) @ P2 ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ G5 @ P2 ) ) ) ).
% K_conjunct_imply
thf(fact_539_K__conjunct__imply,axiom,
! [A3: episte740340785640729014c_fm_a > $o,G5: list_E5775909407618282812c_fm_a,P2: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( stalna5968238273478151382c_fm_a @ G5 ) @ P2 ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ G5 @ P2 ) ) ) ).
% K_conjunct_imply
thf(fact_540_K__conjunct__imply,axiom,
! [A3: epistemic_fm_a > $o,G5: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ G5 @ P2 ) ) ) ).
% K_conjunct_imply
thf(fact_541_K__imply__conjunct,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,G5: list_E2761013248121217820c_fm_a,P2: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ G5 @ P2 ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte5200161025749276397c_fm_a @ ( stalna1247017718033373878c_fm_a @ G5 ) @ P2 ) ) ) ).
% K_imply_conjunct
thf(fact_542_K__imply__conjunct,axiom,
! [A3: episte740340785640729014c_fm_a > $o,G5: list_E5775909407618282812c_fm_a,P2: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ G5 @ P2 ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte260752218777527565c_fm_a @ ( stalna5968238273478151382c_fm_a @ G5 ) @ P2 ) ) ) ).
% K_imply_conjunct
thf(fact_543_K__imply__conjunct,axiom,
! [A3: epistemic_fm_a > $o,G5: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ G5 @ P2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P2 ) ) ) ).
% K_imply_conjunct
thf(fact_544_K__Boole,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,G5: list_E2761013248121217820c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ ( cons_E5857249387053505868c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ episte4474038146033622457c_fm_a ) @ G5 ) @ episte4474038146033622457c_fm_a ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ G5 @ P2 ) ) ) ).
% K_Boole
thf(fact_545_K__Boole,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,G5: list_E5775909407618282812c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ ( cons_E3009779368504322924c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ episte5073044243917183961c_fm_a ) @ G5 ) @ episte5073044243917183961c_fm_a ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ G5 @ P2 ) ) ) ).
% K_Boole
thf(fact_546_K__Boole,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ G5 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ G5 @ P2 ) ) ) ).
% K_Boole
thf(fact_547_distribution,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% distribution
thf(fact_548_distribution,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,I: a,P2: epistemic_fm_a,Q: epistemic_fm_a] : ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% distribution
thf(fact_549_K__mp,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,Q: episte5999154429349999510c_fm_a,G5: list_E2761013248121217820c_fm_a] : ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ ( cons_E5857249387053505868c_fm_a @ P2 @ ( cons_E5857249387053505868c_fm_a @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_550_K__mp,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,Q: episte740340785640729014c_fm_a,G5: list_E5775909407618282812c_fm_a] : ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ ( cons_E3009779368504322924c_fm_a @ P2 @ ( cons_E3009779368504322924c_fm_a @ ( episte260752218777527565c_fm_a @ P2 @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_551_K__mp,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P2 @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_552_K__ImpI,axiom,
! [A3: episte5999154429349999510c_fm_a > $o,P2: episte5999154429349999510c_fm_a,G5: list_E2761013248121217820c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ ( cons_E5857249387053505868c_fm_a @ P2 @ G5 ) @ Q ) )
=> ( episte4141561593244286310c_fm_a @ A3 @ ( episte7011297269853982921c_fm_a @ G5 @ ( episte5200161025749276397c_fm_a @ P2 @ Q ) ) ) ) ).
% K_ImpI
thf(fact_553_K__ImpI,axiom,
! [A3: episte740340785640729014c_fm_a > $o,P2: episte740340785640729014c_fm_a,G5: list_E5775909407618282812c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ ( cons_E3009779368504322924c_fm_a @ P2 @ G5 ) @ Q ) )
=> ( episte3601224840237526918c_fm_a @ A3 @ ( episte1272216372155829993c_fm_a @ G5 @ ( episte260752218777527565c_fm_a @ P2 @ Q ) ) ) ) ).
% K_ImpI
thf(fact_554_K__ImpI,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ G5 ) @ Q ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ G5 @ ( epistemic_Imp_a @ P2 @ Q ) ) ) ) ).
% K_ImpI
thf(fact_555_list_Oinject,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a,Y21: epistemic_fm_a,Y22: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X21 @ X22 )
= ( cons_Epistemic_fm_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_556_list_Oinject,axiom,
! [X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a,Y21: set_Epistemic_fm_a,Y22: list_s580375451141968640c_fm_a] :
( ( ( cons_s4394448069393209776c_fm_a @ X21 @ X22 )
= ( cons_s4394448069393209776c_fm_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_557_Imp__intro,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ Q ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ P2 @ Q ) ) ) ).
% Imp_intro
thf(fact_558_Imp__intro,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ Q ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ P2 @ Q ) ) ) ).
% Imp_intro
thf(fact_559_set__ConsD,axiom,
! [Y: list_s580375451141968640c_fm_a,X2: list_s580375451141968640c_fm_a,Xs: list_l985973790606564998c_fm_a] :
( ( member2328277852289759785c_fm_a @ Y @ ( set_li3422124509070804117c_fm_a @ ( cons_l1626872963913326262c_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member2328277852289759785c_fm_a @ Y @ ( set_li3422124509070804117c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_560_set__ConsD,axiom,
! [Y: set_se5208064806568342746c_fm_a,X2: set_se5208064806568342746c_fm_a,Xs: list_s682303562179440864c_fm_a] :
( ( member3183810400728440963c_fm_a @ Y @ ( set_se6798755039406792943c_fm_a @ ( cons_s7529522790673496976c_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member3183810400728440963c_fm_a @ Y @ ( set_se6798755039406792943c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_561_set__ConsD,axiom,
! [Y: epistemic_fm_a > epistemic_fm_a,X2: epistemic_fm_a > epistemic_fm_a,Xs: list_E1147942331785229941c_fm_a] :
( ( member2634950385374053500c_fm_a @ Y @ ( set_Ep5814651402734953232c_fm_a @ ( cons_E5470485084980704175c_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member2634950385374053500c_fm_a @ Y @ ( set_Ep5814651402734953232c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_562_set__ConsD,axiom,
! [Y: set_Epistemic_fm_a > epistemic_fm_a,X2: set_Epistemic_fm_a > epistemic_fm_a,Xs: list_s3694379548216983701c_fm_a] :
( ( member293847127289681564c_fm_a @ Y @ ( set_se3662515899849057584c_fm_a @ ( cons_s3608211045111937487c_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member293847127289681564c_fm_a @ Y @ ( set_se3662515899849057584c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_563_set__ConsD,axiom,
! [Y: a,X2: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_564_set__ConsD,axiom,
! [Y: epistemic_fm_a,X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_565_set__ConsD,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_566_list_Oset__cases,axiom,
! [E: list_s580375451141968640c_fm_a,A: list_l985973790606564998c_fm_a] :
( ( member2328277852289759785c_fm_a @ E @ ( set_li3422124509070804117c_fm_a @ A ) )
=> ( ! [Z22: list_l985973790606564998c_fm_a] :
( A
!= ( cons_l1626872963913326262c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: list_s580375451141968640c_fm_a,Z22: list_l985973790606564998c_fm_a] :
( ( A
= ( cons_l1626872963913326262c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member2328277852289759785c_fm_a @ E @ ( set_li3422124509070804117c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_567_list_Oset__cases,axiom,
! [E: set_se5208064806568342746c_fm_a,A: list_s682303562179440864c_fm_a] :
( ( member3183810400728440963c_fm_a @ E @ ( set_se6798755039406792943c_fm_a @ A ) )
=> ( ! [Z22: list_s682303562179440864c_fm_a] :
( A
!= ( cons_s7529522790673496976c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_se5208064806568342746c_fm_a,Z22: list_s682303562179440864c_fm_a] :
( ( A
= ( cons_s7529522790673496976c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member3183810400728440963c_fm_a @ E @ ( set_se6798755039406792943c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_568_list_Oset__cases,axiom,
! [E: epistemic_fm_a > epistemic_fm_a,A: list_E1147942331785229941c_fm_a] :
( ( member2634950385374053500c_fm_a @ E @ ( set_Ep5814651402734953232c_fm_a @ A ) )
=> ( ! [Z22: list_E1147942331785229941c_fm_a] :
( A
!= ( cons_E5470485084980704175c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a > epistemic_fm_a,Z22: list_E1147942331785229941c_fm_a] :
( ( A
= ( cons_E5470485084980704175c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member2634950385374053500c_fm_a @ E @ ( set_Ep5814651402734953232c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_569_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_a > epistemic_fm_a,A: list_s3694379548216983701c_fm_a] :
( ( member293847127289681564c_fm_a @ E @ ( set_se3662515899849057584c_fm_a @ A ) )
=> ( ! [Z22: list_s3694379548216983701c_fm_a] :
( A
!= ( cons_s3608211045111937487c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a > epistemic_fm_a,Z22: list_s3694379548216983701c_fm_a] :
( ( A
= ( cons_s3608211045111937487c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member293847127289681564c_fm_a @ E @ ( set_se3662515899849057584c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_570_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_571_list_Oset__cases,axiom,
! [E: epistemic_fm_a,A: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ A ) )
=> ( ! [Z22: list_Epistemic_fm_a] :
( A
!= ( cons_Epistemic_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: list_Epistemic_fm_a] :
( ( A
= ( cons_Epistemic_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_572_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ A ) )
=> ( ! [Z22: list_s580375451141968640c_fm_a] :
( A
!= ( cons_s4394448069393209776c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: list_s580375451141968640c_fm_a] :
( ( A
= ( cons_s4394448069393209776c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_573_list_Oset__intros_I1_J,axiom,
! [X21: list_s580375451141968640c_fm_a,X22: list_l985973790606564998c_fm_a] : ( member2328277852289759785c_fm_a @ X21 @ ( set_li3422124509070804117c_fm_a @ ( cons_l1626872963913326262c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_574_list_Oset__intros_I1_J,axiom,
! [X21: set_se5208064806568342746c_fm_a,X22: list_s682303562179440864c_fm_a] : ( member3183810400728440963c_fm_a @ X21 @ ( set_se6798755039406792943c_fm_a @ ( cons_s7529522790673496976c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_575_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_a > epistemic_fm_a,X22: list_E1147942331785229941c_fm_a] : ( member2634950385374053500c_fm_a @ X21 @ ( set_Ep5814651402734953232c_fm_a @ ( cons_E5470485084980704175c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_576_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_a > epistemic_fm_a,X22: list_s3694379548216983701c_fm_a] : ( member293847127289681564c_fm_a @ X21 @ ( set_se3662515899849057584c_fm_a @ ( cons_s3608211045111937487c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_577_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_578_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X21 @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_579_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a] : ( member536094252920883875c_fm_a @ X21 @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_580_list_Oset__intros_I2_J,axiom,
! [Y: list_s580375451141968640c_fm_a,X22: list_l985973790606564998c_fm_a,X21: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ Y @ ( set_li3422124509070804117c_fm_a @ X22 ) )
=> ( member2328277852289759785c_fm_a @ Y @ ( set_li3422124509070804117c_fm_a @ ( cons_l1626872963913326262c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_581_list_Oset__intros_I2_J,axiom,
! [Y: set_se5208064806568342746c_fm_a,X22: list_s682303562179440864c_fm_a,X21: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ Y @ ( set_se6798755039406792943c_fm_a @ X22 ) )
=> ( member3183810400728440963c_fm_a @ Y @ ( set_se6798755039406792943c_fm_a @ ( cons_s7529522790673496976c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_582_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_a > epistemic_fm_a,X22: list_E1147942331785229941c_fm_a,X21: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ Y @ ( set_Ep5814651402734953232c_fm_a @ X22 ) )
=> ( member2634950385374053500c_fm_a @ Y @ ( set_Ep5814651402734953232c_fm_a @ ( cons_E5470485084980704175c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_583_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_a > epistemic_fm_a,X22: list_s3694379548216983701c_fm_a,X21: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ Y @ ( set_se3662515899849057584c_fm_a @ X22 ) )
=> ( member293847127289681564c_fm_a @ Y @ ( set_se3662515899849057584c_fm_a @ ( cons_s3608211045111937487c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_584_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_585_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_a,X22: list_Epistemic_fm_a,X21: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ X22 ) )
=> ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_586_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a,X21: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ X22 ) )
=> ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_587_not__Cons__self2,axiom,
! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_588_not__Cons__self2,axiom,
! [X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( cons_s4394448069393209776c_fm_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_589_conjunct_Osimps_I2_J,axiom,
! [P2: episte5999154429349999510c_fm_a,Ps: list_E2761013248121217820c_fm_a] :
( ( stalna1247017718033373878c_fm_a @ ( cons_E5857249387053505868c_fm_a @ P2 @ Ps ) )
= ( episte8719284489194433015c_fm_a @ P2 @ ( stalna1247017718033373878c_fm_a @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_590_conjunct_Osimps_I2_J,axiom,
! [P2: episte740340785640729014c_fm_a,Ps: list_E5775909407618282812c_fm_a] :
( ( stalna5968238273478151382c_fm_a @ ( cons_E3009779368504322924c_fm_a @ P2 @ Ps ) )
= ( episte3685526487207141399c_fm_a @ P2 @ ( stalna5968238273478151382c_fm_a @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_591_conjunct_Osimps_I2_J,axiom,
! [P2: epistemic_fm_a,Ps: list_Epistemic_fm_a] :
( ( stalnaker_conjunct_a @ ( cons_Epistemic_fm_a @ P2 @ Ps ) )
= ( epistemic_Con_a @ P2 @ ( stalnaker_conjunct_a @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_592_semantics_Osimps_I5_J,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ P2 @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_593_semantics_Osimps_I5_J,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ P2 @ Q ) )
= ( ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_594_semantics_Osimps_I1_J,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a] :
~ ( episte7081087998767065248c_fm_a @ M @ W3 @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_595_semantics_Osimps_I1_J,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a] :
~ ( episte5333283044364550848c_fm_a @ M @ W3 @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_596_set__subset__Cons,axiom,
! [Xs: list_s580375451141968640c_fm_a,X2: set_Epistemic_fm_a] : ( ord_le7112219575281605754c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_597_set__subset__Cons,axiom,
! [Xs: list_P3099032918374241197c_fm_a,X2: produc2203834439629243303c_fm_a] : ( ord_le8222437384497874983c_fm_a @ ( set_Pr6083076219314900284c_fm_a @ Xs ) @ ( set_Pr6083076219314900284c_fm_a @ ( cons_P6232728021713608541c_fm_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_598_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_a,X2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_599_semantics_Osimps_I4_J,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Con_a @ P2 @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 )
& ( episte7081087998767065248c_fm_a @ M @ W3 @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_600_semantics_Osimps_I4_J,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Con_a @ P2 @ Q ) )
= ( ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 )
& ( episte5333283044364550848c_fm_a @ M @ W3 @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_601_tautology,axiom,
! [P2: epistemic_fm_a,M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P2 )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 ) ) ).
% tautology
thf(fact_602_tautology,axiom,
! [P2: epistemic_fm_a,M: episte94448284482925344t_unit,W3: epistemic_fm_a] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P2 )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 ) ) ).
% tautology
thf(fact_603_imply_Osimps_I2_J,axiom,
! [P2: episte5999154429349999510c_fm_a,Ps: list_E2761013248121217820c_fm_a,Q: episte5999154429349999510c_fm_a] :
( ( episte7011297269853982921c_fm_a @ ( cons_E5857249387053505868c_fm_a @ P2 @ Ps ) @ Q )
= ( episte5200161025749276397c_fm_a @ P2 @ ( episte7011297269853982921c_fm_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_604_imply_Osimps_I2_J,axiom,
! [P2: episte740340785640729014c_fm_a,Ps: list_E5775909407618282812c_fm_a,Q: episte740340785640729014c_fm_a] :
( ( episte1272216372155829993c_fm_a @ ( cons_E3009779368504322924c_fm_a @ P2 @ Ps ) @ Q )
= ( episte260752218777527565c_fm_a @ P2 @ ( episte1272216372155829993c_fm_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_605_imply_Osimps_I2_J,axiom,
! [P2: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ Ps ) @ Q )
= ( epistemic_Imp_a @ P2 @ ( epistemic_imply_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_606_K__imply__head,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Ps: list_Epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ Ps ) @ P2 ) ) ).
% K_imply_head
thf(fact_607_K__imply__Cons,axiom,
! [A3: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_608_K__swap,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a,R2: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ ( cons_Epistemic_fm_a @ Q @ G5 ) ) @ R2 ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P2 @ G5 ) ) @ R2 ) ) ) ).
