TPTP Problem File: SLH0337^1.p
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%------------------------------------------------------------------------------
% 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_00050_001226__5898716_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1644 ( 430 unt; 362 typ; 0 def)
% Number of atoms : 4759 (1019 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 14949 ( 380 ~; 63 |; 396 &;11782 @)
% ( 0 <=>;2328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 54 ( 53 usr)
% Number of type conns : 2884 (2884 >; 0 *; 0 +; 0 <<)
% Number of symbols : 312 ( 309 usr; 27 con; 0-5 aty)
% Number of variables : 4625 ( 550 ^;3862 !; 213 ?;4625 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:21.308
%------------------------------------------------------------------------------
% Could-be-implicit typings (53)
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thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
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thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
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thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
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the_el2173195877760541071c_fm_a: set_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
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wellor5744531135217078139c_fm_a: set_Pr8948481099588399239c_fm_a > $o ).
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member8030264063065380503fm_a_o: ( epistemic_fm_a > epistemic_fm_a > $o ) > set_Ep7530996317732161742fm_a_o > $o ).
thf(sy_c_member_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
member4486839677911940090fm_a_o: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
member_a_a_o: ( a > a > $o ) > set_a_a_o > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
member6642669571620171971c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member5906877432388582473c_fm_a: list_Epistemic_fm_a > set_li769143395467472256c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member2328277852289759785c_fm_a: list_s580375451141968640c_fm_a > set_li4204741992506657632c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member6846535391323571920c_fm_a: produc8009007716763201447c_fm_a > set_Pr4537461906582779015c_fm_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_M_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_M_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J_J_J,type,
member5994151988976616464c_fm_a: produc3797434603744553703c_fm_a > set_Pr4717701496504330695c_fm_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_M_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J_J_J,type,
member5250668776978656404c_fm_a: produc5398904771037890795c_fm_a > set_Pr1281480396656123467c_fm_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mtf__a_J_M_062_It__Epistemic____Logic__Ofm_Itf__a_J_Mtf__a_J_J_J_J,type,
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member2407610774516601616c_fm_a: produc6374449310175442151c_fm_a > set_Pr2829732620705830087c_fm_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_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_Itf__a_J,type,
member_set_a: set_a > set_set_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_a,type,
a3: epistemic_fm_a ).
thf(sy_v_b,type,
b: epistemic_fm_a ).
thf(sy_v_c,type,
c: epistemic_fm_a ).
% Relevant facts (1276)
thf(fact_0_assms_I2_J,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ a3 @ c ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ a3 @ b ) ).
% assms(1)
thf(fact_2_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_3_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_4_eval_Osimps_I4_J,axiom,
! [G: list_char > $o,H: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G @ H @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G @ H @ P )
& ( epistemic_eval_a @ G @ H @ Q ) ) ) ).
% eval.simps(4)
thf(fact_5_eval_Osimps_I5_J,axiom,
! [G: list_char > $o,H: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G @ H @ P )
=> ( epistemic_eval_a @ G @ H @ Q ) ) ) ).
% eval.simps(5)
thf(fact_6_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_7_fm_Orel__distinct_I26_J,axiom,
! [R: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_8_fm_Orel__distinct_I25_J,axiom,
! [R: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_9_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ A2 )
=> ( ! [I: a,P2: epistemic_fm_a,Q2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P2 @ Q2 ) ) ) @ ( epistemic_K_a @ I @ Q2 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P2 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P2 @ A2 ) ) )
=> ~ ! [P2: epistemic_fm_a] :
( ? [I: a] :
( A2
= ( epistemic_K_a @ I @ P2 ) )
=> ~ ( epistemic_AK_a @ A @ P2 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_10_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P3: epistemic_fm_a] :
( ( A4 = P3 )
& ! [G3: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H3 @ P3 ) )
| ? [I2: a,P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P3 ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) ) @ ( epistemic_K_a @ I2 @ Q3 ) ) )
| ? [P3: epistemic_fm_a] :
( ( A4 = P3 )
& ( A3 @ P3 ) )
| ? [P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( ( A4 = Q3 )
& ( epistemic_AK_a @ A3 @ P3 )
& ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) )
| ? [P3: epistemic_fm_a,I2: a] :
( ( A4
= ( epistemic_K_a @ I2 @ P3 ) )
& ( epistemic_AK_a @ A3 @ P3 ) ) ) ) ) ).
% AK.simps
thf(fact_11_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_12_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_13_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_14_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_15_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_16_fm_Orel__distinct_I28_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_17_fm_Orel__distinct_I27_J,axiom,
! [R: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_18_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_19_R2,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I3: a] :
( ( epistemic_AK_a @ A @ P )
=> ( epistemic_AK_a @ A @ ( epistemic_K_a @ I3 @ P ) ) ) ).
% R2
thf(fact_20_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y: a,Z: a] : ( Y = Z ) )
= ( ^ [Y: epistemic_fm_a,Z: epistemic_fm_a] : ( Y = Z ) ) ) ).
% fm.rel_eq
thf(fact_21_fm_Orel__refl,axiom,
! [Ra: a > a > $o,X: epistemic_fm_a] :
( ! [X2: a] : ( Ra @ X2 @ X2 )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ X ) ) ).
% fm.rel_refl
thf(fact_22_K__A2_H,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ Q ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ Q ) ) ) ) ).
% K_A2'
thf(fact_23_K__map,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,I3: a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ Q ) ) ) ) ).
% K_map
thf(fact_24_K__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R2: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ Q @ R2 ) @ ( epistemic_Imp_a @ P @ R2 ) ) ) ) ).
% K_trans
thf(fact_25_R1,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ Q ) ) ) ).
% R1
thf(fact_26_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_27_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_28_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_29_fm_Orel__intros_I4_J,axiom,
! [R: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_30_fm_Orel__inject_I4_J,axiom,
! [R: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_31_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_32_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G4: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H4 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_33_A2,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I3 @ Q ) ) ) ).
% A2
thf(fact_34_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H @ ( epistemic_K_a @ I3 @ P ) )
= ( H @ ( epistemic_K_a @ I3 @ P ) ) ) ).
% eval.simps(6)
thf(fact_35_K__imp__trans_H,axiom,
! [A: epistemic_fm_a > $o,Q: epistemic_fm_a,R2: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ P @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_36_K__imp__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_37_distribution,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I3: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I3 @ Q ) ) ) ).
% distribution
thf(fact_38_duality__taut,axiom,
! [I3: a,P: epistemic_fm_a,Q: epistemic_fm_a,G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_39_K__LK,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) ) ).
% K_LK
thf(fact_40_K__L__dual,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ P ) ) ) ).
% K_L_dual
thf(fact_41_Ax4_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax4_a @ A2 )
=> ~ ! [I: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P2 ) ) ) ) ) ).
% Ax4.cases
thf(fact_42_Ax4_Osimps,axiom,
( epistemic_Ax4_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P3: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P3 ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P3 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_43_Ax4_Ointros,axiom,
! [I3: a,P: epistemic_fm_a] : ( epistemic_Ax4_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ).
% Ax4.intros
thf(fact_44_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_51_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_52_AxT_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxT_a @ A2 )
=> ~ ! [I: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P2 ) @ P2 ) ) ) ).
% AxT.cases
thf(fact_53_AxT_Osimps,axiom,
( epistemic_AxT_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P3: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P3 ) @ P3 ) ) ) ) ).
% AxT.simps
thf(fact_54_AxT_Ointros,axiom,
! [I3: a,P: epistemic_fm_a] : ( epistemic_AxT_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ P ) ) ).