% K_swap
thf(fact_609_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_610_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a,P4: episte94448284482925344t_unit > $o] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_611_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1230141936084514080t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1230141936084514080t_unit,W4: list_s580375451141968640c_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member2328277852289759785c_fm_a @ W4 @ ( episte2985884989665202547t_unit @ M3 ) )
=> ( episte8879809419985483174c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte1230141936084514080t_unit] :
( ( P4 @ M2 )
=> ! [X6: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X6 @ ( episte2985884989665202547t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte8879809419985483174c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte8879809419985483174c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_612_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte507924219821198112t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte507924219821198112t_unit,W4: set_se5208064806568342746c_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member3183810400728440963c_fm_a @ W4 @ ( episte4966606409308169843t_unit @ M3 ) )
=> ( episte2348462963956845184c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte507924219821198112t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X6 @ ( episte4966606409308169843t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte2348462963956845184c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte2348462963956845184c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_613_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte6692431759376177384t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte6692431759376177384t_unit,W4: epistemic_fm_a > epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member2634950385374053500c_fm_a @ W4 @ ( episte7978936019567516915t_unit @ M3 ) )
=> ( episte391013328347312319c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte6692431759376177384t_unit] :
( ( P4 @ M2 )
=> ! [X6: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X6 @ ( episte7978936019567516915t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte391013328347312319c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte391013328347312319c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_614_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1144935709225342120t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1144935709225342120t_unit,W4: set_Epistemic_fm_a > epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member293847127289681564c_fm_a @ W4 @ ( episte2142321691486734515t_unit @ M3 ) )
=> ( episte555002011344058079c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte1144935709225342120t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X6 @ ( episte2142321691486734515t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte555002011344058079c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte555002011344058079c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_615_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W4: a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X6: a] :
( ( member_a @ X6 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580260cs_a_a @ M2 @ X6 @ Xa ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_616_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W4: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_617_strong__soundness,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W4: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_618_mcs__conjunction__mult,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,S: list_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ S ) @ V )
& ( finite3304564945125563331c_fm_a @ ( set_Epistemic_fm_a2 @ S ) ) )
=> ( member6642669571620171971c_fm_a @ ( stalnaker_conjunct_a @ S ) @ V ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_619_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_620_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P2 ) ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( episte4583239219080210381t_unit @ M2 )
=> ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_621_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1230141936084514080t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1230141936084514080t_unit,W4: list_s580375451141968640c_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member2328277852289759785c_fm_a @ W4 @ ( episte2985884989665202547t_unit @ M3 ) )
=> ( episte8879809419985483174c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte1230141936084514080t_unit] :
( ( P4 @ M2 )
=> ! [X6: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X6 @ ( episte2985884989665202547t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte8879809419985483174c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte8879809419985483174c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_622_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte507924219821198112t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte507924219821198112t_unit,W4: set_se5208064806568342746c_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member3183810400728440963c_fm_a @ W4 @ ( episte4966606409308169843t_unit @ M3 ) )
=> ( episte2348462963956845184c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte507924219821198112t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X6 @ ( episte4966606409308169843t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte2348462963956845184c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte2348462963956845184c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_623_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte6692431759376177384t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte6692431759376177384t_unit,W4: epistemic_fm_a > epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member2634950385374053500c_fm_a @ W4 @ ( episte7978936019567516915t_unit @ M3 ) )
=> ( episte391013328347312319c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte6692431759376177384t_unit] :
( ( P4 @ M2 )
=> ! [X6: epistemic_fm_a > epistemic_fm_a] :
( ( member2634950385374053500c_fm_a @ X6 @ ( episte7978936019567516915t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte391013328347312319c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte391013328347312319c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_624_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1144935709225342120t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1144935709225342120t_unit,W4: set_Epistemic_fm_a > epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member293847127289681564c_fm_a @ W4 @ ( episte2142321691486734515t_unit @ M3 ) )
=> ( episte555002011344058079c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte1144935709225342120t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member293847127289681564c_fm_a @ X6 @ ( episte2142321691486734515t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte555002011344058079c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte555002011344058079c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_625_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W4: a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X6: a] :
( ( member_a @ X6 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X6 @ Xa ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_626_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W4: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_627_soundness__imply,axiom,
! [A3: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W4: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A3 @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W4 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Ps @ P2 ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X6 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_628_main_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>K
thf(fact_629_strong__completeness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_630_not__in__set__insert,axiom,
! [X2: list_s580375451141968640c_fm_a,Xs: list_l985973790606564998c_fm_a] :
( ~ ( member2328277852289759785c_fm_a @ X2 @ ( set_li3422124509070804117c_fm_a @ Xs ) )
=> ( ( insert5429159995128816668c_fm_a @ X2 @ Xs )
= ( cons_l1626872963913326262c_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_631_not__in__set__insert,axiom,
! [X2: set_se5208064806568342746c_fm_a,Xs: list_s682303562179440864c_fm_a] :
( ~ ( member3183810400728440963c_fm_a @ X2 @ ( set_se6798755039406792943c_fm_a @ Xs ) )
=> ( ( insert7920459372561142774c_fm_a @ X2 @ Xs )
= ( cons_s7529522790673496976c_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_632_not__in__set__insert,axiom,
! [X2: epistemic_fm_a > epistemic_fm_a,Xs: list_E1147942331785229941c_fm_a] :
( ~ ( member2634950385374053500c_fm_a @ X2 @ ( set_Ep5814651402734953232c_fm_a @ Xs ) )
=> ( ( insert4129014825578047177c_fm_a @ X2 @ Xs )
= ( cons_E5470485084980704175c_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_633_not__in__set__insert,axiom,
! [X2: set_Epistemic_fm_a > epistemic_fm_a,Xs: list_s3694379548216983701c_fm_a] :
( ~ ( member293847127289681564c_fm_a @ X2 @ ( set_se3662515899849057584c_fm_a @ Xs ) )
=> ( ( insert2839750253210371305c_fm_a @ X2 @ Xs )
= ( cons_s3608211045111937487c_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_634_not__in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= ( cons_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_635_not__in__set__insert,axiom,
! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X2 @ Xs )
= ( cons_Epistemic_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_636_not__in__set__insert,axiom,
! [X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X2 @ Xs )
= ( cons_s4394448069393209776c_fm_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_637_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_a] : ( finite3304564945125563331c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_638_List_Ofinite__set,axiom,
! [Xs: list_s580375451141968640c_fm_a] : ( finite7364866884270929315c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) ) ).
% List.finite_set
thf(fact_639_in__set__insert,axiom,
! [X2: list_s580375451141968640c_fm_a,Xs: list_l985973790606564998c_fm_a] :
( ( member2328277852289759785c_fm_a @ X2 @ ( set_li3422124509070804117c_fm_a @ Xs ) )
=> ( ( insert5429159995128816668c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_640_in__set__insert,axiom,
! [X2: set_se5208064806568342746c_fm_a,Xs: list_s682303562179440864c_fm_a] :
( ( member3183810400728440963c_fm_a @ X2 @ ( set_se6798755039406792943c_fm_a @ Xs ) )
=> ( ( insert7920459372561142774c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_641_in__set__insert,axiom,
! [X2: epistemic_fm_a > epistemic_fm_a,Xs: list_E1147942331785229941c_fm_a] :
( ( member2634950385374053500c_fm_a @ X2 @ ( set_Ep5814651402734953232c_fm_a @ Xs ) )
=> ( ( insert4129014825578047177c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_642_in__set__insert,axiom,
! [X2: set_Epistemic_fm_a > epistemic_fm_a,Xs: list_s3694379548216983701c_fm_a] :
( ( member293847127289681564c_fm_a @ X2 @ ( set_se3662515899849057584c_fm_a @ Xs ) )
=> ( ( insert2839750253210371305c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_643_in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_644_in__set__insert,axiom,
! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_645_in__set__insert,axiom,
! [X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_646_finite__list,axiom,
! [A3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ? [Xs2: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_647_finite__list,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ? [Xs2: list_s580375451141968640c_fm_a] :
( ( set_se8855941934996405007c_fm_a @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_648_soundness__Ax5,axiom,
! [P2: epistemic_fm_a,M: episte1144935709225342120t_unit,W3: set_Epistemic_fm_a > epistemic_fm_a] :
( ( epistemic_Ax5_a @ P2 )
=> ( ( episte907865481439963405t_unit @ M )
=> ( ( member293847127289681564c_fm_a @ W3 @ ( episte2142321691486734515t_unit @ M ) )
=> ( episte555002011344058079c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_Ax5
thf(fact_649_soundness__Ax5,axiom,
! [P2: epistemic_fm_a,M: episte6182337868402532512t_unit,W3: a] :
( ( epistemic_Ax5_a @ P2 )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a @ W3 @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_Ax5
thf(fact_650_soundness__Ax5,axiom,
! [P2: epistemic_fm_a,M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P2 )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_Ax5
thf(fact_651_soundness__Ax5,axiom,
! [P2: epistemic_fm_a,M: episte94448284482925344t_unit,W3: epistemic_fm_a] :
( ( epistemic_Ax5_a @ P2 )
=> ( ( episte4583239219080210381t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_Ax5
thf(fact_652_inconsistent__finite,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A3 @ S )
=> ? [S4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S4 @ S )
& ( finite3304564945125563331c_fm_a @ S4 )
& ~ ( episte2285483198712856226tent_a @ A3 @ S4 ) ) ) ).
% inconsistent_finite
thf(fact_653_generalization,axiom,
! [P2: epistemic_fm_a,W3: epistemic_fm_a,M: episte94448284482925344t_unit,I: a] :
( ! [M3: episte94448284482925344t_unit,X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X @ P2 ) )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_K_a @ I @ P2 ) ) ) ) ).
% generalization
thf(fact_654_generalization,axiom,
! [P2: epistemic_fm_a,W3: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I: a] :
( ! [M3: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_K_a @ I @ P2 ) ) ) ) ).
% generalization
thf(fact_655_List_Oinsert__def,axiom,
( insert177310161492556854c_fm_a
= ( ^ [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_656_List_Oinsert__def,axiom,
( insert1868103382419917334c_fm_a
= ( ^ [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] : ( if_lis3811715425919989702c_fm_a @ ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs3 ) ) @ Xs3 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_657_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_658_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_659_finite__Collect__subsets,axiom,
! [A3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B2 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_660_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_661_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_662_finite__Collect__conjI,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
| ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_663_finite__Collect__conjI,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
| ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_664_finite__Collect__disjI,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_665_finite__Collect__disjI,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_666_soundness__AxTB4,axiom,
! [P2: epistemic_fm_a,M: episte94448284482925344t_unit,W3: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_AxTB4
thf(fact_667_soundness__AxTB4,axiom,
! [P2: epistemic_fm_a,M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_AxTB4
thf(fact_668_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_669_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
@ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_670_neg__introspection,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte2438601301504999334t_unit @ M )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_671_neg__introspection,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_672_not__finite__existsD,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ~ ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ? [X_1: set_Epistemic_fm_a] : ( P4 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_673_not__finite__existsD,axiom,
! [P4: epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ? [X_1: epistemic_fm_a] : ( P4 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_674_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_675_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_676_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_677_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
@ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_678_soundness__AxB,axiom,
! [P2: epistemic_fm_a,M: episte94448284482925344t_unit,W3: epistemic_fm_a] :
( ( epistemic_AxB_a @ P2 )
=> ( ( episte2438601301504999334t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_AxB
thf(fact_679_soundness__AxB,axiom,
! [P2: epistemic_fm_a,M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P2 )
=> ( ( episte5478016696552465318t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ P2 ) ) ) ) ).
% soundness_AxB
thf(fact_680_finite__has__minimal2,axiom,
! [A3: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( member536094252920883875c_fm_a @ A @ A3 )
=> ? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ( ord_le3275665582123262618c_fm_a @ X @ A )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa2 @ X )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_681_finite__has__maximal2,axiom,
! [A3: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( member536094252920883875c_fm_a @ A @ A3 )
=> ? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ( ord_le3275665582123262618c_fm_a @ A @ X )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Xa2 )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_682_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( finite3304564945125563331c_fm_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_683_infinite__super,axiom,
! [S: set_Epistemic_fm_a,T2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S @ T2 )
=> ( ~ ( finite3304564945125563331c_fm_a @ S )
=> ~ ( finite3304564945125563331c_fm_a @ T2 ) ) ) ).
% infinite_super
thf(fact_684_finite__subset,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( finite3304564945125563331c_fm_a @ B )
=> ( finite3304564945125563331c_fm_a @ A3 ) ) ) ).
% finite_subset
thf(fact_685_pos__introspection,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P2 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_686_pos__introspection,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P2 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_687_truth,axiom,
! [M: episte94448284482925344t_unit,W3: epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W3 @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ P2 ) ) ) ) ).
% truth
thf(fact_688_truth,axiom,
! [M: episte1560738328020401952t_unit,W3: set_Epistemic_fm_a,I: a,P2: epistemic_fm_a] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W3 @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ P2 ) ) ) ) ).
% truth
thf(fact_689_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_690_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_691_main_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X3 @ P2 ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs @ P2 ) ) ) ) ) ).
% main\<^sub>T
thf(fact_692_strong__completeness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_693_K__conjunction__out__mult,axiom,
! [A3: epistemic_fm_a > $o,I: a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G5 ) ) @ ( epistemic_K_a @ I @ ( stalnaker_conjunct_a @ G5 ) ) ) ) ).
% K_conjunction_out_mult
thf(fact_694_K__conjunction__in__mult,axiom,
! [A3: epistemic_fm_a > $o,I: a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( stalnaker_conjunct_a @ G5 ) ) @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G5 ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_695_list_Omap__ident__strong,axiom,
! [T3: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ T3 ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_696_list_Omap__ident__strong,axiom,
! [T3: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( set_se8855941934996405007c_fm_a @ T3 ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_697_map__idI,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_698_map__idI,axiom,
! [Xs: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_699_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_700_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_701_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_702_eval_Osimps_I3_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Dis_a @ P2 @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P2 )
| ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_703_K__distrib__K__imp,axiom,
! [A3: epistemic_fm_a > $o,I: a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A3 @ ( epistemic_K_a @ I @ ( epistemic_imply_a @ G5 @ Q ) ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G5 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_704_fm_Oexhaust,axiom,
! [Y: epistemic_fm_a] :
( ( Y != epistemic_FF_a )
=> ( ! [X23: list_char] :
( Y
!= ( epistemic_Pro_a @ X23 ) )
=> ( ! [X312: epistemic_fm_a,X322: epistemic_fm_a] :
( Y
!= ( epistemic_Dis_a @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_a,X422: epistemic_fm_a] :
( Y
!= ( epistemic_Con_a @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_a,X522: epistemic_fm_a] :
( Y
!= ( epistemic_Imp_a @ X512 @ X522 ) )
=> ~ ! [X612: a,X622: epistemic_fm_a] :
( Y
!= ( epistemic_K_a @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_705_fm_Oset__cases,axiom,
! [E: epistemic_fm_a,A: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ A ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte740340785640729014c_fm_a] :
( A
!= ( episte5657488632024175118c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A
= ( episte5657488632024175118c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_706_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ A ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte5999154429349999510c_fm_a] :
( A
!= ( episte3013695276341575662c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A
= ( episte3013695276341575662c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_707_fm_Oset__cases,axiom,
! [E: a,A: epistemic_fm_a] :
( ( member_a @ E @ ( epistemic_set_fm_a @ A ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_a] :
( A
!= ( epistemic_K_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: epistemic_fm_a] :
( ( A
= ( epistemic_K_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_708_empty__Collect__eq,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ( bot_bo1868452413526482246c_fm_a
= ( collec2519470961442302949c_fm_a @ P4 ) )
= ( ! [X3: set_Epistemic_fm_a] :
~ ( P4 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_709_empty__Collect__eq,axiom,
! [P4: epistemic_fm_a > $o] :
( ( bot_bo3626323581529592678c_fm_a
= ( collec4904205152690461189c_fm_a @ P4 ) )
= ( ! [X3: epistemic_fm_a] :
~ ( P4 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_710_Collect__empty__eq,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ( ( collec2519470961442302949c_fm_a @ P4 )
= bot_bo1868452413526482246c_fm_a )
= ( ! [X3: set_Epistemic_fm_a] :
~ ( P4 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_711_Collect__empty__eq,axiom,
! [P4: epistemic_fm_a > $o] :
( ( ( collec4904205152690461189c_fm_a @ P4 )
= bot_bo3626323581529592678c_fm_a )
= ( ! [X3: epistemic_fm_a] :
~ ( P4 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_712_all__not__in__conv,axiom,
! [A3: set_Epistemic_fm_a] :
( ( ! [X3: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X3 @ A3 ) )
= ( A3 = bot_bo3626323581529592678c_fm_a ) ) ).
% all_not_in_conv
thf(fact_713_all__not__in__conv,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( ! [X3: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ X3 @ A3 ) )
= ( A3 = bot_bo1868452413526482246c_fm_a ) ) ).
% all_not_in_conv
thf(fact_714_empty__iff,axiom,
! [C: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ C @ bot_bo3626323581529592678c_fm_a ) ).
% empty_iff
thf(fact_715_empty__iff,axiom,
! [C: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ C @ bot_bo1868452413526482246c_fm_a ) ).
% empty_iff
thf(fact_716_subset__empty,axiom,
! [A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ bot_bo3626323581529592678c_fm_a )
= ( A3 = bot_bo3626323581529592678c_fm_a ) ) ).
% subset_empty
thf(fact_717_empty__subsetI,axiom,
! [A3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A3 ) ).
% empty_subsetI
thf(fact_718_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_719_ex__in__conv,axiom,
! [A3: set_Epistemic_fm_a] :
( ( ? [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 ) )
= ( A3 != bot_bo3626323581529592678c_fm_a ) ) ).
% ex_in_conv
thf(fact_720_ex__in__conv,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( ? [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 ) )
= ( A3 != bot_bo1868452413526482246c_fm_a ) ) ).
% ex_in_conv
thf(fact_721_equals0I,axiom,
! [A3: set_Epistemic_fm_a] :
( ! [Y2: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ Y2 @ A3 )
=> ( A3 = bot_bo3626323581529592678c_fm_a ) ) ).
% equals0I
thf(fact_722_equals0I,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ! [Y2: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ Y2 @ A3 )
=> ( A3 = bot_bo1868452413526482246c_fm_a ) ) ).
% equals0I
thf(fact_723_equals0D,axiom,
! [A3: set_Epistemic_fm_a,A: epistemic_fm_a] :
( ( A3 = bot_bo3626323581529592678c_fm_a )
=> ~ ( member6642669571620171971c_fm_a @ A @ A3 ) ) ).
% equals0D
thf(fact_724_equals0D,axiom,
! [A3: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( A3 = bot_bo1868452413526482246c_fm_a )
=> ~ ( member536094252920883875c_fm_a @ A @ A3 ) ) ).
% equals0D
thf(fact_725_emptyE,axiom,
! [A: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ).
% emptyE
thf(fact_726_emptyE,axiom,
! [A: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ).
% emptyE
thf(fact_727_empty__def,axiom,
( bot_bo1868452413526482246c_fm_a
= ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : $false ) ) ).
% empty_def
thf(fact_728_empty__def,axiom,
( bot_bo3626323581529592678c_fm_a
= ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : $false ) ) ).
% empty_def
thf(fact_729_bot_Oextremum,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A ) ).
% bot.extremum
thf(fact_730_bot_Oextremum__unique,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a )
= ( A = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_731_bot_Oextremum__uniqueI,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a )
=> ( A = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_732_subset__emptyI,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le7112219575281605754c_fm_a @ A3 @ bot_bo1868452413526482246c_fm_a ) ) ).
% subset_emptyI
thf(fact_733_subset__emptyI,axiom,
! [A3: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ bot_bo3626323581529592678c_fm_a ) ) ).
% subset_emptyI
thf(fact_734_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_a,X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ X51 ) )
=> ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_735_fm_Oset__intros_I5_J,axiom,
! [Ye: set_Epistemic_fm_a,X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ X51 ) )
=> ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_736_fm_Oset__intros_I5_J,axiom,
! [Ye: a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( member_a @ Ye @ ( epistemic_set_fm_a @ X51 ) )
=> ( member_a @ Ye @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_737_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_a,X52: episte740340785640729014c_fm_a,X51: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ X52 ) )
=> ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_738_fm_Oset__intros_I6_J,axiom,
! [Yf: set_Epistemic_fm_a,X52: episte5999154429349999510c_fm_a,X51: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ X52 ) )
=> ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_739_fm_Oset__intros_I6_J,axiom,
! [Yf: a,X52: epistemic_fm_a,X51: epistemic_fm_a] :
( ( member_a @ Yf @ ( epistemic_set_fm_a @ X52 ) )
=> ( member_a @ Yf @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_740_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] : ( member6642669571620171971c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_741_fm_Oset__intros_I7_J,axiom,
! [X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] : ( member536094252920883875c_fm_a @ X61 @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_742_fm_Oset__intros_I7_J,axiom,
! [X61: a,X62: epistemic_fm_a] : ( member_a @ X61 @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_743_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_a,X62: episte740340785640729014c_fm_a,X61: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ X62 ) )
=> ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_744_fm_Oset__intros_I8_J,axiom,
! [Yg: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a,X61: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ X62 ) )
=> ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_745_fm_Oset__intros_I8_J,axiom,
! [Yg: a,X62: epistemic_fm_a,X61: a] :
( ( member_a @ Yg @ ( epistemic_set_fm_a @ X62 ) )
=> ( member_a @ Yg @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_746_fm_Odistinct_I15_J,axiom,
! [X24: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_747_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_748_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_a,X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ X41 ) )
=> ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_749_fm_Oset__intros_I3_J,axiom,
! [Yc: set_Epistemic_fm_a,X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ X41 ) )
=> ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_750_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_a,X42: episte740340785640729014c_fm_a,X41: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ X42 ) )
=> ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_751_fm_Oset__intros_I4_J,axiom,
! [Yd: set_Epistemic_fm_a,X42: episte5999154429349999510c_fm_a,X41: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ X42 ) )
=> ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_752_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X24 ) ) ).
% fm.distinct(1)
thf(fact_753_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_a,X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ X31 ) )
=> ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_754_fm_Oset__intros_I1_J,axiom,
! [Ya: set_Epistemic_fm_a,X31: episte5999154429349999510c_fm_a,X32: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ X31 ) )
=> ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_755_fm_Oset__intros_I2_J,axiom,
! [Yb: epistemic_fm_a,X32: episte740340785640729014c_fm_a,X31: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ X32 ) )
=> ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_756_fm_Oset__intros_I2_J,axiom,
! [Yb: set_Epistemic_fm_a,X32: episte5999154429349999510c_fm_a,X31: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ X32 ) )
=> ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_757_eval_Osimps_I2_J,axiom,
! [G2: list_char > $o,Uw: epistemic_fm_a > $o,X2: list_char] :
( ( epistemic_eval_a @ G2 @ Uw @ ( epistemic_Pro_a @ X2 ) )
= ( G2 @ X2 ) ) ).
% eval.simps(2)
thf(fact_758_finite__has__maximal,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Xa2 )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_759_finite__has__minimal,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa2 @ X )
=> ( X = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_760_fm_Orel__induct,axiom,
! [R: a > a > $o,X2: epistemic_fm_a,Y: epistemic_fm_a,Q2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R @ X2 @ Y )
=> ( ( Q2 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q2 @ ( epistemic_Pro_a @ A22 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q2 @ A31 @ B31 )
=> ( ( Q2 @ A32 @ B32 )
=> ( Q2 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q2 @ A41 @ B41 )
=> ( ( Q2 @ A42 @ B42 )
=> ( Q2 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q2 @ A51 @ B51 )
=> ( ( Q2 @ A52 @ B52 )
=> ( Q2 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R @ A61 @ B61 )
=> ( ( Q2 @ A62 @ B62 )
=> ( Q2 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q2 @ X2 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_761_fm_Orel__cases,axiom,
! [R: a > a > $o,A: epistemic_fm_a,B3: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ A @ B3 )
=> ( ( ( A = epistemic_FF_a )
=> ( B3 != epistemic_FF_a ) )
=> ( ! [X: list_char] :
( ( A
= ( epistemic_Pro_a @ X ) )
=> ! [Y2: list_char] :
( ( B3
= ( epistemic_Pro_a @ Y2 ) )
=> ( X != Y2 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B3
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B3
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B3
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B3
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_762_inconsistent__subset,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ~ ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P2 @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [V2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V2 ) @ V )
=> ~ ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P2 @ V2 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_763_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P2: epistemic_fm_a,A3: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs4 @ P2 ) ) ) ) ) ).
% strong_completeness
thf(fact_764_insertCI,axiom,
! [A: epistemic_fm_a,B: set_Epistemic_fm_a,B3: epistemic_fm_a] :
( ( ~ ( member6642669571620171971c_fm_a @ A @ B )
=> ( A = B3 ) )
=> ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_765_insertCI,axiom,
! [A: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a] :
( ( ~ ( member536094252920883875c_fm_a @ A @ B )
=> ( A = B3 ) )
=> ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_766_insert__iff,axiom,
! [A: epistemic_fm_a,B3: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B3 @ A3 ) )
= ( ( A = B3 )
| ( member6642669571620171971c_fm_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_767_insert__iff,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ B3 @ A3 ) )
= ( ( A = B3 )
| ( member536094252920883875c_fm_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_768_UnCI,axiom,
! [C: epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ~ ( member6642669571620171971c_fm_a @ C @ B )
=> ( member6642669571620171971c_fm_a @ C @ A3 ) )
=> ( member6642669571620171971c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_769_UnCI,axiom,
! [C: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( ~ ( member536094252920883875c_fm_a @ C @ B )
=> ( member536094252920883875c_fm_a @ C @ A3 ) )
=> ( member536094252920883875c_fm_a @ C @ ( sup_su4305657963326700718c_fm_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_770_Un__iff,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) )
= ( ( member6642669571620171971c_fm_a @ C @ A3 )
| ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_771_Un__iff,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( sup_su4305657963326700718c_fm_a @ A3 @ B ) )
= ( ( member536094252920883875c_fm_a @ C @ A3 )
| ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_772_singletonI,axiom,
! [A: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% singletonI
thf(fact_773_singletonI,axiom,
! [A: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) ).
% singletonI
thf(fact_774_insert__subset,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ A3 ) @ B )
= ( ( member536094252920883875c_fm_a @ X2 @ B )
& ( ord_le7112219575281605754c_fm_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_775_insert__subset,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ A3 ) @ B )
= ( ( member6642669571620171971c_fm_a @ X2 @ B )
& ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_776_Un__subset__iff,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) @ C2 )
= ( ( ord_le3275665582123262618c_fm_a @ A3 @ C2 )
& ( ord_le3275665582123262618c_fm_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_777_singleton__conv2,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ( ^ [Y3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y3 = Z2 )
@ A ) )
= ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) ).
% singleton_conv2
thf(fact_778_singleton__conv2,axiom,
! [A: epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ( ^ [Y3: epistemic_fm_a,Z2: epistemic_fm_a] : ( Y3 = Z2 )
@ A ) )
= ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% singleton_conv2
thf(fact_779_singleton__conv,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( X3 = A ) )
= ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) ).
% singleton_conv
thf(fact_780_singleton__conv,axiom,
! [A: epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( X3 = A ) )
= ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% singleton_conv
thf(fact_781_singleton__insert__inj__eq_H,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B3: epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ A @ A3 )
= ( insert7817948963269374698c_fm_a @ B3 @ bot_bo3626323581529592678c_fm_a ) )
= ( ( A = B3 )
& ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ B3 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_782_singleton__insert__inj__eq,axiom,
! [B3: epistemic_fm_a,A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ B3 @ bot_bo3626323581529592678c_fm_a )
= ( insert7817948963269374698c_fm_a @ A @ A3 ) )
= ( ( A = B3 )
& ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ B3 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_783_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a2 @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_784_UnE,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) )
=> ( ~ ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% UnE
thf(fact_785_UnE,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( sup_su4305657963326700718c_fm_a @ A3 @ B ) )
=> ( ~ ( member536094252920883875c_fm_a @ C @ A3 )
=> ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% UnE
thf(fact_786_UnI1,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( member6642669571620171971c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_787_UnI1,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ A3 )
=> ( member536094252920883875c_fm_a @ C @ ( sup_su4305657963326700718c_fm_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_788_UnI2,axiom,
! [C: epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ B )
=> ( member6642669571620171971c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_789_UnI2,axiom,
! [C: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ B )
=> ( member536094252920883875c_fm_a @ C @ ( sup_su4305657963326700718c_fm_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_790_insertE,axiom,
! [A: epistemic_fm_a,B3: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B3 @ A3 ) )
=> ( ( A != B3 )
=> ( member6642669571620171971c_fm_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_791_insertE,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ B3 @ A3 ) )
=> ( ( A != B3 )
=> ( member536094252920883875c_fm_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_792_insertI1,axiom,
! [A: epistemic_fm_a,B: set_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ A @ B ) ) ).
% insertI1
thf(fact_793_insertI1,axiom,
! [A: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] : ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ A @ B ) ) ).
% insertI1
thf(fact_794_insertI2,axiom,
! [A: epistemic_fm_a,B: set_Epistemic_fm_a,B3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ B )
=> ( member6642669571620171971c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_795_insertI2,axiom,
! [A: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ B )
=> ( member536094252920883875c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_796_Set_Oset__insert,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ~ ! [B5: set_Epistemic_fm_a] :
( ( A3
= ( insert7817948963269374698c_fm_a @ X2 @ B5 ) )
=> ( member6642669571620171971c_fm_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_797_Set_Oset__insert,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ~ ! [B5: set_se5208064806568342746c_fm_a] :
( ( A3
= ( insert7129737658439086282c_fm_a @ X2 @ B5 ) )
=> ( member536094252920883875c_fm_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_798_insert__ident,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ~ ( member6642669571620171971c_fm_a @ X2 @ B )
=> ( ( ( insert7817948963269374698c_fm_a @ X2 @ A3 )
= ( insert7817948963269374698c_fm_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_799_insert__ident,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ~ ( member536094252920883875c_fm_a @ X2 @ B )
=> ( ( ( insert7129737658439086282c_fm_a @ X2 @ A3 )
= ( insert7129737658439086282c_fm_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_800_insert__absorb,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( insert7817948963269374698c_fm_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_801_insert__absorb,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( insert7129737658439086282c_fm_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_802_insert__eq__iff,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B3: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ~ ( member6642669571620171971c_fm_a @ B3 @ B )
=> ( ( ( insert7817948963269374698c_fm_a @ A @ A3 )
= ( insert7817948963269374698c_fm_a @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A3 = B ) )
& ( ( A != B3 )
=> ? [C3: set_Epistemic_fm_a] :
( ( A3
= ( insert7817948963269374698c_fm_a @ B3 @ C3 ) )
& ~ ( member6642669571620171971c_fm_a @ B3 @ C3 )
& ( B
= ( insert7817948963269374698c_fm_a @ A @ C3 ) )
& ~ ( member6642669571620171971c_fm_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_803_insert__eq__iff,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ~ ( member536094252920883875c_fm_a @ B3 @ B )
=> ( ( ( insert7129737658439086282c_fm_a @ A @ A3 )
= ( insert7129737658439086282c_fm_a @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A3 = B ) )
& ( ( A != B3 )
=> ? [C3: set_se5208064806568342746c_fm_a] :
( ( A3
= ( insert7129737658439086282c_fm_a @ B3 @ C3 ) )
& ~ ( member536094252920883875c_fm_a @ B3 @ C3 )
& ( B
= ( insert7129737658439086282c_fm_a @ A @ C3 ) )
& ~ ( member536094252920883875c_fm_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_804_mk__disjoint__insert,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ A3 )
=> ? [B5: set_Epistemic_fm_a] :
( ( A3
= ( insert7817948963269374698c_fm_a @ A @ B5 ) )
& ~ ( member6642669571620171971c_fm_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_805_mk__disjoint__insert,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ A3 )
=> ? [B5: set_se5208064806568342746c_fm_a] :
( ( A3
= ( insert7129737658439086282c_fm_a @ A @ B5 ) )
& ~ ( member536094252920883875c_fm_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_806_insert__def,axiom,
( insert7129737658439086282c_fm_a
= ( ^ [A2: set_Epistemic_fm_a] :
( sup_su4305657963326700718c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( X3 = A2 ) ) ) ) ) ).
% insert_def
thf(fact_807_insert__def,axiom,
( insert7817948963269374698c_fm_a
= ( ^ [A2: epistemic_fm_a] :
( sup_su1367922730591523534c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( X3 = A2 ) ) ) ) ) ).