% AxT.intros
thf(fact_55_Imp__intro,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_56_semantics_Osimps_I1_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
~ ( episte7081087998767065248c_fm_a @ M @ W @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_57_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_58_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_59_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_60_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_61_semantics_Osimps_I5_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_62_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_63_semantics_Osimps_I4_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Con_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
& ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_64_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G4: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H4 @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_65_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_66_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_67_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_68_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_69_fm_Orel__distinct_I5_J,axiom,
! [R: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_70_fm_Orel__distinct_I6_J,axiom,
! [R: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_71_K4__L,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% K4_L
thf(fact_72_T__L,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I3: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% T_L
thf(fact_73_AxB_Ointros,axiom,
! [P: epistemic_fm_a,I3: a] : ( epistemic_AxB_a @ ( 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.intros
thf(fact_74_AxB_Osimps,axiom,
( epistemic_AxB_a
= ( ^ [A4: epistemic_fm_a] :
? [P3: epistemic_fm_a,I2: a] :
( A4
= ( 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.simps
thf(fact_75_AxB_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxB_a @ A2 )
=> ~ ! [P2: epistemic_fm_a,I: a] :
( A2
!= ( 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.cases
thf(fact_76_Ax5_Ointros,axiom,
! [I3: a,P: epistemic_fm_a] : ( epistemic_Ax5_a @ ( 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.intros
thf(fact_77_Ax5_Osimps,axiom,
( epistemic_Ax5_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P3: epistemic_fm_a] :
( A4
= ( 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.simps
thf(fact_78_Ax5_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax5_a @ A2 )
=> ~ ! [I: a,P2: epistemic_fm_a] :
( A2
!= ( 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.cases
thf(fact_79_KB4__5,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ 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 ) ) ) ) ) ) ).
% KB4_5
thf(fact_80_K5__L,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ P ) ) ) ) ).
% K5_L
thf(fact_81_S5_H__4,axiom,
! [A: epistemic_fm_a > $o,I3: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_82_S5_H__B,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I3: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_83_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_84_order__refl,axiom,
! [X: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X @ X ) ).
% order_refl
thf(fact_85_order__refl,axiom,
! [X: a > a > $o] : ( ord_less_eq_a_a_o @ X @ X ) ).
% order_refl
thf(fact_86_order__refl,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ X @ X ) ).
% order_refl
thf(fact_87_order__refl,axiom,
! [X: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X @ X ) ).
% order_refl
thf(fact_88_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_89_dual__order_Orefl,axiom,
! [A2: a > a > $o] : ( ord_less_eq_a_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_90_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_91_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_92_le__funD,axiom,
! [F: epistemic_fm_a > $o,G: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_93_le__funD,axiom,
! [F: a > a > $o,G: a > a > $o,X: a] :
( ( ord_less_eq_a_a_o @ F @ G )
=> ( ord_less_eq_a_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_94_le__funD,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,G: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ F @ G )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_95_le__funE,axiom,
! [F: epistemic_fm_a > $o,G: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_96_le__funE,axiom,
! [F: a > a > $o,G: a > a > $o,X: a] :
( ( ord_less_eq_a_a_o @ F @ G )
=> ( ord_less_eq_a_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_97_le__funE,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,G: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ F @ G )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_98_le__funI,axiom,
! [F: epistemic_fm_a > $o,G: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G ) ) ).
% le_funI
thf(fact_99_le__funI,axiom,
! [F: a > a > $o,G: a > a > $o] :
( ! [X2: a] : ( ord_less_eq_a_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_a_a_o @ F @ G ) ) ).
% le_funI
thf(fact_100_le__funI,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,G: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le3934200179093585166fm_a_o @ F @ G ) ) ).
% le_funI
thf(fact_101_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_102_le__fun__def,axiom,
( ord_less_eq_a_a_o
= ( ^ [F2: a > a > $o,G3: a > a > $o] :
! [X3: a] : ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_103_le__fun__def,axiom,
( ord_le3934200179093585166fm_a_o
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a > $o,G3: epistemic_fm_a > epistemic_fm_a > $o] :
! [X3: epistemic_fm_a] : ( ord_le4043730696559282883fm_a_o @ ( F2 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_104_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X )
=> ( Q4 @ X ) ) ) ).
% predicate1D
thf(fact_105_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X: epistemic_fm_a,Q4: epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X ) ) ) ).
% rev_predicate1D
thf(fact_106_verit__comp__simplify1_I2_J,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_107_verit__comp__simplify1_I2_J,axiom,
! [A2: a > a > $o] : ( ord_less_eq_a_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_108_verit__comp__simplify1_I2_J,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_109_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_110_fm_Orel__mono,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ R @ Ra )
=> ( ord_le6583030472306871822fm_a_o @ ( episte7774795710028497888c_fm_a @ R ) @ ( episte7774795710028497888c_fm_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_111_fm_Orel__mono,axiom,
! [R: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_112_order__antisym__conv,axiom,
! [Y2: epistemic_fm_a > $o,X: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_113_order__antisym__conv,axiom,
! [Y2: a > a > $o,X: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y2 @ X )
=> ( ( ord_less_eq_a_a_o @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_114_order__antisym__conv,axiom,
! [Y2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ Y2 @ X )
=> ( ( ord_le3934200179093585166fm_a_o @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_115_order__antisym__conv,axiom,
! [Y2: set_Epistemic_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_116_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_ord__le__eq__subst,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_121_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > a > a > $o,C: a > a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_122_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > a > a > $o,C: a > a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_123_ord__le__eq__subst,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_ord__le__eq__subst,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,F: ( epistemic_fm_a > epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_127_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_128_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_129_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_130_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: ( a > a > $o ) > set_Epistemic_fm_a,B: a > a > $o,C: a > a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_131_ord__eq__le__subst,axiom,
! [A2: a > a > $o,F: set_Epistemic_fm_a > a > a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
! [A2: a > a > $o,F: ( epistemic_fm_a > $o ) > a > a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: ( a > a > $o ) > epistemic_fm_a > $o,B: a > a > $o,C: a > a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A2: a > a > $o,F: ( a > a > $o ) > a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_order__eq__refl,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( X = Y2 )
=> ( ord_le4043730696559282883fm_a_o @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_137_order__eq__refl,axiom,
! [X: a > a > $o,Y2: a > a > $o] :
( ( X = Y2 )
=> ( ord_less_eq_a_a_o @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_138_order__eq__refl,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X = Y2 )
=> ( ord_le3934200179093585166fm_a_o @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_139_order__eq__refl,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( X = Y2 )
=> ( ord_le3275665582123262618c_fm_a @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_140_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_141_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_142_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_143_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_144_order__subst2,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_145_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > a > a > $o,C: a > a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_less_eq_a_a_o @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_146_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > a > a > $o,C: a > a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_less_eq_a_a_o @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_147_order__subst2,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_148_order__subst2,axiom,
! [A2: a > a > $o,B: a > a > $o,F: ( a > a > $o ) > a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ord_less_eq_a_a_o @ ( F @ B ) @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_149_order__subst2,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,F: ( epistemic_fm_a > epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_150_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_151_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_152_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_153_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_154_order__subst1,axiom,
! [A2: a > a > $o,F: set_Epistemic_fm_a > a > a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_a_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_155_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: ( a > a > $o ) > set_Epistemic_fm_a,B: a > a > $o,C: a > a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_156_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: ( a > a > $o ) > epistemic_fm_a > $o,B: a > a > $o,C: a > a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_157_order__subst1,axiom,
! [A2: a > a > $o,F: ( epistemic_fm_a > $o ) > a > a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_158_order__subst1,axiom,
! [A2: a > a > $o,F: ( a > a > $o ) > a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_a_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_159_order__subst1,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ord_le3934200179093585166fm_a_o @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3934200179093585166fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_160_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B2 )
& ( ord_le4043730696559282883fm_a_o @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_161_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: a > a > $o,Z: a > a > $o] : ( Y = Z ) )
= ( ^ [A4: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A4 @ B2 )
& ( ord_less_eq_a_a_o @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_162_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > epistemic_fm_a > $o,Z: epistemic_fm_a > epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [A4: epistemic_fm_a > epistemic_fm_a > $o,B2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A4 @ B2 )
& ( ord_le3934200179093585166fm_a_o @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_163_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_164_antisym,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_165_antisym,axiom,
! [A2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_166_antisym,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_167_antisym,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_168_dual__order_Otrans,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_169_dual__order_Otrans,axiom,
! [B: a > a > $o,A2: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( ( ord_less_eq_a_a_o @ C @ B )
=> ( ord_less_eq_a_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_170_dual__order_Otrans,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( ( ord_le3934200179093585166fm_a_o @ C @ B )
=> ( ord_le3934200179093585166fm_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_171_dual__order_Otrans,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_172_dual__order_Oantisym,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_173_dual__order_Oantisym,axiom,
! [B: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_174_dual__order_Oantisym,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_175_dual__order_Oantisym,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A4 )
& ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_dual__order_Oeq__iff,axiom,
( ( ^ [Y: a > a > $o,Z: a > a > $o] : ( Y = Z ) )
= ( ^ [A4: a > a > $o,B2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B2 @ A4 )
& ( ord_less_eq_a_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_178_dual__order_Oeq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > epistemic_fm_a > $o,Z: epistemic_fm_a > epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [A4: epistemic_fm_a > epistemic_fm_a > $o,B2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B2 @ A4 )
& ( ord_le3934200179093585166fm_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_179_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A4 )
& ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_180_order__trans,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ Y2 @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_181_order__trans,axiom,
! [X: a > a > $o,Y2: a > a > $o,Z2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ Y2 )
=> ( ( ord_less_eq_a_a_o @ Y2 @ Z2 )
=> ( ord_less_eq_a_a_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_182_order__trans,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o,Z2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X @ Y2 )
=> ( ( ord_le3934200179093585166fm_a_o @ Y2 @ Z2 )
=> ( ord_le3934200179093585166fm_a_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_183_order__trans,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Y2 @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_184_order_Otrans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_185_order_Otrans,axiom,
! [A2: a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ord_less_eq_a_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_186_order_Otrans,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ C )
=> ( ord_le3934200179093585166fm_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_187_order_Otrans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_188_order__antisym,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_189_order__antisym,axiom,
! [X: a > a > $o,Y2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ Y2 )
=> ( ( ord_less_eq_a_a_o @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_190_order__antisym,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X @ Y2 )
=> ( ( ord_le3934200179093585166fm_a_o @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_191_order__antisym,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_192_ord__le__eq__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( B = C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_193_ord__le__eq__trans,axiom,
! [A2: a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_a_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_194_ord__le__eq__trans,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3934200179093585166fm_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_195_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_196_ord__eq__le__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2 = B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_197_ord__eq__le__trans,axiom,
! [A2: a > a > $o,B: a > a > $o,C: a > a > $o] :
( ( A2 = B )
=> ( ( ord_less_eq_a_a_o @ B @ C )
=> ( ord_less_eq_a_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_198_ord__eq__le__trans,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A2 = B )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ C )
=> ( ord_le3934200179093585166fm_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_199_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2 = B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_200_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [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_201_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: a > a > $o,Z: a > a > $o] : ( Y = Z ) )
= ( ^ [X3: a > a > $o,Y4: a > a > $o] :
( ( ord_less_eq_a_a_o @ X3 @ Y4 )
& ( ord_less_eq_a_a_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_202_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_a > epistemic_fm_a > $o,Z: epistemic_fm_a > epistemic_fm_a > $o] : ( Y = Z ) )
= ( ^ [X3: epistemic_fm_a > epistemic_fm_a > $o,Y4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X3 @ Y4 )
& ( ord_le3934200179093585166fm_a_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_203_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y = Z ) )
= ( ^ [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_204_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ! [Y3: epistemic_fm_a > $o] :
( ( P4 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ Y3 @ X ) )
=> ( ( order_253494837916242250fm_a_o @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_205_Greatest__equality,axiom,
! [P4: ( a > a > $o ) > $o,X: a > a > $o] :
( ( P4 @ X )
=> ( ! [Y3: a > a > $o] :
( ( P4 @ Y3 )
=> ( ord_less_eq_a_a_o @ Y3 @ X ) )
=> ( ( order_Greatest_a_a_o @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_206_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ! [Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ Y3 )
=> ( ord_le3934200179093585166fm_a_o @ Y3 @ X ) )
=> ( ( order_3254504553663415623fm_a_o @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_207_Greatest__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( ! [Y3: set_Epistemic_fm_a] :
( ( P4 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ Y3 @ X ) )
=> ( ( order_4585748725732241747c_fm_a @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_208_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X )
=> ( ! [Y3: epistemic_fm_a > $o] :
( ( P4 @ Y3 )
=> ( ord_le4043730696559282883fm_a_o @ Y3 @ X ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_253494837916242250fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_209_GreatestI2__order,axiom,
! [P4: ( a > a > $o ) > $o,X: a > a > $o,Q4: ( a > a > $o ) > $o] :
( ( P4 @ X )
=> ( ! [Y3: a > a > $o] :
( ( P4 @ Y3 )
=> ( ord_less_eq_a_a_o @ Y3 @ X ) )
=> ( ! [X2: a > a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: a > a > $o] :
( ( P4 @ Y5 )
=> ( ord_less_eq_a_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_Greatest_a_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_210_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > epistemic_fm_a > $o,Q4: ( epistemic_fm_a > epistemic_fm_a > $o ) > $o] :
( ( P4 @ X )
=> ( ! [Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ Y3 )
=> ( ord_le3934200179093585166fm_a_o @ Y3 @ X ) )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le3934200179093585166fm_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_3254504553663415623fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_211_GreatestI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X: set_Epistemic_fm_a,Q4: set_Epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ! [Y3: set_Epistemic_fm_a] :
( ( P4 @ Y3 )
=> ( ord_le3275665582123262618c_fm_a @ Y3 @ X ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_4585748725732241747c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_212_le__rel__bool__arg__iff,axiom,
( ord_le2986116216980733324fm_a_o
= ( ^ [X4: $o > epistemic_fm_a > $o,Y6: $o > epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le4043730696559282883fm_a_o @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_213_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a_a_o
= ( ^ [X4: $o > a > a > $o,Y6: $o > a > a > $o] :
( ( ord_less_eq_a_a_o @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a_a_o @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_214_le__rel__bool__arg__iff,axiom,
( ord_le4036451250170401435fm_a_o
= ( ^ [X4: $o > epistemic_fm_a > epistemic_fm_a > $o,Y6: $o > epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3934200179093585166fm_a_o @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_215_le__rel__bool__arg__iff,axiom,
( ord_le1382570192702674727c_fm_a
= ( ^ [X4: $o > set_Epistemic_fm_a,Y6: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3275665582123262618c_fm_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_216_fm_Opred__mono,axiom,
! [P4: a > $o,Pa: a > $o] :
( ( ord_less_eq_a_o @ P4 @ Pa )
=> ( ord_le4043730696559282883fm_a_o @ ( epistemic_pred_fm_a @ P4 ) @ ( epistemic_pred_fm_a @ Pa ) ) ) ).
% fm.pred_mono
thf(fact_217_fm_Opred__mono,axiom,
! [P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Pa )
=> ( ord_le7240545311762076327fm_a_o @ ( episte3114992851182798545c_fm_a @ P4 ) @ ( episte3114992851182798545c_fm_a @ Pa ) ) ) ).
% fm.pred_mono
thf(fact_218_fm_Orel__induct,axiom,
! [R: a > a > $o,X: epistemic_fm_a,Y2: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R @ X @ Y2 )
=> ( ( Q4 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q4 @ ( epistemic_Pro_a @ A22 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q4 @ A31 @ B31 )
=> ( ( Q4 @ A32 @ B32 )
=> ( Q4 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q4 @ A41 @ B41 )
=> ( ( Q4 @ A42 @ B42 )
=> ( Q4 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q4 @ A51 @ B51 )
=> ( ( Q4 @ A52 @ B52 )
=> ( Q4 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X @ Y2 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_219_fm_Orel__cases,axiom,
! [R: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B != epistemic_FF_a ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_a @ X2 ) )
=> ! [Y3: list_char] :
( ( B
= ( epistemic_Pro_a @ Y3 ) )
=> ( X2 != Y3 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_220_S5_H__S5,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ P ) ) ).