% insert_def
thf(fact_808_insert__Collect,axiom,
! [A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( insert7129737658439086282c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a
@ ^ [U2: set_Epistemic_fm_a] :
( ( U2 != A )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_809_insert__Collect,axiom,
! [A: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( insert7817948963269374698c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a
@ ^ [U2: epistemic_fm_a] :
( ( U2 != A )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_810_insert__compr,axiom,
( insert7129737658439086282c_fm_a
= ( ^ [A2: set_Epistemic_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A2 )
| ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_811_insert__compr,axiom,
( insert7817948963269374698c_fm_a
= ( ^ [A2: epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A2 )
| ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_812_Collect__disj__eq,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su4305657963326700718c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_813_Collect__disj__eq,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_814_Un__def,axiom,
( sup_su4305657963326700718c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A4 )
| ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_815_Un__def,axiom,
( sup_su1367922730591523534c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A4 )
| ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_816_bot__empty__eq,axiom,
( bot_bo6433428028861825271fm_a_o
= ( ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) ) ) ).
% bot_empty_eq
thf(fact_817_bot__empty__eq,axiom,
( bot_bo1951224918602374935fm_a_o
= ( ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ bot_bo1868452413526482246c_fm_a ) ) ) ).
% bot_empty_eq
thf(fact_818_bot__set__def,axiom,
( bot_bo1868452413526482246c_fm_a
= ( collec2519470961442302949c_fm_a @ bot_bo1951224918602374935fm_a_o ) ) ).
% bot_set_def
thf(fact_819_bot__set__def,axiom,
( bot_bo3626323581529592678c_fm_a
= ( collec4904205152690461189c_fm_a @ bot_bo6433428028861825271fm_a_o ) ) ).
% bot_set_def
thf(fact_820_fm_Orel__inject_I5_J,axiom,
! [R: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_821_fm_Orel__intros_I5_J,axiom,
! [R: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_822_fm_Orel__inject_I6_J,axiom,
! [R: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_823_fm_Orel__intros_I6_J,axiom,
! [R: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_824_fm_Octr__transfer_I1_J,axiom,
! [R: a > a > $o] : ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_825_fm_Orel__cong,axiom,
! [X2: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Xa3: episte740340785640729014c_fm_a,R: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa3 ) )
=> ( ( R @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R @ X2 @ Y )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_826_fm_Orel__cong,axiom,
! [X2: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte5999154429349999510c_fm_a,Xa3: episte5999154429349999510c_fm_a,R: epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa3 ) )
=> ( ( R @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte4430291480098810304c_fm_a @ R @ X2 @ Y )
= ( episte4430291480098810304c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_827_fm_Orel__cong,axiom,
! [X2: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte740340785640729014c_fm_a,Xa3: episte740340785640729014c_fm_a,R: set_Epistemic_fm_a > epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa3 ) )
=> ( ( R @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte6543039790609369600c_fm_a @ R @ X2 @ Y )
= ( episte6543039790609369600c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_828_fm_Orel__cong,axiom,
! [X2: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte5999154429349999510c_fm_a,Xa3: episte5999154429349999510c_fm_a,R: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa3 ) )
=> ( ( R @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte8246226516352692192c_fm_a @ R @ X2 @ Y )
= ( episte8246226516352692192c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_829_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,X2: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte7774795710028497888c_fm_a @ R @ X2 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X2 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X2 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_830_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_a > set_Epistemic_fm_a > $o,X2: episte740340785640729014c_fm_a,Y: episte5999154429349999510c_fm_a,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte4430291480098810304c_fm_a @ R @ X2 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X2 ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte4430291480098810304c_fm_a @ Ra @ X2 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_831_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_a > epistemic_fm_a > $o,X2: episte5999154429349999510c_fm_a,Y: episte740340785640729014c_fm_a,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte6543039790609369600c_fm_a @ R @ X2 @ Y )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X2 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte6543039790609369600c_fm_a @ Ra @ X2 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_832_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,X2: episte5999154429349999510c_fm_a,Y: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte8246226516352692192c_fm_a @ R @ X2 @ Y )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X2 ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X2 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_833_fm_Orel__refl__strong,axiom,
! [X2: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X2 @ X2 ) ) ).
% fm.rel_refl_strong
thf(fact_834_fm_Orel__refl__strong,axiom,
! [X2: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ! [Z4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X2 @ X2 ) ) ).
% fm.rel_refl_strong
thf(fact_835_subset__Un__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A4 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_836_subset__UnE,axiom,
! [C2: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) )
=> ~ ! [A5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A5 @ A3 )
=> ! [B6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B6 @ B )
=> ( C2
!= ( sup_su1367922730591523534c_fm_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_837_Un__absorb2,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A3 )
=> ( ( sup_su1367922730591523534c_fm_a @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_838_Un__absorb1,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( sup_su1367922730591523534c_fm_a @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_839_Un__upper2,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) ) ).
% Un_upper2
thf(fact_840_Un__upper1,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) ) ).
% Un_upper1
thf(fact_841_Un__least,axiom,
! [A3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_842_Un__mono,axiom,
! [A3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A3 @ B ) @ ( sup_su1367922730591523534c_fm_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_843_singletonD,axiom,
! [B3: epistemic_fm_a,A: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B3 @ ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_844_singletonD,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_845_singleton__iff,axiom,
! [B3: epistemic_fm_a,A: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B3 @ ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_846_singleton__iff,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_847_insert__subsetI,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,X4: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( ord_le7112219575281605754c_fm_a @ X4 @ A3 )
=> ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ X4 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_848_insert__subsetI,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,X4: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X4 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ X4 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_849_subset__insertI2,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,B3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_850_subset__insertI,axiom,
! [B: set_Epistemic_fm_a,A: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ A @ B ) ) ).
% subset_insertI
thf(fact_851_subset__insert,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( ord_le7112219575281605754c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ X2 @ B ) )
= ( ord_le7112219575281605754c_fm_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_852_subset__insert,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ B ) )
= ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_853_insert__mono,axiom,
! [C2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,A: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ C2 ) @ ( insert7817948963269374698c_fm_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_854_Collect__conv__if2,axiom,
! [P4: set_Epistemic_fm_a > $o,A: set_Epistemic_fm_a] :
( ( ( P4 @ A )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( A = X3 )
& ( P4 @ X3 ) ) )
= ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( A = X3 )
& ( P4 @ X3 ) ) )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_conv_if2
thf(fact_855_Collect__conv__if2,axiom,
! [P4: epistemic_fm_a > $o,A: epistemic_fm_a] :
( ( ( P4 @ A )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( A = X3 )
& ( P4 @ X3 ) ) )
= ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( A = X3 )
& ( P4 @ X3 ) ) )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_conv_if2
thf(fact_856_Collect__conv__if,axiom,
! [P4: set_Epistemic_fm_a > $o,A: set_Epistemic_fm_a] :
( ( ( P4 @ A )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A )
& ( P4 @ X3 ) ) )
= ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A )
& ( P4 @ X3 ) ) )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_conv_if
thf(fact_857_Collect__conv__if,axiom,
! [P4: epistemic_fm_a > $o,A: epistemic_fm_a] :
( ( ( P4 @ A )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A )
& ( P4 @ X3 ) ) )
= ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A )
& ( P4 @ X3 ) ) )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_conv_if
thf(fact_858_consistent__disjuncts,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Dis_a @ P2 @ Q ) @ V )
=> ( ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P2 @ bot_bo3626323581529592678c_fm_a ) @ V ) )
| ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_859_maximal__def,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ S )
= ( ! [P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ P @ S ) ) ) ) ).
% maximal_def
thf(fact_860_fm_Orel__distinct_I30_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_861_fm_Orel__distinct_I29_J,axiom,
! [R: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_862_fm_Orel__distinct_I8_J,axiom,
! [R: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_863_fm_Orel__distinct_I7_J,axiom,
! [R: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_864_fm_Orel__distinct_I10_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_865_fm_Orel__distinct_I9_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_866_subset__singletonD,axiom,
! [A3: set_Epistemic_fm_a,X2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) )
=> ( ( A3 = bot_bo3626323581529592678c_fm_a )
| ( A3
= ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singletonD
thf(fact_867_subset__singleton__iff,axiom,
! [X4: set_Epistemic_fm_a,A: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X4 @ ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) )
= ( ( X4 = bot_bo3626323581529592678c_fm_a )
| ( X4
= ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_868_consistent__consequent,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( member6642669571620171971c_fm_a @ P2 @ V )
=> ( ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q ) )
=> ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_869_consistent__consequent_H,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( member6642669571620171971c_fm_a @ P2 @ V )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Imp_a @ P2 @ Q ) )
=> ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_870_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X51 ) @ ( epistemic_set_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_871_truth__lemma,axiom,
! [A3: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P2 @ V )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) )
@ V
@ P2 ) ) ) ) ).
% truth_lemma
thf(fact_872_finite__subset__induct_H,axiom,
! [F3: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ F3 )
=> ( ( ord_le7112219575281605754c_fm_a @ F3 @ A3 )
=> ( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a,F4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ F4 )
=> ( ( member536094252920883875c_fm_a @ A6 @ A3 )
=> ( ( ord_le7112219575281605754c_fm_a @ F4 @ A3 )
=> ( ~ ( member536094252920883875c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7129737658439086282c_fm_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_873_finite__subset__induct_H,axiom,
! [F3: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ F3 )
=> ( ( ord_le3275665582123262618c_fm_a @ F3 @ A3 )
=> ( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ! [A6: epistemic_fm_a,F4: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ F4 )
=> ( ( member6642669571620171971c_fm_a @ A6 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ F4 @ A3 )
=> ( ~ ( member6642669571620171971c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7817948963269374698c_fm_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_874_finite__subset__induct,axiom,
! [F3: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ F3 )
=> ( ( ord_le7112219575281605754c_fm_a @ F3 @ A3 )
=> ( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a,F4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ F4 )
=> ( ( member536094252920883875c_fm_a @ A6 @ A3 )
=> ( ~ ( member536094252920883875c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7129737658439086282c_fm_a @ A6 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_875_finite__subset__induct,axiom,
! [F3: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ F3 )
=> ( ( ord_le3275665582123262618c_fm_a @ F3 @ A3 )
=> ( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ! [A6: epistemic_fm_a,F4: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ F4 )
=> ( ( member6642669571620171971c_fm_a @ A6 @ A3 )
=> ( ~ ( member6642669571620171971c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7817948963269374698c_fm_a @ A6 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_876_reflexive_092_060_094sub_062T,axiom,
! [A3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A3 )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_877_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A3 )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_878_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A3 )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_879_exists__finite__inconsistent,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,V: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [W2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W2 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ W2 )
=> ( ( finite3304564945125563331c_fm_a @ W2 )
=> ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W2 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_880_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A3 )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_881_inconsistent__imply,axiom,
! [A3: epistemic_fm_a > $o,P2: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ ( set_Epistemic_fm_a2 @ G5 ) ) )
=> ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ G5 @ P2 ) ) ) ).
% inconsistent_imply
thf(fact_882_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P2: epistemic_fm_a,A3: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P2 ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A3 @ P2 ) ) ) ).
% completeness
thf(fact_883_le__sup__iff,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) @ Z3 )
= ( ( ord_le3275665582123262618c_fm_a @ X2 @ Z3 )
& ( ord_le3275665582123262618c_fm_a @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_884_sup_Obounded__iff,axiom,
! [B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B3 @ C ) @ A )
= ( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
& ( ord_le3275665582123262618c_fm_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_885_sup__set__def,axiom,
( sup_su4305657963326700718c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ( sup_su2585399871592001967fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A4 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_886_sup__set__def,axiom,
( sup_su1367922730591523534c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ( sup_su6697315146409554831fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A4 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_887_sup__Un__eq,axiom,
! [R: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( sup_su6697315146409554831fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ R )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ S ) )
= ( ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_888_sup__Un__eq,axiom,
! [R: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( sup_su2585399871592001967fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ R )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ S ) )
= ( ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ ( sup_su4305657963326700718c_fm_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_889_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_890_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_891_sup_Oabsorb__iff2,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A2 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_892_sup_Oabsorb__iff1,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B4: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A2 @ B4 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_893_sup_Ocobounded2,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B3 @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_894_sup_Ocobounded1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_895_sup_Oorder__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B4: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( A2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_896_sup_OboundedI,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_897_sup_OboundedE,axiom,
! [B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B3 @ C ) @ A )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ~ ( ord_le3275665582123262618c_fm_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_898_sup__absorb2,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
=> ( ( sup_su1367922730591523534c_fm_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_899_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X2 )
=> ( ( sup_su1367922730591523534c_fm_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_900_sup_Oabsorb2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_901_sup_Oabsorb1,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_902_sup__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X )
=> ( ( ord_le3275665582123262618c_fm_a @ Z4 @ X )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y2 @ Z4 ) @ X ) ) )
=> ( ( sup_su1367922730591523534c_fm_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_903_sup_OorderI,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( A
= ( sup_su1367922730591523534c_fm_a @ A @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ B3 @ A ) ) ).
% sup.orderI
thf(fact_904_sup_OorderE,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( A
= ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_905_le__iff__sup,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X3: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_906_sup__least,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z3 @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ Y @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_907_sup__mono,axiom,
! [A: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,D2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ D2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_908_sup_Omono,axiom,
! [C: set_Epistemic_fm_a,A: set_Epistemic_fm_a,D2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ D2 @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_909_le__supI2,axiom,
! [X2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_910_le__supI1,axiom,
! [X2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_911_sup__ge2,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_912_sup__ge1,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X2 @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_913_le__supI,axiom,
! [A: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_914_le__supE,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B3 ) @ X2 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A @ X2 )
=> ~ ( ord_le3275665582123262618c_fm_a @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_915_inf__sup__ord_I3_J,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X2 @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_916_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_917_extendS__def,axiom,
! [A3: epistemic_fm_a > $o,N: epistemic_fm_a,Prev: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ S @ N @ Prev )
= ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) ) )
& ( ~ ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_918_MCS_Omaximal__def,axiom,
! [Consistent: set_se5208064806568342746c_fm_a > $o,S: set_se5208064806568342746c_fm_a] :
( ( maxima4972129789629145062c_fm_a @ Consistent )
=> ( ( maxima7173642526191862697c_fm_a @ Consistent @ S )
= ( ! [P: set_Epistemic_fm_a] :
( ( Consistent @ ( sup_su4305657963326700718c_fm_a @ ( insert7129737658439086282c_fm_a @ P @ bot_bo1868452413526482246c_fm_a ) @ S ) )
=> ( member536094252920883875c_fm_a @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_919_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima1924290459274269190c_fm_a @ Consistent )
=> ( ( maxima3264069584562520521c_fm_a @ Consistent @ S )
= ( ! [P: epistemic_fm_a] :
( ( Consistent @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_920_MCS__axioms,axiom,
! [A3: epistemic_fm_a > $o] : ( maxima1924290459274269190c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) ) ).
% MCS_axioms
thf(fact_921_is__singletonI_H,axiom,
! [A3: set_Epistemic_fm_a] :
( ( A3 != bot_bo3626323581529592678c_fm_a )
=> ( ! [X: epistemic_fm_a,Y2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( ( member6642669571620171971c_fm_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_sin4923117394748398094c_fm_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_922_is__singletonI_H,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ( member536094252920883875c_fm_a @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_sin1790460086249228270c_fm_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_923_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a,S2: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S2 @ S )
=> ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_924_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S4 @ S )
& ( finite3304564945125563331c_fm_a @ S4 )
& ~ ( Consistent @ S4 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_925_subset__singleton__iff__Uniq,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( ? [A2: set_Epistemic_fm_a] : ( ord_le7112219575281605754c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A2 @ bot_bo1868452413526482246c_fm_a ) ) )
= ( uniq_s5323624820373029887c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_926_subset__singleton__iff__Uniq,axiom,
! [A3: set_Epistemic_fm_a] :
( ( ? [A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) )
= ( uniq_Epistemic_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_927_Inf__fin_Osubset__imp,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ B ) @ ( lattic7927669795667837335c_fm_a @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_928_Sup__fin_Osubset__imp,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A3 ) @ ( lattic3176627891459829297c_fm_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_929_Inf__fin__le__Sup__fin,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A3 ) @ ( lattic3176627891459829297c_fm_a @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_930_Sup__fin_OcoboundedI,axiom,
! [A3: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( lattic3176627891459829297c_fm_a @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_931_Inf__fin_OcoboundedI,axiom,
! [A3: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_932_Inf__fin_OboundedE,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ ( lattic7927669795667837335c_fm_a @ A3 ) )
=> ! [A7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A7 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_933_Inf__fin_OboundedI,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ A6 ) )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( lattic7927669795667837335c_fm_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_934_Sup__fin_OboundedE,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A3 ) @ X2 )
=> ! [A7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A7 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_935_Sup__fin_OboundedI,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ A6 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A3 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_936_Inf__fin_Obounded__iff,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ ( lattic7927669795667837335c_fm_a @ A3 ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_937_Sup__fin_Obounded__iff,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A3 ) @ X2 )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_938_sup__Inf2__distrib,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B )
=> ( ( B != bot_bo1868452413526482246c_fm_a )
=> ( ( sup_su1367922730591523534c_fm_a @ ( lattic7927669795667837335c_fm_a @ A3 ) @ ( lattic7927669795667837335c_fm_a @ B ) )
= ( lattic7927669795667837335c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( Uu2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B4 ) )
& ( member536094252920883875c_fm_a @ A2 @ A3 )
& ( member536094252920883875c_fm_a @ B4 @ B ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_939_sup__Inf1__distrib,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( sup_su1367922730591523534c_fm_a @ X2 @ ( lattic7927669795667837335c_fm_a @ A3 ) )
= ( lattic7927669795667837335c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A2: set_Epistemic_fm_a] :
( ( Uu2
= ( sup_su1367922730591523534c_fm_a @ X2 @ A2 ) )
& ( member536094252920883875c_fm_a @ A2 @ A3 ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_940_finite__Collect__bounded__ex,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
? [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
& ( Q2 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( Q2 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_941_finite__Collect__bounded__ex,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
? [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
& ( Q2 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( Q2 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_942_finite__Collect__bounded__ex,axiom,
! [P4: epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
? [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
& ( Q2 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( Q2 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_943_finite__Collect__bounded__ex,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
? [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
& ( Q2 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( Q2 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_944_finite__image__set,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_945_finite__image__set,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_946_finite__image__set,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_947_finite__image__set,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) ) ) ) ).
% finite_image_set
thf(fact_948_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_949_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a > epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_950_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a,Y4: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_951_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a,Y4: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_952_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_953_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q2 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_954_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a,Y4: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_955_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q2 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a,Y4: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q2 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_956_Collect__insert,axiom,
! [R: epistemic_fm_a > set_Epistemic_fm_a,J1: epistemic_fm_a,J: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J2: epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member6642669571620171971c_fm_a @ J2 @ ( insert7817948963269374698c_fm_a @ J1 @ J ) ) ) )
= ( insert7129737658439086282c_fm_a @ ( R @ J1 )
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J2: epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member6642669571620171971c_fm_a @ J2 @ J ) ) ) ) ) ).
% Collect_insert
thf(fact_957_Collect__insert,axiom,
! [R: set_Epistemic_fm_a > set_Epistemic_fm_a,J1: set_Epistemic_fm_a,J: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J2: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member536094252920883875c_fm_a @ J2 @ ( insert7129737658439086282c_fm_a @ J1 @ J ) ) ) )
= ( insert7129737658439086282c_fm_a @ ( R @ J1 )
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J2: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member536094252920883875c_fm_a @ J2 @ J ) ) ) ) ) ).
% Collect_insert
thf(fact_958_Collect__insert,axiom,
! [R: epistemic_fm_a > epistemic_fm_a,J1: epistemic_fm_a,J: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J2: epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member6642669571620171971c_fm_a @ J2 @ ( insert7817948963269374698c_fm_a @ J1 @ J ) ) ) )
= ( insert7817948963269374698c_fm_a @ ( R @ J1 )
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J2: epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member6642669571620171971c_fm_a @ J2 @ J ) ) ) ) ) ).
% Collect_insert
thf(fact_959_Collect__insert,axiom,
! [R: set_Epistemic_fm_a > epistemic_fm_a,J1: set_Epistemic_fm_a,J: set_se5208064806568342746c_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J2: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member536094252920883875c_fm_a @ J2 @ ( insert7129737658439086282c_fm_a @ J1 @ J ) ) ) )
= ( insert7817948963269374698c_fm_a @ ( R @ J1 )
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J2: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J2 ) )
& ( member536094252920883875c_fm_a @ J2 @ J ) ) ) ) ) ).
% Collect_insert
thf(fact_960_set__Cons__def,axiom,
( set_Co4868574198862222544c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,XS: set_li769143395467472256c_fm_a] :
( collec5191077796991884427c_fm_a
@ ^ [Z5: list_Epistemic_fm_a] :
? [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] :
( ( Z5
= ( cons_Epistemic_fm_a @ X3 @ Xs3 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A4 )
& ( member5906877432388582473c_fm_a @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_961_set__Cons__def,axiom,
( set_Co7170608787392608944c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,XS: set_li4204741992506657632c_fm_a] :
( collec3923057941238076011c_fm_a
@ ^ [Z5: list_s580375451141968640c_fm_a] :
? [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] :
( ( Z5
= ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) )
& ( member536094252920883875c_fm_a @ X3 @ A4 )
& ( member2328277852289759785c_fm_a @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_962_inf__Sup1__distrib,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( inf_in2882328776738922472c_fm_a @ X2 @ ( lattic3176627891459829297c_fm_a @ A3 ) )
= ( lattic3176627891459829297c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A2: set_Epistemic_fm_a] :
( ( Uu2
= ( inf_in2882328776738922472c_fm_a @ X2 @ A2 ) )
& ( member536094252920883875c_fm_a @ A2 @ A3 ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_963_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) )
= ( ( ord_le3275665582123262618c_fm_a @ C2 @ A3 )
& ( ord_le3275665582123262618c_fm_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_964_Int__insert__right__if1,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A @ B ) )
= ( insert7817948963269374698c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_965_Int__insert__right__if1,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A @ B ) )
= ( insert7129737658439086282c_fm_a @ A @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_966_Int__insert__right__if0,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A @ B ) )
= ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_967_Int__insert__right__if0,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A @ B ) )
= ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_968_Int__insert__left__if1,axiom,
! [A: epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ C2 )
=> ( ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ B ) @ C2 )
= ( insert7817948963269374698c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_969_Int__insert__left__if1,axiom,
! [A: set_Epistemic_fm_a,C2: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ C2 )
=> ( ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ B ) @ C2 )
= ( insert7129737658439086282c_fm_a @ A @ ( inf_in1884693029477671368c_fm_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_970_Int__insert__left__if0,axiom,
! [A: epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ A @ C2 )
=> ( ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ B ) @ C2 )
= ( inf_in2882328776738922472c_fm_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_971_Int__insert__left__if0,axiom,
! [A: set_Epistemic_fm_a,C2: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ A @ C2 )
=> ( ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ B ) @ C2 )
= ( inf_in1884693029477671368c_fm_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_972_inf_Obounded__iff,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ B3 @ C ) )
= ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
& ( ord_le3275665582123262618c_fm_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_973_le__inf__iff,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z3 ) )
= ( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
& ( ord_le3275665582123262618c_fm_a @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_974_insert__disjoint_I1_J,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ A3 ) @ B )
= bot_bo3626323581529592678c_fm_a )
= ( ~ ( member6642669571620171971c_fm_a @ A @ B )
& ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% insert_disjoint(1)
thf(fact_975_insert__disjoint_I1_J,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ A3 ) @ B )
= bot_bo1868452413526482246c_fm_a )
= ( ~ ( member536094252920883875c_fm_a @ A @ B )
& ( ( inf_in1884693029477671368c_fm_a @ A3 @ B )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% insert_disjoint(1)
thf(fact_976_insert__disjoint_I2_J,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( bot_bo3626323581529592678c_fm_a
= ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ A3 ) @ B ) )
= ( ~ ( member6642669571620171971c_fm_a @ A @ B )
& ( bot_bo3626323581529592678c_fm_a
= ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_977_insert__disjoint_I2_J,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( bot_bo1868452413526482246c_fm_a
= ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ A3 ) @ B ) )
= ( ~ ( member536094252920883875c_fm_a @ A @ B )
& ( bot_bo1868452413526482246c_fm_a
= ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_978_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_a,A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ( inf_in2882328776738922472c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ A @ A3 ) )
= bot_bo3626323581529592678c_fm_a )
= ( ~ ( member6642669571620171971c_fm_a @ A @ B )
& ( ( inf_in2882328776738922472c_fm_a @ B @ A3 )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% disjoint_insert(1)
thf(fact_979_disjoint__insert_I1_J,axiom,
! [B: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( ( inf_in1884693029477671368c_fm_a @ B @ ( insert7129737658439086282c_fm_a @ A @ A3 ) )
= bot_bo1868452413526482246c_fm_a )
= ( ~ ( member536094252920883875c_fm_a @ A @ B )
& ( ( inf_in1884693029477671368c_fm_a @ B @ A3 )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% disjoint_insert(1)
thf(fact_980_disjoint__insert_I2_J,axiom,
! [A3: set_Epistemic_fm_a,B3: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( bot_bo3626323581529592678c_fm_a
= ( inf_in2882328776738922472c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ B3 @ B ) ) )
= ( ~ ( member6642669571620171971c_fm_a @ B3 @ A3 )
& ( bot_bo3626323581529592678c_fm_a
= ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_981_disjoint__insert_I2_J,axiom,
! [A3: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( bot_bo1868452413526482246c_fm_a
= ( inf_in1884693029477671368c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ B3 @ B ) ) )
= ( ~ ( member536094252920883875c_fm_a @ B3 @ A3 )
& ( bot_bo1868452413526482246c_fm_a
= ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_982_inf_OcoboundedI2,axiom,
! [B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ C )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_983_inf_OcoboundedI1,axiom,
! [A: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_984_inf_Oabsorb__iff2,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B4: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ A2 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_985_inf_Oabsorb__iff1,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ A2 @ B4 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_986_inf_Ocobounded2,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_987_inf_Ocobounded1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_988_inf_Oorder__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( A2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_989_inf__greatest,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Z3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_990_inf_OboundedI,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ C )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_991_inf_OboundedE,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ B3 @ C ) )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ~ ( ord_le3275665582123262618c_fm_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_992_inf__absorb2,axiom,
! [Y: set_Epistemic_fm_a,X2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X2 )
=> ( ( inf_in2882328776738922472c_fm_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_993_inf__absorb1,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
=> ( ( inf_in2882328776738922472c_fm_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_994_inf_Oabsorb2,axiom,
! [B3: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( ( inf_in2882328776738922472c_fm_a @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_995_inf_Oabsorb1,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_996_le__iff__inf,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X3: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_997_inf__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ X @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in2882328776738922472c_fm_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_998_inf_OorderI,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( A
= ( inf_in2882328776738922472c_fm_a @ A @ B3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B3 ) ) ).
% inf.orderI
thf(fact_999_inf_OorderE,axiom,
! [A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B3 )
=> ( A
= ( inf_in2882328776738922472c_fm_a @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_1000_le__infI2,axiom,
! [B3: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ X2 ) ) ).
% le_infI2
thf(fact_1001_le__infI1,axiom,
! [A: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ X2 ) ) ).
% le_infI1
thf(fact_1002_inf__mono,axiom,
! [A: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,D2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B3 @ D2 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) @ ( inf_in2882328776738922472c_fm_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1003_le__infI,axiom,
! [X2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ B3 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_1004_le__infE,axiom,
! [X2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ A @ B3 ) )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ X2 @ A )
=> ~ ( ord_le3275665582123262618c_fm_a @ X2 @ B3 ) ) ) ).
% le_infE
thf(fact_1005_inf__le2,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1006_inf__le1,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1007_inf__sup__ord_I1_J,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1008_inf__sup__ord_I2_J,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1009_disjoint__iff,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= bot_bo3626323581529592678c_fm_a )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
=> ~ ( member6642669571620171971c_fm_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1010_disjoint__iff,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( inf_in1884693029477671368c_fm_a @ A3 @ B )
= bot_bo1868452413526482246c_fm_a )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
=> ~ ( member536094252920883875c_fm_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1011_Int__emptyI,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ~ ( member6642669571620171971c_fm_a @ X @ B ) )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= bot_bo3626323581529592678c_fm_a ) ) ).
% Int_emptyI
thf(fact_1012_Int__emptyI,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ~ ( member536094252920883875c_fm_a @ X @ B ) )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ B )
= bot_bo1868452413526482246c_fm_a ) ) ).
% Int_emptyI
thf(fact_1013_Int__mono,axiom,
! [A3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) @ ( inf_in2882328776738922472c_fm_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1014_Int__lower1,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_1015_Int__lower2,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_1016_Int__absorb1,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1017_Int__absorb2,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_1018_Int__greatest,axiom,
! [C2: set_Epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ C2 @ B )
=> ( ord_le3275665582123262618c_fm_a @ C2 @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1019_Int__Collect__mono,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ( P4 @ X )
=> ( Q2 @ X ) ) )
=> ( ord_le7112219575281605754c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A3 @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ ( inf_in1884693029477671368c_fm_a @ B @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1020_Int__Collect__mono,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( ( P4 @ X )
=> ( Q2 @ X ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A3 @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ ( inf_in2882328776738922472c_fm_a @ B @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1021_Int__insert__right,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A @ B ) )
= ( insert7817948963269374698c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) )
& ( ~ ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A @ B ) )
= ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1022_Int__insert__right,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A @ B ) )
= ( insert7129737658439086282c_fm_a @ A @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) )
& ( ~ ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A @ B ) )
= ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1023_Int__insert__left,axiom,
! [A: epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ A @ C2 )
=> ( ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ B ) @ C2 )
= ( insert7817948963269374698c_fm_a @ A @ ( inf_in2882328776738922472c_fm_a @ B @ C2 ) ) ) )
& ( ~ ( member6642669571620171971c_fm_a @ A @ C2 )
=> ( ( inf_in2882328776738922472c_fm_a @ ( insert7817948963269374698c_fm_a @ A @ B ) @ C2 )
= ( inf_in2882328776738922472c_fm_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1024_Int__insert__left,axiom,
! [A: set_Epistemic_fm_a,C2: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( member536094252920883875c_fm_a @ A @ C2 )
=> ( ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ B ) @ C2 )
= ( insert7129737658439086282c_fm_a @ A @ ( inf_in1884693029477671368c_fm_a @ B @ C2 ) ) ) )
& ( ~ ( member536094252920883875c_fm_a @ A @ C2 )
=> ( ( inf_in1884693029477671368c_fm_a @ ( insert7129737658439086282c_fm_a @ A @ B ) @ C2 )
= ( inf_in1884693029477671368c_fm_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1025_lfp__greatest,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ! [U3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ U3 ) @ U3 )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ U3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ ( comple693844135356717107c_fm_a @ F ) ) ) ).
% lfp_greatest
thf(fact_1026_lfp__lowerbound,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ A3 ) @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ A3 ) ) ).
% lfp_lowerbound
thf(fact_1027_lfp__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z6: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ Z6 ) @ ( G2 @ Z6 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ ( comple693844135356717107c_fm_a @ G2 ) ) ) ).
% lfp_mono
thf(fact_1028_lfp__lfp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,W4: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ W4 @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X @ W4 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( comple693844135356717107c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( comple693844135356717107c_fm_a @ ( F @ X3 ) ) )
= ( comple693844135356717107c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( F @ X3 @ X3 ) ) ) ) ).
% lfp_lfp
thf(fact_1029_distrib__sup__le,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z3 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X2 @ Y ) @ ( sup_su1367922730591523534c_fm_a @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1030_distrib__inf__le,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X2 @ Y ) @ ( inf_in2882328776738922472c_fm_a @ X2 @ Z3 ) ) @ ( inf_in2882328776738922472c_fm_a @ X2 @ ( sup_su1367922730591523534c_fm_a @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1031_Un__Int__assoc__eq,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ( sup_su1367922730591523534c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) @ C2 )
= ( inf_in2882328776738922472c_fm_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ B @ C2 ) ) )
= ( ord_le3275665582123262618c_fm_a @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1032_inf__Sup2__distrib,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( A3 != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B )
=> ( ( B != bot_bo1868452413526482246c_fm_a )
=> ( ( inf_in2882328776738922472c_fm_a @ ( lattic3176627891459829297c_fm_a @ A3 ) @ ( lattic3176627891459829297c_fm_a @ B ) )
= ( lattic3176627891459829297c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( Uu2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B4 ) )
& ( member536094252920883875c_fm_a @ A2 @ A3 )
& ( member536094252920883875c_fm_a @ B4 @ B ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_1033_IntI,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( ( member6642669571620171971c_fm_a @ C @ B )
=> ( member6642669571620171971c_fm_a @ C @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_1034_IntI,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ A3 )
=> ( ( member536094252920883875c_fm_a @ C @ B )
=> ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_1035_Int__iff,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) )
= ( ( member6642669571620171971c_fm_a @ C @ A3 )
& ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_1036_Int__iff,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) )
= ( ( member536094252920883875c_fm_a @ C @ A3 )
& ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_1037_inf__Int__eq,axiom,
! [R: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( inf_in3606484609122063093fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ R )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ S ) )
= ( ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1038_inf__Int__eq,axiom,
! [R: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( inf_in5938521974262266133fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ R )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ S ) )
= ( ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ ( inf_in1884693029477671368c_fm_a @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1039_IntE,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) )
=> ~ ( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ~ ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% IntE
thf(fact_1040_IntE,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) )
=> ~ ( ( member536094252920883875c_fm_a @ C @ A3 )
=> ~ ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% IntE
thf(fact_1041_IntD1,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) )
=> ( member6642669571620171971c_fm_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1042_IntD1,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) )
=> ( member536094252920883875c_fm_a @ C @ A3 ) ) ).