% S5'_S5
thf(fact_221_S5__S5_H,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ P ) ) ).
% S5_S5'
thf(fact_222_fm_Orel__distinct_I19_J,axiom,
! [R: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_223_fm_Orel__distinct_I20_J,axiom,
! [R: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_224_predicate2I,axiom,
! [P4: a > a > $o,Q4: a > a > $o] :
( ! [X2: a,Y3: a] :
( ( P4 @ X2 @ Y3 )
=> ( Q4 @ X2 @ Y3 ) )
=> ( ord_less_eq_a_a_o @ P4 @ Q4 ) ) ).
% predicate2I
thf(fact_225_predicate2I,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a,Y3: epistemic_fm_a] :
( ( P4 @ X2 @ Y3 )
=> ( Q4 @ X2 @ Y3 ) )
=> ( ord_le3934200179093585166fm_a_o @ P4 @ Q4 ) ) ).
% predicate2I
thf(fact_226_fm_Oinject_I1_J,axiom,
! [X22: list_char,Y22: list_char] :
( ( ( epistemic_Pro_a @ X22 )
= ( epistemic_Pro_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% fm.inject(1)
thf(fact_227_fm_Oinject_I2_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( ( epistemic_Dis_a @ X31 @ X32 )
= ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_228_fm_Opred__inject_I5_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Imp_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(5)
thf(fact_229_fm_Opred__inject_I6_J,axiom,
! [P4: a > $o,A2: a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_K_a @ A2 @ Aa ) )
= ( ( P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(6)
thf(fact_230_fm_Opred__inject_I4_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Con_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(4)
thf(fact_231_fm_Opred__inject_I3_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Dis_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(3)
thf(fact_232_predicate2D,axiom,
! [P4: a > a > $o,Q4: a > a > $o,X: a,Y2: a] :
( ( ord_less_eq_a_a_o @ P4 @ Q4 )
=> ( ( P4 @ X @ Y2 )
=> ( Q4 @ X @ Y2 ) ) ) ).
% predicate2D
thf(fact_233_predicate2D,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,Q4: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a,Y2: epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X @ Y2 )
=> ( Q4 @ X @ Y2 ) ) ) ).
% predicate2D
thf(fact_234_rev__predicate2D,axiom,
! [P4: a > a > $o,X: a,Y2: a,Q4: a > a > $o] :
( ( P4 @ X @ Y2 )
=> ( ( ord_less_eq_a_a_o @ P4 @ Q4 )
=> ( Q4 @ X @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_235_rev__predicate2D,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a,Y2: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( P4 @ X @ Y2 )
=> ( ( ord_le3934200179093585166fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_236_fm_Opred__True,axiom,
( ( epistemic_pred_fm_a
@ ^ [Uu2: a] : $true )
= ( ^ [Uu2: epistemic_fm_a] : $true ) ) ).
% fm.pred_True
thf(fact_237_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_238_fm_Orel__distinct_I12_J,axiom,
! [R: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X22 ) ) ).
% fm.rel_distinct(12)
thf(fact_239_fm_Orel__distinct_I11_J,axiom,
! [R: a > a > $o,X22: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_240_fm_Odistinct_I15_J,axiom,
! [X22: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_241_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_242_fm_Odistinct_I17_J,axiom,
! [X22: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_243_fm_Orel__inject_I2_J,axiom,
! [R: a > a > $o,X22: list_char,Y22: list_char] :
( ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_Pro_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% fm.rel_inject(2)
thf(fact_244_fm_Orel__intros_I2_J,axiom,
! [X22: list_char,Y22: list_char,R: a > a > $o] :
( ( X22 = Y22 )
=> ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_Pro_a @ Y22 ) ) ) ).
% fm.rel_intros(2)
thf(fact_245_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_246_fm_Orel__inject_I3_J,axiom,
! [R: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_247_fm_Orel__intros_I3_J,axiom,
! [R: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_248_fm_Odistinct_I1_J,axiom,
! [X22: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X22 ) ) ).
% fm.distinct(1)
thf(fact_249_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_250_semantics_Osimps_I3_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
| ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_251_fm_Odistinct_I13_J,axiom,
! [X22: list_char,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_252_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_253_fm_Opred__inject_I1_J,axiom,
! [P4: a > $o] : ( epistemic_pred_fm_a @ P4 @ epistemic_FF_a ) ).
% fm.pred_inject(1)
thf(fact_254_eval_Osimps_I2_J,axiom,
! [G: list_char > $o,Uw: epistemic_fm_a > $o,X: list_char] :
( ( epistemic_eval_a @ G @ Uw @ ( epistemic_Pro_a @ X ) )
= ( G @ X ) ) ).
% eval.simps(2)
thf(fact_255_eval_Osimps_I3_J,axiom,
! [G: list_char > $o,H: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G @ H @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G @ H @ P )
| ( epistemic_eval_a @ G @ H @ Q ) ) ) ).
% eval.simps(3)
thf(fact_256_fm_Oexhaust,axiom,
! [Y2: epistemic_fm_a] :
( ( Y2 != epistemic_FF_a )
=> ( ! [X23: list_char] :
( Y2
!= ( epistemic_Pro_a @ X23 ) )
=> ( ! [X312: epistemic_fm_a,X322: epistemic_fm_a] :
( Y2
!= ( epistemic_Dis_a @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_a,X422: epistemic_fm_a] :
( Y2
!= ( epistemic_Con_a @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_a,X522: epistemic_fm_a] :
( Y2
!= ( epistemic_Imp_a @ X512 @ X522 ) )
=> ~ ! [X612: a,X622: epistemic_fm_a] :
( Y2
!= ( epistemic_K_a @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_257_fm_Orel__distinct_I16_J,axiom,
! [R: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X22 ) ) ).
% fm.rel_distinct(16)
thf(fact_258_fm_Orel__distinct_I15_J,axiom,
! [R: a > a > $o,X22: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_259_fm_Orel__distinct_I18_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a,X22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X22 ) ) ).
% fm.rel_distinct(18)
thf(fact_260_fm_Orel__distinct_I17_J,axiom,
! [R: a > a > $o,X22: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_261_fm_Orel__distinct_I22_J,axiom,
! [R: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_262_fm_Orel__distinct_I21_J,axiom,
! [R: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_263_fm_Orel__distinct_I24_J,axiom,
! [R: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_264_fm_Orel__distinct_I23_J,axiom,
! [R: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_265_fm_Orel__distinct_I2_J,axiom,
! [R: a > a > $o,Y22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ Y22 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_266_fm_Orel__distinct_I1_J,axiom,
! [R: a > a > $o,Y22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ ( epistemic_Pro_a @ Y22 ) ) ).
% fm.rel_distinct(1)
thf(fact_267_fm_Orel__distinct_I4_J,axiom,
! [R: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_268_fm_Orel__distinct_I3_J,axiom,
! [R: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_269_fm_Orel__distinct_I14_J,axiom,
! [R: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X22: list_char] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X22 ) ) ).