% IntD1
thf(fact_1043_IntD2,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( inf_in2882328776738922472c_fm_a @ A3 @ B ) )
=> ( member6642669571620171971c_fm_a @ C @ B ) ) ).
% IntD2
thf(fact_1044_IntD2,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A3 @ B ) )
=> ( member536094252920883875c_fm_a @ C @ B ) ) ).
% IntD2
thf(fact_1045_inf__set__def,axiom,
( inf_in1884693029477671368c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ( inf_in5938521974262266133fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A4 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1046_inf__set__def,axiom,
( inf_in2882328776738922472c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ( inf_in3606484609122063093fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A4 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1047_Collect__conj__eq,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in1884693029477671368c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1048_Collect__conj__eq,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in2882328776738922472c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_1049_Int__Collect,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X2 @ ( inf_in1884693029477671368c_fm_a @ A3 @ ( collec2519470961442302949c_fm_a @ P4 ) ) )
= ( ( member536094252920883875c_fm_a @ X2 @ A3 )
& ( P4 @ X2 ) ) ) ).
% Int_Collect
thf(fact_1050_Int__Collect,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X2 @ ( inf_in2882328776738922472c_fm_a @ A3 @ ( collec4904205152690461189c_fm_a @ P4 ) ) )
= ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
& ( P4 @ X2 ) ) ) ).
% Int_Collect
thf(fact_1051_Int__def,axiom,
( inf_in1884693029477671368c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A4 )
& ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_1052_Int__def,axiom,
( inf_in2882328776738922472c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A4 )
& ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_1053_inter__Set__filter,axiom,
! [B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B )
=> ( ( inf_in2882328776738922472c_fm_a @ A3 @ B )
= ( filter6053540574317342667c_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_1054_inter__Set__filter,axiom,
! [B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ B )
=> ( ( inf_in1884693029477671368c_fm_a @ A3 @ B )
= ( filter7881760030630619563c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_1055_image__eqI,axiom,
! [B3: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,X2: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1056_image__eqI,axiom,
! [B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,X2: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ B3 @ ( image_1263732536703906021c_fm_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1057_image__eqI,axiom,
! [B3: epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ B3 @ ( image_3376480847214465317c_fm_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1058_image__eqI,axiom,
! [B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ B3 @ ( image_8582343580727714565c_fm_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1059_member__filter,axiom,
! [X2: epistemic_fm_a,P4: epistemic_fm_a > $o,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( filter6053540574317342667c_fm_a @ P4 @ A3 ) )
= ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
& ( P4 @ X2 ) ) ) ).
% member_filter
thf(fact_1060_member__filter,axiom,
! [X2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( filter7881760030630619563c_fm_a @ P4 @ A3 ) )
= ( ( member536094252920883875c_fm_a @ X2 @ A3 )
& ( P4 @ X2 ) ) ) ).
% member_filter
thf(fact_1061_all__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ( P4 @ B2 ) ) )
= ( ! [B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A3 )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_1062_image__mono,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A3 ) @ ( image_4449434806354059013c_fm_a @ F @ B ) ) ) ).
% image_mono
thf(fact_1063_image__subsetI,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( member536094252920883875c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1064_image__subsetI,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( member536094252920883875c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1065_image__subsetI,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( member6642669571620171971c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1066_image__subsetI,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( member6642669571620171971c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_1067_subset__imageE,axiom,
! [B: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ~ ! [C4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C4 @ A3 )
=> ( B
!= ( image_4449434806354059013c_fm_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1068_subset__image__iff,axiom,
! [B: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A3 )
& ( B
= ( image_4449434806354059013c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1069_imageI,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ ( image_4449434806354059013c_fm_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1070_imageI,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ ( image_1263732536703906021c_fm_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1071_imageI,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ ( image_3376480847214465317c_fm_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1072_imageI,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ ( image_8582343580727714565c_fm_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_1073_rev__image__eqI,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B3: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member6642669571620171971c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1074_rev__image__eqI,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member536094252920883875c_fm_a @ B3 @ ( image_1263732536703906021c_fm_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1075_rev__image__eqI,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B3: epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member6642669571620171971c_fm_a @ B3 @ ( image_3376480847214465317c_fm_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1076_rev__image__eqI,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member536094252920883875c_fm_a @ B3 @ ( image_8582343580727714565c_fm_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1077_imageE,axiom,
! [B3: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ~ ! [X: epistemic_fm_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member6642669571620171971c_fm_a @ X @ A3 ) ) ) ).
% imageE
thf(fact_1078_imageE,axiom,
! [B3: epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ B3 @ ( image_3376480847214465317c_fm_a @ F @ A3 ) )
=> ~ ! [X: set_Epistemic_fm_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member536094252920883875c_fm_a @ X @ A3 ) ) ) ).
% imageE
thf(fact_1079_imageE,axiom,
! [B3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ ( image_1263732536703906021c_fm_a @ F @ A3 ) )
=> ~ ! [X: epistemic_fm_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member6642669571620171971c_fm_a @ X @ A3 ) ) ) ).
% imageE
thf(fact_1080_imageE,axiom,
! [B3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ ( image_8582343580727714565c_fm_a @ F @ A3 ) )
=> ~ ! [X: set_Epistemic_fm_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member536094252920883875c_fm_a @ X @ A3 ) ) ) ).
% imageE
thf(fact_1081_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( image_8582343580727714565c_fm_a @ F @ A3 ) )
& ( P4 @ X3 ) ) )
= ( image_8582343580727714565c_fm_a @ F
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1082_Compr__image__eq,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( image_1263732536703906021c_fm_a @ F @ A3 ) )
& ( P4 @ X3 ) ) )
= ( image_1263732536703906021c_fm_a @ F
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1083_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,P4: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( image_3376480847214465317c_fm_a @ F @ A3 ) )
& ( P4 @ X3 ) ) )
= ( image_3376480847214465317c_fm_a @ F
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1084_Compr__image__eq,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
& ( P4 @ X3 ) ) )
= ( image_4449434806354059013c_fm_a @ F
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1085_Set_Ofilter__def,axiom,
( filter7881760030630619563c_fm_a
= ( ^ [P5: set_Epistemic_fm_a > $o,A4: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [A2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ A4 )
& ( P5 @ A2 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1086_Set_Ofilter__def,axiom,
( filter6053540574317342667c_fm_a
= ( ^ [P5: epistemic_fm_a > $o,A4: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [A2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A4 )
& ( P5 @ A2 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1087_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( member536094252920883875c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1088_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( member536094252920883875c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1089_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( member6642669571620171971c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1090_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( member6642669571620171971c_fm_a @ ( F @ X ) @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1091_setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) )
= ( image_8582343580727714565c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) ) ).
% setcompr_eq_image
thf(fact_1092_setcompr__eq__image,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) )
= ( image_1263732536703906021c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) ) ).
% setcompr_eq_image
thf(fact_1093_setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) )
= ( image_3376480847214465317c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) ) ).
% setcompr_eq_image
thf(fact_1094_setcompr__eq__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( P4 @ X3 ) ) )
= ( image_4449434806354059013c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) ) ).
% setcompr_eq_image
thf(fact_1095_Setcompr__eq__image,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A3 ) ) )
= ( image_1263732536703906021c_fm_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_1096_Setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member536094252920883875c_fm_a @ X3 @ A3 ) ) )
= ( image_8582343580727714565c_fm_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_1097_Setcompr__eq__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A3 ) ) )
= ( image_4449434806354059013c_fm_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_1098_Setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member536094252920883875c_fm_a @ X3 @ A3 ) ) )
= ( image_3376480847214465317c_fm_a @ F @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_1099_all__finite__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B2: set_Epistemic_fm_a] :
( ( ( finite3304564945125563331c_fm_a @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) ) )
=> ( P4 @ B2 ) ) )
= ( ! [B2: set_Epistemic_fm_a] :
( ( ( finite3304564945125563331c_fm_a @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A3 ) )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1100_ex__finite__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ? [B2: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
& ( P4 @ B2 ) ) )
= ( ? [B2: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A3 )
& ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1101_finite__subset__image,axiom,
! [B: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ? [C4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C4 @ A3 )
& ( finite3304564945125563331c_fm_a @ C4 )
& ( B
= ( image_4449434806354059013c_fm_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1102_Diff__iff,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( minus_975701618739183585c_fm_a @ A3 @ B ) )
= ( ( member6642669571620171971c_fm_a @ C @ A3 )
& ~ ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1103_Diff__iff,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) )
= ( ( member536094252920883875c_fm_a @ C @ A3 )
& ~ ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1104_DiffI,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( ~ ( member6642669571620171971c_fm_a @ C @ B )
=> ( member6642669571620171971c_fm_a @ C @ ( minus_975701618739183585c_fm_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1105_DiffI,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ A3 )
=> ( ~ ( member536094252920883875c_fm_a @ C @ B )
=> ( member536094252920883875c_fm_a @ C @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1106_Diff__insert0,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( minus_975701618739183585c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ B ) )
= ( minus_975701618739183585c_fm_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1107_Diff__insert0,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( minus_2796033211270405569c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ X2 @ B ) )
= ( minus_2796033211270405569c_fm_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1108_insert__Diff1,axiom,
! [X2: epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B )
=> ( ( minus_975701618739183585c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ A3 ) @ B )
= ( minus_975701618739183585c_fm_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1109_insert__Diff1,axiom,
! [X2: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B )
=> ( ( minus_2796033211270405569c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ A3 ) @ B )
= ( minus_2796033211270405569c_fm_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1110_Diff__eq__empty__iff,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( minus_975701618739183585c_fm_a @ A3 @ B )
= bot_bo3626323581529592678c_fm_a )
= ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1111_Diff__mono,axiom,
! [A3: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ D @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ B ) @ ( minus_975701618739183585c_fm_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1112_Diff__subset,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_1113_double__diff,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C2 )
=> ( ( minus_975701618739183585c_fm_a @ B @ ( minus_975701618739183585c_fm_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1114_set__diff__eq,axiom,
( minus_2796033211270405569c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A4 )
& ~ ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1115_set__diff__eq,axiom,
( minus_975701618739183585c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A4 )
& ~ ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1116_DiffD2,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( minus_975701618739183585c_fm_a @ A3 @ B ) )
=> ~ ( member6642669571620171971c_fm_a @ C @ B ) ) ).
% DiffD2
thf(fact_1117_DiffD2,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) )
=> ~ ( member536094252920883875c_fm_a @ C @ B ) ) ).
% DiffD2
thf(fact_1118_DiffD1,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( minus_975701618739183585c_fm_a @ A3 @ B ) )
=> ( member6642669571620171971c_fm_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1119_DiffD1,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) )
=> ( member536094252920883875c_fm_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1120_DiffE,axiom,
! [C: epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ C @ ( minus_975701618739183585c_fm_a @ A3 @ B ) )
=> ~ ( ( member6642669571620171971c_fm_a @ C @ A3 )
=> ( member6642669571620171971c_fm_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1121_DiffE,axiom,
! [C: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) )
=> ~ ( ( member536094252920883875c_fm_a @ C @ A3 )
=> ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1122_insert__Diff__if,axiom,
! [X2: epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ X2 @ B )
=> ( ( minus_975701618739183585c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ A3 ) @ B )
= ( minus_975701618739183585c_fm_a @ A3 @ B ) ) )
& ( ~ ( member6642669571620171971c_fm_a @ X2 @ B )
=> ( ( minus_975701618739183585c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ A3 ) @ B )
= ( insert7817948963269374698c_fm_a @ X2 @ ( minus_975701618739183585c_fm_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1123_insert__Diff__if,axiom,
! [X2: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( ( member536094252920883875c_fm_a @ X2 @ B )
=> ( ( minus_2796033211270405569c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ A3 ) @ B )
= ( minus_2796033211270405569c_fm_a @ A3 @ B ) ) )
& ( ~ ( member536094252920883875c_fm_a @ X2 @ B )
=> ( ( minus_2796033211270405569c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ A3 ) @ B )
= ( insert7129737658439086282c_fm_a @ X2 @ ( minus_2796033211270405569c_fm_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1124_Diff__partition,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( sup_su1367922730591523534c_fm_a @ A3 @ ( minus_975701618739183585c_fm_a @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_1125_Diff__subset__conv,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ B ) @ C2 )
= ( ord_le3275665582123262618c_fm_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1126_subset__Diff__insert,axiom,
! [A3: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a,C2: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ ( minus_2796033211270405569c_fm_a @ B @ ( insert7129737658439086282c_fm_a @ X2 @ C2 ) ) )
= ( ( ord_le7112219575281605754c_fm_a @ A3 @ ( minus_2796033211270405569c_fm_a @ B @ C2 ) )
& ~ ( member536094252920883875c_fm_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1127_subset__Diff__insert,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X2: epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( minus_975701618739183585c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ X2 @ C2 ) ) )
= ( ( ord_le3275665582123262618c_fm_a @ A3 @ ( minus_975701618739183585c_fm_a @ B @ C2 ) )
& ~ ( member6642669571620171971c_fm_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1128_in__image__insert__iff,axiom,
! [B: set_se8609989359752948410c_fm_a,X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ! [C4: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ C4 @ B )
=> ~ ( member536094252920883875c_fm_a @ X2 @ C4 ) )
=> ( ( member3183810400728440963c_fm_a @ A3 @ ( image_5146850558160414469c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 ) @ B ) )
= ( ( member536094252920883875c_fm_a @ X2 @ A3 )
& ( member3183810400728440963c_fm_a @ ( minus_2796033211270405569c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ X2 @ bot_bo1868452413526482246c_fm_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1129_in__image__insert__iff,axiom,
! [B: set_se5208064806568342746c_fm_a,X2: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ! [C4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ C4 @ B )
=> ~ ( member6642669571620171971c_fm_a @ X2 @ C4 ) )
=> ( ( member536094252920883875c_fm_a @ A3 @ ( image_8582343580727714565c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 ) @ B ) )
= ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
& ( member536094252920883875c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1130_Diff__insert__absorb,axiom,
! [X2: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ( minus_975701618739183585c_fm_a @ ( insert7817948963269374698c_fm_a @ X2 @ A3 ) @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1131_Diff__insert__absorb,axiom,
! [X2: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ( minus_2796033211270405569c_fm_a @ ( insert7129737658439086282c_fm_a @ X2 @ A3 ) @ ( insert7129737658439086282c_fm_a @ X2 @ bot_bo1868452413526482246c_fm_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1132_insert__Diff,axiom,
! [A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ A3 )
=> ( ( insert7817948963269374698c_fm_a @ A @ ( minus_975701618739183585c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1133_insert__Diff,axiom,
! [A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ A3 )
=> ( ( insert7129737658439086282c_fm_a @ A @ ( minus_2796033211270405569c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1134_remdups_Osimps_I2_J,axiom,
! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( remdup2862625192277219637c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs ) )
= ( remdup2862625192277219637c_fm_a @ Xs ) ) )
& ( ~ ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( remdup2862625192277219637c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs ) )
= ( cons_Epistemic_fm_a @ X2 @ ( remdup2862625192277219637c_fm_a @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1135_remdups_Osimps_I2_J,axiom,
! [X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( remdup9148899285324835605c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X2 @ Xs ) )
= ( remdup9148899285324835605c_fm_a @ Xs ) ) )
& ( ~ ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( remdup9148899285324835605c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X2 @ Xs ) )
= ( cons_s4394448069393209776c_fm_a @ X2 @ ( remdup9148899285324835605c_fm_a @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1136_subset__insert__iff,axiom,
! [A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ X2 @ B ) )
= ( ( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ord_le7112219575281605754c_fm_a @ ( minus_2796033211270405569c_fm_a @ A3 @ ( insert7129737658439086282c_fm_a @ X2 @ bot_bo1868452413526482246c_fm_a ) ) @ B ) )
& ( ~ ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( ord_le7112219575281605754c_fm_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1137_subset__insert__iff,axiom,
! [A3: set_Epistemic_fm_a,X2: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ B ) )
= ( ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) @ B ) )
& ( ~ ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1138_Diff__single__insert,axiom,
! [A3: set_Epistemic_fm_a,X2: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( minus_975701618739183585c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) @ B )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ ( insert7817948963269374698c_fm_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1139_remove__induct,axiom,
! [P4: set_se5208064806568342746c_fm_a > $o,B: set_se5208064806568342746c_fm_a] :
( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ( ~ ( finite7364866884270929315c_fm_a @ B )
=> ( P4 @ B ) )
=> ( ! [A8: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A8 )
=> ( ( A8 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le7112219575281605754c_fm_a @ A8 @ B )
=> ( ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ A8 )
=> ( P4 @ ( minus_2796033211270405569c_fm_a @ A8 @ ( insert7129737658439086282c_fm_a @ X6 @ bot_bo1868452413526482246c_fm_a ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B ) ) ) ) ).
% remove_induct
thf(fact_1140_remove__induct,axiom,
! [P4: set_Epistemic_fm_a > $o,B: set_Epistemic_fm_a] :
( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ( ~ ( finite3304564945125563331c_fm_a @ B )
=> ( P4 @ B ) )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A8 )
=> ( ( A8 != bot_bo3626323581529592678c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ A8 @ B )
=> ( ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ A8 )
=> ( P4 @ ( minus_975701618739183585c_fm_a @ A8 @ ( insert7817948963269374698c_fm_a @ X6 @ bot_bo3626323581529592678c_fm_a ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B ) ) ) ) ).
% remove_induct
thf(fact_1141_finite__remove__induct,axiom,
! [B: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ B )
=> ( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ! [A8: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A8 )
=> ( ( A8 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le7112219575281605754c_fm_a @ A8 @ B )
=> ( ! [X6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X6 @ A8 )
=> ( P4 @ ( minus_2796033211270405569c_fm_a @ A8 @ ( insert7129737658439086282c_fm_a @ X6 @ bot_bo1868452413526482246c_fm_a ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1142_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ B )
=> ( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A8 )
=> ( ( A8 != bot_bo3626323581529592678c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ A8 @ B )
=> ( ! [X6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X6 @ A8 )
=> ( P4 @ ( minus_975701618739183585c_fm_a @ A8 @ ( insert7817948963269374698c_fm_a @ X6 @ bot_bo3626323581529592678c_fm_a ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1143_diff__shunt__var,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ( minus_975701618739183585c_fm_a @ X2 @ Y )
= bot_bo3626323581529592678c_fm_a )
= ( ord_le3275665582123262618c_fm_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1144_minus__set__def,axiom,
( minus_2796033211270405569c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ( minus_1533017855062696540fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A4 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1145_minus__set__def,axiom,
( minus_975701618739183585c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ( minus_1148766553201301564fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A4 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1146_member__remove,axiom,
! [X2: epistemic_fm_a,Y: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( remove5552923366785645183c_fm_a @ Y @ A3 ) )
= ( ( member6642669571620171971c_fm_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1147_member__remove,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( remove824461714993407071c_fm_a @ Y @ A3 ) )
= ( ( member536094252920883875c_fm_a @ X2 @ A3 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1148_removeAll__id,axiom,
! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( remove817376121289147686c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1149_removeAll__id,axiom,
! [X2: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( remove5657790513793422086c_fm_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1150_subset__code_I2_J,axiom,
! [A3: set_se5208064806568342746c_fm_a,Ys: list_s580375451141968640c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A3 @ ( coset_774667214334347501c_fm_a @ Ys ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Ys ) )
=> ~ ( member536094252920883875c_fm_a @ X3 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1151_subset__code_I2_J,axiom,
! [A3: set_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( coset_Epistemic_fm_a @ Ys ) )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ~ ( member6642669571620171971c_fm_a @ X3 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1152_flat__lub__def,axiom,
( partia1668821614471696070c_fm_a
= ( ^ [B4: set_Epistemic_fm_a,A4: set_se5208064806568342746c_fm_a] :
( if_set2375185401397246656c_fm_a @ ( ord_le7112219575281605754c_fm_a @ A4 @ ( insert7129737658439086282c_fm_a @ B4 @ bot_bo1868452413526482246c_fm_a ) ) @ B4
@ ( the_se1939870641335875201c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ ( minus_2796033211270405569c_fm_a @ A4 @ ( insert7129737658439086282c_fm_a @ B4 @ bot_bo1868452413526482246c_fm_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1153_flat__lub__def,axiom,
( partia6511039438745469158c_fm_a
= ( ^ [B4: epistemic_fm_a,A4: set_Epistemic_fm_a] :
( if_Epistemic_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ ( insert7817948963269374698c_fm_a @ B4 @ bot_bo3626323581529592678c_fm_a ) ) @ B4
@ ( the_Epistemic_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ ( minus_975701618739183585c_fm_a @ A4 @ ( insert7817948963269374698c_fm_a @ B4 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1154_Pow__iff,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A3 @ ( pow_Epistemic_fm_a @ B ) )
= ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_1155_PowI,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( member536094252920883875c_fm_a @ A3 @ ( pow_Epistemic_fm_a @ B ) ) ) ).
% PowI
thf(fact_1156_finite__inverse__image__gen,axiom,
! [A3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,D: set_se5208064806568342746c_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( inj_on2489648849568095377c_fm_a @ F @ D )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [J2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J2 @ D )
& ( member6642669571620171971c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1157_finite__inverse__image__gen,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,D: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( inj_on5140746211723312753c_fm_a @ F @ D )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [J2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J2 @ D )
& ( member536094252920883875c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1158_finite__inverse__image__gen,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( inj_on1950397435265702513c_fm_a @ F @ D )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [J2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J2 @ D )
& ( member6642669571620171971c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1159_finite__inverse__image__gen,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( inj_on376900539057536081c_fm_a @ F @ D )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [J2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J2 @ D )
& ( member536094252920883875c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1160_Powp__Pow__eq,axiom,
! [A3: set_Epistemic_fm_a] :
( ( powp_Epistemic_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 ) )
= ( ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ ( pow_Epistemic_fm_a @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1161_Powp__Pow__eq,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ( powp_s8172341239314103577c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 ) )
= ( ^ [X3: set_se5208064806568342746c_fm_a] : ( member3183810400728440963c_fm_a @ X3 @ ( pow_se8934638293255475223c_fm_a @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_1162_Pow__bottom,axiom,
! [B: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ bot_bo3626323581529592678c_fm_a @ ( pow_Epistemic_fm_a @ B ) ) ).
% Pow_bottom
thf(fact_1163_Pow__top,axiom,
! [A3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ A3 @ ( pow_Epistemic_fm_a @ A3 ) ) ).
% Pow_top
thf(fact_1164_Pow__def,axiom,
( pow_Epistemic_fm_a
= ( ^ [A4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% Pow_def
thf(fact_1165_Pow__mono,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ord_le7112219575281605754c_fm_a @ ( pow_Epistemic_fm_a @ A3 ) @ ( pow_Epistemic_fm_a @ B ) ) ) ).
% Pow_mono
thf(fact_1166_PowD,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A3 @ ( pow_Epistemic_fm_a @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ).
% PowD
thf(fact_1167_inj__on__image__mem__iff,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( inj_on2489648849568095377c_fm_a @ F @ B )
=> ( ( member536094252920883875c_fm_a @ A @ B )
=> ( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( member6642669571620171971c_fm_a @ ( F @ A ) @ ( image_3376480847214465317c_fm_a @ F @ A3 ) )
= ( member536094252920883875c_fm_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1168_inj__on__image__mem__iff,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( inj_on5140746211723312753c_fm_a @ F @ B )
=> ( ( member536094252920883875c_fm_a @ A @ B )
=> ( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( ( member536094252920883875c_fm_a @ ( F @ A ) @ ( image_8582343580727714565c_fm_a @ F @ A3 ) )
= ( member536094252920883875c_fm_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1169_inj__on__image__mem__iff,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a,A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( inj_on1950397435265702513c_fm_a @ F @ B )
=> ( ( member6642669571620171971c_fm_a @ A @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( member6642669571620171971c_fm_a @ ( F @ A ) @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
= ( member6642669571620171971c_fm_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1170_inj__on__image__mem__iff,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( inj_on376900539057536081c_fm_a @ F @ B )
=> ( ( member6642669571620171971c_fm_a @ A @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ( member536094252920883875c_fm_a @ ( F @ A ) @ ( image_1263732536703906021c_fm_a @ F @ A3 ) )
= ( member6642669571620171971c_fm_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1171_arg__min__inj__eq,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o,A: set_Epistemic_fm_a] :
( ( inj_on5140746211723312753c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( P4 @ A )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ ( F @ Y2 ) ) )
=> ( ( lattic3567457394704492570c_fm_a @ F @ P4 )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1172_arg__min__inj__eq,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,P4: epistemic_fm_a > $o,A: epistemic_fm_a] :
( ( inj_on376900539057536081c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( P4 @ A )
=> ( ! [Y2: epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ ( F @ Y2 ) ) )
=> ( ( lattic7538965924720691706c_fm_a @ F @ P4 )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_1173_finite__surj__inj,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ( inj_on1950397435265702513c_fm_a @ F @ A3 ) ) ) ).
% finite_surj_inj
thf(fact_1174_endo__inj__surj,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A3 ) @ A3 )
=> ( ( inj_on1950397435265702513c_fm_a @ F @ A3 )
=> ( ( image_4449434806354059013c_fm_a @ F @ A3 )
= A3 ) ) ) ) ).
% endo_inj_surj
thf(fact_1175_subset__image__inj,axiom,
! [S: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,T2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S @ ( image_4449434806354059013c_fm_a @ F @ T2 ) )
= ( ? [U4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ U4 @ T2 )
& ( inj_on1950397435265702513c_fm_a @ F @ U4 )
& ( S
= ( image_4449434806354059013c_fm_a @ F @ U4 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1176_one__set__greater,axiom,
! [A3: set_Epistemic_fm_a,A9: set_Epistemic_fm_a] :
( ? [F5: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F5 @ A3 ) @ A9 )
& ( inj_on1950397435265702513c_fm_a @ F5 @ A3 ) )
| ? [G4: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ G4 @ A9 ) @ A3 )
& ( inj_on1950397435265702513c_fm_a @ G4 @ A9 ) ) ) ).
% one_set_greater
thf(fact_1177_vimage__eq,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage5801615914250455739c_fm_a @ F @ B ) )
= ( member6642669571620171971c_fm_a @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_1178_vimage__eq,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage666644291036497051c_fm_a @ F @ B ) )
= ( member536094252920883875c_fm_a @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_1179_vimage__eq,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage2779392601547056347c_fm_a @ F @ B ) )
= ( member6642669571620171971c_fm_a @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_1180_vimage__eq,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage6198526822262906555c_fm_a @ F @ B ) )
= ( member536094252920883875c_fm_a @ ( F @ A ) @ B ) ) ).
% vimage_eq
thf(fact_1181_vimageI,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: epistemic_fm_a,B3: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( F @ A )
= B3 )
=> ( ( member6642669571620171971c_fm_a @ B3 @ B )
=> ( member6642669571620171971c_fm_a @ A @ ( vimage5801615914250455739c_fm_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1182_vimageI,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B3: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ( F @ A )
= B3 )
=> ( ( member6642669571620171971c_fm_a @ B3 @ B )
=> ( member536094252920883875c_fm_a @ A @ ( vimage2779392601547056347c_fm_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1183_vimageI,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: epistemic_fm_a,B3: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( F @ A )
= B3 )
=> ( ( member536094252920883875c_fm_a @ B3 @ B )
=> ( member6642669571620171971c_fm_a @ A @ ( vimage666644291036497051c_fm_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1184_vimageI,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,B3: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ( F @ A )
= B3 )
=> ( ( member536094252920883875c_fm_a @ B3 @ B )
=> ( member536094252920883875c_fm_a @ A @ ( vimage6198526822262906555c_fm_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1185_vimage__Collect__eq,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( vimage6198526822262906555c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] : ( P4 @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1186_vimage__Collect__eq,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( vimage666644291036497051c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a
@ ^ [Y4: epistemic_fm_a] : ( P4 @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1187_vimage__Collect__eq,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( vimage2779392601547056347c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] : ( P4 @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1188_vimage__Collect__eq,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( vimage5801615914250455739c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a
@ ^ [Y4: epistemic_fm_a] : ( P4 @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1189_vimage__def,axiom,
( vimage2779392601547056347c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% vimage_def
thf(fact_1190_vimage__def,axiom,
( vimage6198526822262906555c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > set_Epistemic_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% vimage_def
thf(fact_1191_vimage__def,axiom,
( vimage5801615914250455739c_fm_a
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a,B2: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% vimage_def
thf(fact_1192_vimage__def,axiom,
( vimage666644291036497051c_fm_a
= ( ^ [F2: epistemic_fm_a > set_Epistemic_fm_a,B2: set_se5208064806568342746c_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( member536094252920883875c_fm_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% vimage_def
thf(fact_1193_vimage__Collect,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ ( F @ X ) )
= ( Q2 @ X ) )
=> ( ( vimage6198526822262906555c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_1194_vimage__Collect,axiom,
! [P4: set_Epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ ( F @ X ) )
= ( Q2 @ X ) )
=> ( ( vimage666644291036497051c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_1195_vimage__Collect,axiom,
! [P4: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ ( F @ X ) )
= ( Q2 @ X ) )
=> ( ( vimage2779392601547056347c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_1196_vimage__Collect,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ ( F @ X ) )
= ( Q2 @ X ) )
=> ( ( vimage5801615914250455739c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_1197_vimageI2,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ ( F @ A ) @ A3 )
=> ( member6642669571620171971c_fm_a @ A @ ( vimage5801615914250455739c_fm_a @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_1198_vimageI2,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ ( F @ A ) @ A3 )
=> ( member536094252920883875c_fm_a @ A @ ( vimage2779392601547056347c_fm_a @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_1199_vimageI2,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ ( F @ A ) @ A3 )
=> ( member6642669571620171971c_fm_a @ A @ ( vimage666644291036497051c_fm_a @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_1200_vimageI2,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ ( F @ A ) @ A3 )
=> ( member536094252920883875c_fm_a @ A @ ( vimage6198526822262906555c_fm_a @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_1201_vimageE,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage5801615914250455739c_fm_a @ F @ B ) )
=> ( member6642669571620171971c_fm_a @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_1202_vimageE,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage666644291036497051c_fm_a @ F @ B ) )
=> ( member536094252920883875c_fm_a @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_1203_vimageE,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage2779392601547056347c_fm_a @ F @ B ) )
=> ( member6642669571620171971c_fm_a @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_1204_vimageE,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage6198526822262906555c_fm_a @ F @ B ) )
=> ( member536094252920883875c_fm_a @ ( F @ A ) @ B ) ) ).
% vimageE
thf(fact_1205_vimageD,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage5801615914250455739c_fm_a @ F @ A3 ) )
=> ( member6642669571620171971c_fm_a @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_1206_vimageD,axiom,
! [A: epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ A @ ( vimage666644291036497051c_fm_a @ F @ A3 ) )
=> ( member536094252920883875c_fm_a @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_1207_vimageD,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage2779392601547056347c_fm_a @ F @ A3 ) )
=> ( member6642669571620171971c_fm_a @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_1208_vimageD,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( vimage6198526822262906555c_fm_a @ F @ A3 ) )
=> ( member536094252920883875c_fm_a @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_1209_subset__vimage__iff,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( vimage5801615914250455739c_fm_a @ F @ B ) )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
=> ( member6642669571620171971c_fm_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1210_subset__vimage__iff,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( vimage666644291036497051c_fm_a @ F @ B ) )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
=> ( member536094252920883875c_fm_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1211_vimage__mono,axiom,
! [A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( vimage5801615914250455739c_fm_a @ F @ A3 ) @ ( vimage5801615914250455739c_fm_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_1212_image__subset__iff__subset__vimage,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A3 ) @ B )
= ( ord_le3275665582123262618c_fm_a @ A3 @ ( vimage5801615914250455739c_fm_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1213_empty__Shift,axiom,
! [Kl: set_li769143395467472256c_fm_a,K: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ Kl )
=> ( ( member6642669571620171971c_fm_a @ K @ ( bNF_Gr1093487135560555701c_fm_a @ Kl @ nil_Epistemic_fm_a ) )
=> ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( bNF_Gr8437504134799245625c_fm_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1214_empty__Shift,axiom,
! [Kl: set_li4204741992506657632c_fm_a,K: set_Epistemic_fm_a] :
( ( member2328277852289759785c_fm_a @ nil_se102576565402781184c_fm_a @ Kl )
=> ( ( member536094252920883875c_fm_a @ K @ ( bNF_Gr1807846264492387989c_fm_a @ Kl @ nil_se102576565402781184c_fm_a ) )
=> ( member2328277852289759785c_fm_a @ nil_se102576565402781184c_fm_a @ ( bNF_Gr7707547485936239385c_fm_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1215_subset__code_I3_J,axiom,
~ ( ord_le3275665582123262618c_fm_a @ ( coset_Epistemic_fm_a @ nil_Epistemic_fm_a ) @ ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% subset_code(3)
thf(fact_1216_conjunct_Osimps_I1_J,axiom,
( ( stalnaker_conjunct_a @ nil_Epistemic_fm_a )
= ( epistemic_Imp_a @ epistemic_FF_a @ epistemic_FF_a ) ) ).