% fm.rel_distinct(14)
thf(fact_270_fm_Orel__distinct_I13_J,axiom,
! [R: a > a > $o,X22: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R @ ( epistemic_Pro_a @ X22 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_271_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A3 )
@ ^ [X3: a] : ( member_a @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_272_less__eq__set__def,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ord_le1568123225352354019fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_273_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_274_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_275_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_276_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_277_eq__subset,axiom,
! [P4: a > a > $o] :
( ord_less_eq_a_a_o
@ ^ [Y: a,Z: a] : ( Y = Z )
@ ^ [A4: a,B2: a] :
( ( P4 @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% eq_subset
thf(fact_278_eq__subset,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o] :
( ord_le3934200179093585166fm_a_o
@ ^ [Y: epistemic_fm_a,Z: epistemic_fm_a] : ( Y = Z )
@ ^ [A4: epistemic_fm_a,B2: epistemic_fm_a] :
( ( P4 @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% eq_subset
thf(fact_279_semantics_Osimps_I2_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,X: list_char] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Pro_a @ X ) )
= ( episte2398645135750866164t_unit @ M @ W @ X ) ) ).
% semantics.simps(2)
thf(fact_280_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A2: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ A2 ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte5999154429349999510c_fm_a] :
( A2
!= ( episte3013695276341575662c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte3013695276341575662c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_281_fm_Oset__cases,axiom,
! [E: epistemic_fm_a,A2: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ A2 ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte740340785640729014c_fm_a] :
( A2
!= ( episte5657488632024175118c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte5657488632024175118c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_282_fm_Oset__cases,axiom,
! [E: a,A2: epistemic_fm_a] :
( ( member_a @ E @ ( epistemic_set_fm_a @ A2 ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_a] :
( A2
!= ( epistemic_K_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_283_ge__eq__refl,axiom,
! [R: a > a > $o,X: a] :
( ( ord_less_eq_a_a_o
@ ^ [Y: a,Z: a] : ( Y = Z )
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_284_ge__eq__refl,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le3934200179093585166fm_a_o
@ ^ [Y: epistemic_fm_a,Z: epistemic_fm_a] : ( Y = Z )
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_285_refl__ge__eq,axiom,
! [R: a > a > $o] :
( ! [X2: a] : ( R @ X2 @ X2 )
=> ( ord_less_eq_a_a_o
@ ^ [Y: a,Z: a] : ( Y = Z )
@ R ) ) ).
% refl_ge_eq
thf(fact_286_refl__ge__eq,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( R @ X2 @ X2 )
=> ( ord_le3934200179093585166fm_a_o
@ ^ [Y: epistemic_fm_a,Z: epistemic_fm_a] : ( Y = Z )
@ R ) ) ).
% refl_ge_eq
thf(fact_287_predicate2D__conj,axiom,
! [P4: a > a > $o,Q4: a > a > $o,R: $o,X: a,Y2: a] :
( ( ( ord_less_eq_a_a_o @ P4 @ Q4 )
& R )
=> ( R
& ( ( P4 @ X @ Y2 )
=> ( Q4 @ X @ Y2 ) ) ) ) ).
% predicate2D_conj
thf(fact_288_predicate2D__conj,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,Q4: epistemic_fm_a > epistemic_fm_a > $o,R: $o,X: epistemic_fm_a,Y2: epistemic_fm_a] :
( ( ( ord_le3934200179093585166fm_a_o @ P4 @ Q4 )
& R )
=> ( R
& ( ( P4 @ X @ Y2 )
=> ( Q4 @ X @ Y2 ) ) ) ) ).
% predicate2D_conj
thf(fact_289_S5__S5_H__assms,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_290_subsetI,axiom,
! [A: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% subsetI
thf(fact_291_subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ X2 @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_292_subsetI,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ X2 @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_293_subset__antisym,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_294_in__mono,axiom,
! [A: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_295_in__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( member536094252920883875c_fm_a @ X @ A )
=> ( member536094252920883875c_fm_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_296_in__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,X: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( member6642669571620171971c_fm_a @ X @ A )
=> ( member6642669571620171971c_fm_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_297_subsetD,axiom,
! [A: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_298_subsetD,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( member536094252920883875c_fm_a @ C @ A )
=> ( member536094252920883875c_fm_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_299_subsetD,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( member6642669571620171971c_fm_a @ C @ A )
=> ( member6642669571620171971c_fm_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_300_equalityE,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ~ ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ) ).
% equalityE
thf(fact_301_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_302_subset__eq,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
=> ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_303_subset__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
=> ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_304_equalityD1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% equalityD1
thf(fact_305_equalityD2,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ).
% equalityD2
thf(fact_306_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_307_subset__iff,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
! [T: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ T @ A3 )
=> ( member536094252920883875c_fm_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_308_subset__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
! [T: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ T @ A3 )
=> ( member6642669571620171971c_fm_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_309_subset__refl,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_310_Collect__mono,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_311_Collect__mono,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_312_subset__trans,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_313_set__eq__subset,axiom,
( ( ^ [Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y = Z ) )
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_314_Collect__subset,axiom,
! [A: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_315_Collect__subset,axiom,
! [A: 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 @ A )
& ( P4 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_316_Collect__subset,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_317_Collect__mono__iff,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_318_Collect__mono__iff,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) )
= ( ! [X3: epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_319_K__imply__weaken,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,Ps2: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Ps2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_320_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 ) )
=> ! [G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R2 ) @ ( epistemic_imply_a @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_fm_Orel__refl__strong,axiom,
! [X: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_334_fm_Orel__refl__strong,axiom,
! [X: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_335_fm_Orel__refl__strong,axiom,
! [X: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_336_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,X: episte5999154429349999510c_fm_a,Y2: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte8246226516352692192c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_337_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_a > epistemic_fm_a > $o,X: episte5999154429349999510c_fm_a,Y2: episte740340785640729014c_fm_a,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte6543039790609369600c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte6543039790609369600c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_338_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_a > set_Epistemic_fm_a > $o,X: episte740340785640729014c_fm_a,Y2: episte5999154429349999510c_fm_a,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte4430291480098810304c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte4430291480098810304c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_339_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,X: episte740340785640729014c_fm_a,Y2: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte7774795710028497888c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_340_fm_Orel__mono__strong,axiom,
! [R: set_Epistemic_fm_a > a > $o,X: episte5999154429349999510c_fm_a,Y2: epistemic_fm_a,Ra: set_Epistemic_fm_a > a > $o] :
( ( episte7682788968413056228fm_a_a @ R @ X @ Y2 )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte7682788968413056228fm_a_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_341_fm_Orel__mono__strong,axiom,
! [R: epistemic_fm_a > a > $o,X: episte740340785640729014c_fm_a,Y2: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R @ X @ Y2 )
=> ( ! [Z3: epistemic_fm_a,Yb: a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_342_fm_Orel__mono__strong,axiom,
! [R: a > set_Epistemic_fm_a > $o,X: epistemic_fm_a,Y2: episte5999154429349999510c_fm_a,Ra: a > set_Epistemic_fm_a > $o] :
( ( episte6665775379439401564c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: a,Yb: set_Epistemic_fm_a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ X ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte6665775379439401564c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_343_fm_Orel__mono__strong,axiom,
! [R: a > epistemic_fm_a > $o,X: epistemic_fm_a,Y2: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R @ X @ Y2 )
=> ( ! [Z3: a,Yb: epistemic_fm_a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_344_fm_Orel__mono__strong,axiom,
! [R: a > a > $o,X: epistemic_fm_a,Y2: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R @ X @ Y2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ X ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Y2 ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_345_fm_Orel__cong,axiom,
! [X: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y2: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte8246226516352692192c_fm_a @ R @ X @ Y2 )
= ( episte8246226516352692192c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_346_fm_Orel__cong,axiom,
! [X: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y2: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R: set_Epistemic_fm_a > epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte6543039790609369600c_fm_a @ R @ X @ Y2 )
= ( episte6543039790609369600c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_347_fm_Orel__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y2: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R: epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte4430291480098810304c_fm_a @ R @ X @ Y2 )
= ( episte4430291480098810304c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_348_fm_Orel__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y2: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R @ X @ Y2 )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_349_fm_Orel__cong,axiom,
! [X: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y2: epistemic_fm_a,Xa: epistemic_fm_a,R: set_Epistemic_fm_a > a > $o,Ra: set_Epistemic_fm_a > a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: set_Epistemic_fm_a,Yb: a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte7682788968413056228fm_a_a @ R @ X @ Y2 )
= ( episte7682788968413056228fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_350_fm_Orel__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y2: epistemic_fm_a,Xa: epistemic_fm_a,R: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb: a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R @ X @ Y2 )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_351_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y2: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R: a > set_Epistemic_fm_a > $o,Ra: a > set_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: a,Yb: set_Epistemic_fm_a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte6665775379439401564c_fm_a @ R @ X @ Y2 )
= ( episte6665775379439401564c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_352_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y2: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: a,Yb: epistemic_fm_a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R @ X @ Y2 )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_353_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y2: epistemic_fm_a,Xa: epistemic_fm_a,R: a > a > $o,Ra: a > a > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a @ Yb @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R @ X @ Y2 )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_354_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_355_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_356_fm_Oset__intros_I4_J,axiom,