% conjunct.simps(1)
thf(fact_1217_UNIV__I,axiom,
! [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ top_to7796028867103199306c_fm_a ) ).
% UNIV_I
thf(fact_1218_UNIV__I,axiom,
! [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ top_to5398033090887092778c_fm_a ) ).
% UNIV_I
thf(fact_1219_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collec2519470961442302949c_fm_a
@ ^ [S5: set_Epistemic_fm_a] : P4 )
= top_to5398033090887092778c_fm_a ) )
& ( ~ P4
=> ( ( collec2519470961442302949c_fm_a
@ ^ [S5: set_Epistemic_fm_a] : P4 )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_const
thf(fact_1220_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collec4904205152690461189c_fm_a
@ ^ [S5: epistemic_fm_a] : P4 )
= top_to7796028867103199306c_fm_a ) )
& ( ~ P4
=> ( ( collec4904205152690461189c_fm_a
@ ^ [S5: epistemic_fm_a] : P4 )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_const
thf(fact_1221_finite__Collect__not,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
=> ( ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
~ ( P4 @ X3 ) ) )
= ( finite7364866884270929315c_fm_a @ top_to5398033090887092778c_fm_a ) ) ) ).
% finite_Collect_not
thf(fact_1222_finite__Collect__not,axiom,
! [P4: epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
=> ( ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
~ ( P4 @ X3 ) ) )
= ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a ) ) ) ).
% finite_Collect_not
thf(fact_1223_infinite__iff__countable__subset,axiom,
! [S: set_Epistemic_fm_a] :
( ( ~ ( finite3304564945125563331c_fm_a @ S ) )
= ( ? [F2: nat > epistemic_fm_a] :
( ( inj_on6046519297833526039c_fm_a @ F2 @ top_top_set_nat )
& ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1224_infinite__countable__subset,axiom,
! [S: set_Epistemic_fm_a] :
( ~ ( finite3304564945125563331c_fm_a @ S )
=> ? [F5: nat > epistemic_fm_a] :
( ( inj_on6046519297833526039c_fm_a @ F5 @ top_top_set_nat )
& ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_1225_top__greatest,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ top_to7796028867103199306c_fm_a ) ).
% top_greatest
thf(fact_1226_top_Oextremum__unique,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A )
= ( A = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_unique
thf(fact_1227_top_Oextremum__uniqueI,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A )
=> ( A = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_uniqueI
thf(fact_1228_subset__UNIV,axiom,
! [A3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A3 @ top_to7796028867103199306c_fm_a ) ).
% subset_UNIV
thf(fact_1229_finite__inverse__image,axiom,
! [A3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( inj_on2489648849568095377c_fm_a @ F @ top_to5398033090887092778c_fm_a )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [J2: set_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_1230_finite__inverse__image,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( inj_on5140746211723312753c_fm_a @ F @ top_to5398033090887092778c_fm_a )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [J2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_1231_finite__inverse__image,axiom,
! [A3: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A3 )
=> ( ( inj_on1950397435265702513c_fm_a @ F @ top_to7796028867103199306c_fm_a )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [J2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_1232_finite__inverse__image,axiom,
! [A3: set_se5208064806568342746c_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A3 )
=> ( ( inj_on376900539057536081c_fm_a @ F @ top_to7796028867103199306c_fm_a )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [J2: epistemic_fm_a] : ( member536094252920883875c_fm_a @ ( F @ J2 ) @ A3 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_1233_UNIV__eq__I,axiom,
! [A3: set_Epistemic_fm_a] :
( ! [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( top_to7796028867103199306c_fm_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1234_UNIV__eq__I,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ! [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ A3 )
=> ( top_to5398033090887092778c_fm_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1235_UNIV__witness,axiom,
? [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ top_to7796028867103199306c_fm_a ) ).
% UNIV_witness
thf(fact_1236_UNIV__witness,axiom,
? [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ top_to5398033090887092778c_fm_a ) ).
% UNIV_witness
thf(fact_1237_UNIV__def,axiom,
( top_to5398033090887092778c_fm_a
= ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : $true ) ) ).
% UNIV_def
thf(fact_1238_UNIV__def,axiom,
( top_to7796028867103199306c_fm_a
= ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : $true ) ) ).
% UNIV_def
thf(fact_1239_inj__image__subset__iff,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( inj_on1950397435265702513c_fm_a @ F @ top_to7796028867103199306c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A3 ) @ ( image_4449434806354059013c_fm_a @ F @ B ) )
= ( ord_le3275665582123262618c_fm_a @ A3 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1240_vimage__subsetD,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( ( image_4449434806354059013c_fm_a @ F @ top_to7796028867103199306c_fm_a )
= top_to7796028867103199306c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( vimage5801615914250455739c_fm_a @ F @ B ) @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A3 ) ) ) ) ).
% vimage_subsetD
thf(fact_1241_MCS__def,axiom,
( maxima1924290459274269190c_fm_a
= ( ^ [Consistent2: set_Epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a )
& ! [S3: set_Epistemic_fm_a,S6: set_Epistemic_fm_a] :
( ( Consistent2 @ S3 )
=> ( ( ord_le3275665582123262618c_fm_a @ S6 @ S3 )
=> ( Consistent2 @ S6 ) ) )
& ! [S3: set_Epistemic_fm_a] :
( ~ ( Consistent2 @ S3 )
=> ? [S6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S6 @ S3 )
& ( finite3304564945125563331c_fm_a @ S6 )
& ~ ( Consistent2 @ S6 ) ) ) ) ) ) ).
% MCS_def
thf(fact_1242_MCS_Ointro,axiom,
! [Consistent: set_Epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a )
=> ( ! [S7: set_Epistemic_fm_a,S4: set_Epistemic_fm_a] :
( ( Consistent @ S7 )
=> ( ( ord_le3275665582123262618c_fm_a @ S4 @ S7 )
=> ( Consistent @ S4 ) ) )
=> ( ! [S7: set_Epistemic_fm_a] :
( ~ ( Consistent @ S7 )
=> ? [S8: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S8 @ S7 )
& ( finite3304564945125563331c_fm_a @ S8 )
& ~ ( Consistent @ S8 ) ) )
=> ( maxima1924290459274269190c_fm_a @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_1243_vimage__subsetI,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,B: set_Epistemic_fm_a,A3: set_Epistemic_fm_a] :
( ( inj_on1950397435265702513c_fm_a @ F @ top_to7796028867103199306c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( vimage5801615914250455739c_fm_a @ F @ B ) @ A3 ) ) ) ).
% vimage_subsetI
thf(fact_1244_If__the__inv__into__in__Func,axiom,
! [G2: epistemic_fm_a > epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X2: epistemic_fm_a] :
( ( inj_on1950397435265702513c_fm_a @ G2 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) )
=> ( member2634950385374053500c_fm_a
@ ^ [I3: epistemic_fm_a] : ( if_Epistemic_fm_a @ ( member6642669571620171971c_fm_a @ I3 @ ( image_4449434806354059013c_fm_a @ G2 @ C2 ) ) @ ( the_in5018756904006898867c_fm_a @ C2 @ G2 @ I3 ) @ X2 )
@ ( bNF_We4818972062080617492c_fm_a @ top_to7796028867103199306c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1245_If__the__inv__into__in__Func,axiom,
! [G2: epistemic_fm_a > set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X2: epistemic_fm_a] :
( ( inj_on376900539057536081c_fm_a @ G2 @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) )
=> ( member293847127289681564c_fm_a
@ ^ [I3: set_Epistemic_fm_a] : ( if_Epistemic_fm_a @ ( member536094252920883875c_fm_a @ I3 @ ( image_1263732536703906021c_fm_a @ G2 @ C2 ) ) @ ( the_in3164076125808357011c_fm_a @ C2 @ G2 @ I3 ) @ X2 )
@ ( bNF_We594613761336656436c_fm_a @ top_to5398033090887092778c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ ( insert7817948963269374698c_fm_a @ X2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1246_top__empty__eq,axiom,
( top_to6710954824118802707fm_a_o
= ( ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ top_to7796028867103199306c_fm_a ) ) ) ).
% top_empty_eq
thf(fact_1247_top__empty__eq,axiom,
( top_to5479180331574806323fm_a_o
= ( ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ top_to5398033090887092778c_fm_a ) ) ) ).
% top_empty_eq
thf(fact_1248_top__set__def,axiom,
( top_to5398033090887092778c_fm_a
= ( collec2519470961442302949c_fm_a @ top_to5479180331574806323fm_a_o ) ) ).
% top_set_def
thf(fact_1249_top__set__def,axiom,
( top_to7796028867103199306c_fm_a
= ( collec4904205152690461189c_fm_a @ top_to6710954824118802707fm_a_o ) ) ).
% top_set_def
thf(fact_1250_the__inv__into__into,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,X2: epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( inj_on2489648849568095377c_fm_a @ F @ A3 )
=> ( ( member6642669571620171971c_fm_a @ X2 @ ( image_3376480847214465317c_fm_a @ F @ A3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( member536094252920883875c_fm_a @ ( the_in5276824436318916307c_fm_a @ A3 @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1251_the__inv__into__into,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A3: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( inj_on5140746211723312753c_fm_a @ F @ A3 )
=> ( ( member536094252920883875c_fm_a @ X2 @ ( image_8582343580727714565c_fm_a @ F @ A3 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A3 @ B )
=> ( member536094252920883875c_fm_a @ ( the_in2681013610681362611c_fm_a @ A3 @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1252_the__inv__into__into,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A3: set_Epistemic_fm_a,X2: epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( inj_on1950397435265702513c_fm_a @ F @ A3 )
=> ( ( member6642669571620171971c_fm_a @ X2 @ ( image_4449434806354059013c_fm_a @ F @ A3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( member6642669571620171971c_fm_a @ ( the_in5018756904006898867c_fm_a @ A3 @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1253_the__inv__into__into,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A3: set_Epistemic_fm_a,X2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( inj_on376900539057536081c_fm_a @ F @ A3 )
=> ( ( member536094252920883875c_fm_a @ X2 @ ( image_1263732536703906021c_fm_a @ F @ A3 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A3 @ B )
=> ( member6642669571620171971c_fm_a @ ( the_in3164076125808357011c_fm_a @ A3 @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1254_canonical__model_I1_J,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ S )
=> ( ( member6642669571620171971c_fm_a @ P2 @ S )
=> ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ W5 ) ) )
@ ^ [I3: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V3 )
@ product_Unity ) )
@ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S )
@ P2 ) ) ) ).
% canonical_model(1)
thf(fact_1255_chains__extend,axiom,
! [C: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a,Z3: set_Epistemic_fm_a] :
( ( member3183810400728440963c_fm_a @ C @ ( chains830531721465917930c_fm_a @ S ) )
=> ( ( member536094252920883875c_fm_a @ Z3 @ S )
=> ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ C )
=> ( ord_le3275665582123262618c_fm_a @ X @ Z3 ) )
=> ( member3183810400728440963c_fm_a @ ( sup_su4305657963326700718c_fm_a @ ( insert7129737658439086282c_fm_a @ Z3 @ bot_bo1868452413526482246c_fm_a ) @ C ) @ ( chains830531721465917930c_fm_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1256_MCS_OExtend__subset,axiom,
! [Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima1924290459274269190c_fm_a @ Consistent )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ Consistent @ S ) ) ) ).
% MCS.Extend_subset
thf(fact_1257_Extend__subset,axiom,
! [S: set_Epistemic_fm_a,A3: epistemic_fm_a > $o] : ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ).
% Extend_subset
thf(fact_1258_MCS_Omaximal__Extend,axiom,
! [Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima1924290459274269190c_fm_a @ Consistent )
=> ( maxima3264069584562520521c_fm_a @ Consistent @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ Consistent @ S ) ) ) ).
% MCS.maximal_Extend
thf(fact_1259_MCS__Lim__Ord__axioms,axiom,
! [A3: epistemic_fm_a > $o] : ( maxima8116622252184345237c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) ) ).
% MCS_Lim_Ord_axioms
thf(fact_1260_maximal__Extend,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ).
% maximal_Extend
thf(fact_1261_canonical__model_I3_J,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ S )
=> ( ( member6642669571620171971c_fm_a @ P2 @ S )
=> ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ) ) ).
% canonical_model(3)
thf(fact_1262_consistent__Extend,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ S )
=> ( episte2285483198712856226tent_a @ A3 @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ) ).
% consistent_Extend
thf(fact_1263_canonical__model_I2_J,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A3 @ S )
=> ( ( member6642669571620171971c_fm_a @ P2 @ S )
=> ( episte2285483198712856226tent_a @ A3 @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ) ) ).
% canonical_model(2)
thf(fact_1264_chainsD,axiom,
! [C: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a,X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( member3183810400728440963c_fm_a @ C @ ( chains830531721465917930c_fm_a @ S ) )
=> ( ( member536094252920883875c_fm_a @ X2 @ C )
=> ( ( member536094252920883875c_fm_a @ Y @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Y )
| ( ord_le3275665582123262618c_fm_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1265_Zorn__Lemma2,axiom,
! [A3: set_se5208064806568342746c_fm_a] :
( ! [X: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X @ ( chains830531721465917930c_fm_a @ A3 ) )
=> ? [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
& ! [Xb: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xb @ X )
=> ( ord_le3275665582123262618c_fm_a @ Xb @ Xa2 ) ) ) )
=> ? [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Xa2 )
=> ( Xa2 = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1266_MCS_Omaximal__maximal_H,axiom,
! [Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima1924290459274269190c_fm_a @ Consistent )
=> ( ( maxima3264069584562520521c_fm_a @ Consistent @ S )
= ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ Consistent @ S ) ) ) ).
% MCS.maximal_maximal'
thf(fact_1267_maximal_H__Extend,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ).
% maximal'_Extend
thf(fact_1268_maximal__maximal_H,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A3 ) @ S )
= ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ).
% maximal_maximal'
thf(fact_1269_maximal_H__def,axiom,
! [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ( episte2285483198712856226tent_a @ A3 @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ X3 @ S ) ) ) ) ) ).
% maximal'_def
thf(fact_1270_mono__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,S9: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ R2 @ S9 )
=> ( ord_le3275665582123262618c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ ( field_Epistemic_fm_a @ S9 ) ) ) ).
% mono_Field
thf(fact_1271_chain__subset__def,axiom,
( chain_7805378426857191923c_fm_a
= ( ^ [C3: set_se5208064806568342746c_fm_a] :
! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ C3 )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ C3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y4 )
| ( ord_le3275665582123262618c_fm_a @ Y4 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1272_MCS__Lim__Ord_OExtend__subset_H,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( ( field_Epistemic_fm_a @ R2 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ R2 @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_subset'
thf(fact_1273_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( maxima2621630783216038545c_fm_a @ R2 @ Consistent @ S )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( Consistent @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ X3 @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_1274_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,Consistent: set_se5208064806568342746c_fm_a > $o,S: set_se5208064806568342746c_fm_a] :
( ( maxima8505690773950114933c_fm_a @ R2 @ Consistent )
=> ( ( maxima2311692462071403633c_fm_a @ R2 @ Consistent @ S )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( Consistent @ ( sup_su4305657963326700718c_fm_a @ ( insert7129737658439086282c_fm_a @ X3 @ bot_bo1868452413526482246c_fm_a ) @ S ) )
=> ( member536094252920883875c_fm_a @ X3 @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_1275_Extend__subset_H,axiom,
! [S: set_Epistemic_fm_a,A3: epistemic_fm_a > $o] :
( ( ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) )
!= bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ) ).
% Extend_subset'
thf(fact_1276_Extend__bound,axiom,
! [N: epistemic_fm_a,A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S @ N ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A3 ) @ S ) ) ) ).
% Extend_bound
% Helper facts (9)
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X2: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X2: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X2: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X2: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X2: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X2: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X2: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [P3: epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ P3 @ w )
| ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ v ) ) ).
%------------------------------------------------------------------------------