! [Yd: a,X42: epistemic_fm_a,X41: epistemic_fm_a] :
( ( member_a @ Yd @ ( epistemic_set_fm_a @ X42 ) )
=> ( member_a @ Yd @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_357_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_358_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_359_fm_Oset__intros_I3_J,axiom,
! [Yc: a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( member_a @ Yc @ ( epistemic_set_fm_a @ X41 ) )
=> ( member_a @ Yc @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_360_fm_Oset__intros_I2_J,axiom,
! [Yb2: set_Epistemic_fm_a,X32: episte5999154429349999510c_fm_a,X31: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ X32 ) )
=> ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_361_fm_Oset__intros_I2_J,axiom,
! [Yb2: epistemic_fm_a,X32: episte740340785640729014c_fm_a,X31: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ X32 ) )
=> ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_362_fm_Oset__intros_I2_J,axiom,
! [Yb2: a,X32: epistemic_fm_a,X31: epistemic_fm_a] :
( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ X32 ) )
=> ( member_a @ Yb2 @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_363_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_364_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_365_fm_Oset__intros_I1_J,axiom,
! [Ya: a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( member_a @ Ya @ ( epistemic_set_fm_a @ X31 ) )
=> ( member_a @ Ya @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_366_fm_Opred__cong,axiom,
! [X: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,P4: set_Epistemic_fm_a > $o,Pa: set_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( episte1367365995573110961c_fm_a @ P4 @ X )
= ( episte1367365995573110961c_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_367_fm_Opred__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( episte3114992851182798545c_fm_a @ P4 @ X )
= ( episte3114992851182798545c_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_368_fm_Opred__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,P4: a > $o,Pa: a > $o] :
( ( X = Ya )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( epistemic_pred_fm_a @ P4 @ X )
= ( epistemic_pred_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_369_fm_Opred__mono__strong,axiom,
! [P4: set_Epistemic_fm_a > $o,X: episte5999154429349999510c_fm_a,Pa: set_Epistemic_fm_a > $o] :
( ( episte1367365995573110961c_fm_a @ P4 @ X )
=> ( ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( episte1367365995573110961c_fm_a @ Pa @ X ) ) ) ).
% fm.pred_mono_strong
thf(fact_370_fm_Opred__mono__strong,axiom,
! [P4: epistemic_fm_a > $o,X: episte740340785640729014c_fm_a,Pa: epistemic_fm_a > $o] :
( ( episte3114992851182798545c_fm_a @ P4 @ X )
=> ( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( episte3114992851182798545c_fm_a @ Pa @ X ) ) ) ).
% fm.pred_mono_strong
thf(fact_371_fm_Opred__mono__strong,axiom,
! [P4: a > $o,X: epistemic_fm_a,Pa: a > $o] :
( ( epistemic_pred_fm_a @ P4 @ X )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( epistemic_set_fm_a @ X ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( epistemic_pred_fm_a @ Pa @ X ) ) ) ).
% fm.pred_mono_strong
thf(fact_372_K__right__mp,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ ( epistemic_Imp_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_373_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_374_subset__code_I1_J,axiom,
! [Xs: list_s580375451141968640c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) @ B4 )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( member536094252920883875c_fm_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_375_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ B4 )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_376_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_377_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X5: a] :
( ( member_a @ X5 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte295617885132580260cs_a_a @ M2 @ X5 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_378_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte5333283044364550848c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_379_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_380_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_381_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X5: a] :
( ( member_a @ X5 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X5 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_382_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_383_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_384_main_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_385_strong__completeness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) )
=> ? [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 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_386_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ S2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_387_consistent__hereditary,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S )
=> ( episte2285483198712856226tent_a @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_388_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_389_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte4583239219080210381t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_390_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_391_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I3: a] :
( ! [M3: episte6182337868402532512t_unit,X2: a] :
( ( member_a @ X2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X2 @ P ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I3 @ P ) ) ) ) ).
% generalization
thf(fact_392_generalization,axiom,
! [P: epistemic_fm_a,W: epistemic_fm_a,M: episte94448284482925344t_unit,I3: a] :
( ! [M3: episte94448284482925344t_unit,X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X2 @ P ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_K_a @ I3 @ P ) ) ) ) ).
% generalization
thf(fact_393_generalization,axiom,
! [P: epistemic_fm_a,W: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I3: a] :
( ! [M3: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I3 @ P ) ) ) ) ).
% generalization
thf(fact_394_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_395_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_396_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W2 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_397_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_398_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [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 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_399_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_400_strong__soundness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_401_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M2 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_402_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_403_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [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 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_404_strong__completeness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [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 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_405_main_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_406_refl__Euclid__equiv,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_407_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_408_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_409_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_410_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_411_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_412_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_413_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_414_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_415_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_416_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_417_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_418_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_419_neg__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I3: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_420_neg__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte2438601301504999334t_unit @ M )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_421_neg__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_422_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_423_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_424_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_425_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_426_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_427_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_428_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_429_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_430_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_431_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_432_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_433_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte2438601301504999334t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_434_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5478016696552465318t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_435_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_436_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_437_pos__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I3: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_438_pos__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_439_pos__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_440_truth,axiom,
! [M: episte6182337868402532512t_unit,W: a,I3: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_441_truth,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_442_truth,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I3: a,P: epistemic_fm_a] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_443_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_444_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ? [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 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_445_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: 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 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_446_AxB__symmetric_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I3: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ V ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ W3 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_447_Ax5__Euclidean,axiom,
! [A: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I3: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ U )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ U )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ U ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ U ) ) ) ) )
=> ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_448_Ax4__transitive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I3: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ V ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ W3 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_449_AxT__reflexive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,I3: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I3 @ P3 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_450_conj__subset__def,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) )
= ( ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_451_conj__subset__def,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) )
= ( ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_452_prop__restrict,axiom,
! [X: a,Z4: set_a,X6: set_a,P4: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X6 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_453_prop__restrict,axiom,
! [X: set_Epistemic_fm_a,Z4: set_se5208064806568342746c_fm_a,X6: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X @ Z4 )
=> ( ( ord_le7112219575281605754c_fm_a @ Z4
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X6 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_454_prop__restrict,axiom,
! [X: epistemic_fm_a,Z4: set_Epistemic_fm_a,X6: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X @ Z4 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z4
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ X6 )
& ( P4 @ X3 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_455_Collect__restrict,axiom,
! [X6: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X6 )
& ( P4 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_456_Collect__restrict,axiom,
! [X6: 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 @ X6 )
& ( P4 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_457_Collect__restrict,axiom,
! [X6: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ X6 )
& ( P4 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_458_ax__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_459_consequent__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ V )
=> ( member6642669571620171971c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_460_deriv__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( epistemic_AK_a @ A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_461_maximal__extension,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V @ W4 )
=> ( ( episte2285483198712856226tent_a @ A @ W4 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) ) ) ).
% maximal_extension
thf(fact_462_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_463_subset__Collect__iff,axiom,
! [B4: set_a,A: set_a,P4: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_464_subset__Collect__iff,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ B4
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ B4 )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_465_subset__Collect__iff,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ B4
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ B4 )
=> ( P4 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_466_subset__CollectI,axiom,
! [B4: set_a,A: set_a,Q4: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B4 )
& ( Q4 @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_467_subset__CollectI,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,Q4: set_Epistemic_fm_a > $o,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ B4 )
& ( Q4 @ X3 ) ) )
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_468_subset__CollectI,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,Q4: epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ B4 )
& ( Q4 @ X3 ) ) )
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_469_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G5 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% K_Boole
thf(fact_470_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_471_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X3 ) )
=> ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_472_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
= ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X3 ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_473_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_474_frame_Oext__inject,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W7: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W6 @ K @ More )
= ( episte2888590659910966568t_unit @ W7 @ K2 @ More2 ) )
= ( ( W6 = W7 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_475_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_a > list_char > $o,More2: product_unit] :
( ( ( episte8239586592105053771t_unit @ Pi @ More )
= ( episte8239586592105053771t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_476_subset__empty,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a )
= ( A = bot_bo3626323581529592678c_fm_a ) ) ).
% subset_empty
thf(fact_477_empty__subsetI,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A ) ).
% empty_subsetI
thf(fact_478_frame_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W6 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_479_kripke_Ocases,axiom,
! [R2: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o] :
( R2
!= ( episte2888590659910966568t_unit @ W8 @ K3 @ ( episte8239586592105053771t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_480_kripke_Oext__induct,axiom,
! [P4: episte1193835314949844379t_unit > $o,R2: episte1193835314949844379t_unit] :
( ! [Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] : ( P4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R2 ) ) ).
% kripke.ext_induct
thf(fact_481_frame_Ocases__scheme,axiom,
! [R2: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R2
!= ( episte2888590659910966568t_unit @ W8 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_482_kripke_Ocases__scheme,axiom,
! [R2: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R2
!= ( episte2888590659910966568t_unit @ W8 @ K3 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_483_empty__def,axiom,
( bot_bo1868452413526482246c_fm_a
= ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : $false ) ) ).
% empty_def
thf(fact_484_empty__def,axiom,
( bot_bo3626323581529592678c_fm_a
= ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : $false ) ) ).
% empty_def
thf(fact_485_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_a > list_char > $o,W: set_Epistemic_fm_a,W3: set_se5208064806568342746c_fm_a,R2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ ( Pi4 @ W ) @ ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R2 @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R2 @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_486_kripke_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_487_bot_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
=> ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_488_bot_Oextremum__uniqueI,axiom,
! [A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ bot_bot_a_a_o )
=> ( A2 = bot_bot_a_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_489_bot_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ bot_bo8256154652621639642fm_a_o )
=> ( A2 = bot_bo8256154652621639642fm_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_490_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
=> ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_491_bot_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
= ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_unique
thf(fact_492_bot_Oextremum__unique,axiom,
! [A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ bot_bot_a_a_o )
= ( A2 = bot_bot_a_a_o ) ) ).
% bot.extremum_unique
thf(fact_493_bot_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ bot_bo8256154652621639642fm_a_o )
= ( A2 = bot_bo8256154652621639642fm_a_o ) ) ).
% bot.extremum_unique
thf(fact_494_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
= ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_495_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ bot_bo6433428028861825271fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_496_bot_Oextremum,axiom,
! [A2: a > a > $o] : ( ord_less_eq_a_a_o @ bot_bot_a_a_o @ A2 ) ).
% bot.extremum
thf(fact_497_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ bot_bo8256154652621639642fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_498_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_499_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_500_subset__emptyI,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
~ ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le7112219575281605754c_fm_a @ A @ bot_bo1868452413526482246c_fm_a ) ) ).
% subset_emptyI
thf(fact_501_subset__emptyI,axiom,
! [A: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% subset_emptyI
thf(fact_502_frame_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W6 @ K @ More ) )
= W6 ) ).
% frame.select_convs(1)
thf(fact_503_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_a,X: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_504_imply_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q )
= ( epistemic_Imp_a @ P @ ( epistemic_imply_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_505_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a,R2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ Q @ G5 ) ) @ R2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G5 ) ) @ R2 ) ) ) ).
% K_swap
thf(fact_506_K__imply__Cons,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_507_K__imply__head,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ P ) ) ).
% K_imply_head
thf(fact_508_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_509_fm_Osimps_I90_J,axiom,
( ( episte9089240958480457552c_fm_a @ episte5073044243917183961c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(90)
thf(fact_510_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_511_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) )
@ V
@ P ) ) ) ) ).
% truth_lemma
thf(fact_512_fm_Osimps_I91_J,axiom,
! [X22: list_char] :
( ( episte9089240958480457552c_fm_a @ ( episte3759128466173231372c_fm_a @ X22 ) )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(91)
thf(fact_513_fm_Osimps_I91_J,axiom,
! [X22: list_char] :
( ( epistemic_set_fm_a @ ( epistemic_Pro_a @ X22 ) )
= bot_bot_set_a ) ).
% fm.simps(91)
thf(fact_514_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G5 ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_515_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_516_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a,R2: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G5 ) @ R2 ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G5 ) @ R2 ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ R2 ) ) ) ) ) ).
% K_DisE
thf(fact_517_K__DisL,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P5: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P5 @ Ps ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Dis_a @ P @ P5 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_518_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_519_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_520_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_521_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_522_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
& ( member536094252920883875c_fm_a @ X3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Z5 ) ) )
=> ( member536094252920883875c_fm_a @ X3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_523_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( member536094252920883875c_fm_a @ X3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X3 ) ) ) ) ) ).
% reflexive_def
thf(fact_524_kripke_Osurjective,axiom,
! [R2: episte1560738328020401952t_unit] :
( R2
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R2 ) @ ( episte6250069432388174439t_unit @ R2 ) @ ( episte8239586592105053771t_unit @ ( episte2398645135750866164t_unit @ R2 ) @ ( episte5479201149095757850t_unit @ R2 ) ) ) ) ).
% kripke.surjective
thf(fact_525_frame_Osurjective,axiom,
! [R2: episte1560738328020401952t_unit] :
( R2
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R2 ) @ ( episte6250069432388174439t_unit @ R2 ) @ ( episte3309513806868946049t_unit @ R2 ) ) ) ).
% frame.surjective
thf(fact_526_kripke_Oequality,axiom,
! [R2: episte1560738328020401952t_unit,R3: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R2 )
= ( episte8072386903178013299t_unit @ R3 ) )
=> ( ( ( episte6250069432388174439t_unit @ R2 )
= ( episte6250069432388174439t_unit @ R3 ) )
=> ( ( ( episte2398645135750866164t_unit @ R2 )
= ( episte2398645135750866164t_unit @ R3 ) )
=> ( ( ( episte5479201149095757850t_unit @ R2 )
= ( episte5479201149095757850t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ) ).
% kripke.equality
thf(fact_527_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte9120385895580347753c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_528_kripke_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_529_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte1857908288096881731t_unit @ Pi2 @ ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_530_frame_Oselect__convs_I3_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W6 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_531_frame_Oequality,axiom,
! [R2: episte1560738328020401952t_unit,R3: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R2 )
= ( episte8072386903178013299t_unit @ R3 ) )
=> ( ( ( episte6250069432388174439t_unit @ R2 )
= ( episte6250069432388174439t_unit @ R3 ) )
=> ( ( ( episte3309513806868946049t_unit @ R2 )
= ( episte3309513806868946049t_unit @ R3 ) )
=> ( R2 = R3 ) ) ) ) ).
% frame.equality
thf(fact_532_inconsistent__subset,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [V3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V3 ) @ V )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V3 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_533_Shift__def,axiom,
( bNF_Gr8437504134799245625c_fm_a
= ( ^ [Kl: set_li769143395467472256c_fm_a,K4: epistemic_fm_a] :
( collec5191077796991884427c_fm_a
@ ^ [Kl2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ K4 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_534_canonical__model_I1_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X3: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X3 ) @ V2 )
@ product_Unity ) )
@ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S )
@ P ) ) ) ).
% canonical_model(1)
thf(fact_535_insert__subset,axiom,
! [X: a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_536_insert__subset,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X @ A ) @ B4 )
= ( ( member536094252920883875c_fm_a @ X @ B4 )
& ( ord_le7112219575281605754c_fm_a @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_537_insert__subset,axiom,
! [X: epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X @ A ) @ B4 )
= ( ( member6642669571620171971c_fm_a @ X @ B4 )
& ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_538_Un__subset__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_539_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ C2 )
= ( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
& ( ord_le3275665582123262618c_fm_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_540_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collec2519470961442302949c_fm_a
@ ^ [S4: set_Epistemic_fm_a] : P4 )
= top_to5398033090887092778c_fm_a ) )
& ( ~ P4
=> ( ( collec2519470961442302949c_fm_a
@ ^ [S4: set_Epistemic_fm_a] : P4 )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_const
thf(fact_541_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collec4904205152690461189c_fm_a
@ ^ [S4: epistemic_fm_a] : P4 )
= top_to7796028867103199306c_fm_a ) )
& ( ~ P4
=> ( ( collec4904205152690461189c_fm_a
@ ^ [S4: epistemic_fm_a] : P4 )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_const
thf(fact_542_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y: a,Z: a] : ( Y = Z )
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_543_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ( ^ [Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y = Z )
@ A2 ) )
= ( insert7129737658439086282c_fm_a @ A2 @ bot_bo1868452413526482246c_fm_a ) ) ).
% singleton_conv2
thf(fact_544_singleton__conv2,axiom,
! [A2: epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ( ^ [Y: epistemic_fm_a,Z: epistemic_fm_a] : ( Y = Z )
@ A2 ) )
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) ).
% singleton_conv2
thf(fact_545_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_546_singleton__conv,axiom,
! [A2: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( X3 = A2 ) )
= ( insert7129737658439086282c_fm_a @ A2 @ bot_bo1868452413526482246c_fm_a ) ) ).
% singleton_conv
thf(fact_547_singleton__conv,axiom,
! [A2: epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( X3 = A2 ) )
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) ).
% singleton_conv
thf(fact_548_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_549_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ A2 @ A )
= ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_550_singleton__insert__inj__eq,axiom,
! [B: a,A2: a,A: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_551_singleton__insert__inj__eq,axiom,
! [B: epistemic_fm_a,A2: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a )
= ( insert7817948963269374698c_fm_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_552_fm_Osimps_I95_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) )
= ( insert7817948963269374698c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_553_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_554_canonical__model_I3_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ) ).
% canonical_model(3)
thf(fact_555_maximal__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% maximal_Extend
thf(fact_556_consistent__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ).
% consistent_Extend
thf(fact_557_canonical__model_I2_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ) ).
% canonical_model(2)
thf(fact_558_Collect__disj__eq,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q4 @ X3 ) ) )
= ( sup_su4305657963326700718c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_disj_eq
thf(fact_559_Collect__disj__eq,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q4 @ X3 ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_disj_eq
thf(fact_560_Collect__disj__eq,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P4 @ X3 )
| ( Q4 @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) ) ) ).
% Collect_disj_eq
thf(fact_561_insert__Collect,axiom,
! [A2: a,P4: a > $o] :
( ( insert_a @ A2 @ ( collect_a @ P4 ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A2 )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_562_insert__Collect,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( insert7129737658439086282c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( collec2519470961442302949c_fm_a
@ ^ [U2: set_Epistemic_fm_a] :
( ( U2 != A2 )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_563_insert__Collect,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( insert7817948963269374698c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( collec4904205152690461189c_fm_a
@ ^ [U2: epistemic_fm_a] :
( ( U2 != A2 )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_564_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B3: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A4 )
| ( member_a @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_565_insert__compr,axiom,
( insert7129737658439086282c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B3: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A4 )
| ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_566_insert__compr,axiom,
( insert7817948963269374698c_fm_a
= ( ^ [A4: epistemic_fm_a,B3: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A4 )
| ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_567_insert__def,axiom,
( insert7129737658439086282c_fm_a
= ( ^ [A4: set_Epistemic_fm_a] :
( sup_su4305657963326700718c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_568_insert__def,axiom,
( insert7817948963269374698c_fm_a
= ( ^ [A4: epistemic_fm_a] :
( sup_su1367922730591523534c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_569_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_570_UNIV__def,axiom,
( top_to5398033090887092778c_fm_a
= ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : $true ) ) ).
% UNIV_def
thf(fact_571_UNIV__def,axiom,
( top_to7796028867103199306c_fm_a
= ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : $true ) ) ).
% UNIV_def
thf(fact_572_Un__def,axiom,
( sup_su4305657963326700718c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A3 )
| ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_573_Un__def,axiom,
( sup_su1367922730591523534c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
| ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_574_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
| ( member_a @ X3 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_575_Extend__subset,axiom,
! [S: set_Epistemic_fm_a,A: epistemic_fm_a > $o] : ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% Extend_subset
thf(fact_576_top_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
=> ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_577_top_Oextremum__uniqueI,axiom,
! [A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ top_top_a_a_o @ A2 )
=> ( A2 = top_top_a_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_578_top_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ top_to8092407255657902782fm_a_o @ A2 )
=> ( A2 = top_to8092407255657902782fm_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_579_top_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
=> ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_uniqueI
thf(fact_580_top_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
= ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_unique
thf(fact_581_top_Oextremum__unique,axiom,
! [A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ top_top_a_a_o @ A2 )
= ( A2 = top_top_a_a_o ) ) ).
% top.extremum_unique
thf(fact_582_top_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ top_to8092407255657902782fm_a_o @ A2 )
= ( A2 = top_to8092407255657902782fm_a_o ) ) ).
% top.extremum_unique
thf(fact_583_top_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
= ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_unique
thf(fact_584_top__greatest,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ top_to6710954824118802707fm_a_o ) ).
% top_greatest
thf(fact_585_top__greatest,axiom,
! [A2: a > a > $o] : ( ord_less_eq_a_a_o @ A2 @ top_top_a_a_o ) ).
% top_greatest
thf(fact_586_top__greatest,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ A2 @ top_to8092407255657902782fm_a_o ) ).
% top_greatest
thf(fact_587_top__greatest,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ top_to7796028867103199306c_fm_a ) ).
% top_greatest
thf(fact_588_Un__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_589_Un__mono,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ ( sup_su1367922730591523534c_fm_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_590_Un__least,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_591_Un__least,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_592_Un__upper1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% Un_upper1
thf(fact_593_Un__upper1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ).
% Un_upper1
thf(fact_594_Un__upper2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% Un_upper2
thf(fact_595_Un__upper2,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ).
% Un_upper2
thf(fact_596_Un__absorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_597_Un__absorb1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_598_Un__absorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_599_Un__absorb2,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_600_subset__UnE,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B4 )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_601_subset__UnE,axiom,
! [C2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) )
=> ~ ! [A5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A5 @ A )
=> ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ B4 )
=> ( C2
!= ( sup_su1367922730591523534c_fm_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_602_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_603_subset__Un__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_604_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_605_insert__mono,axiom,
! [C2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ A2 @ C2 ) @ ( insert7817948963269374698c_fm_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_606_subset__insert,axiom,
! [X: a,A: set_a,B4: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_607_subset__insert,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ X @ B4 ) )
= ( ord_le7112219575281605754c_fm_a @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_608_subset__insert,axiom,
! [X: epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X @ B4 ) )
= ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_609_subset__insertI,axiom,
! [B4: set_a,A2: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_610_subset__insertI,axiom,
! [B4: set_Epistemic_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ ( insert7817948963269374698c_fm_a @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_611_subset__insertI2,axiom,
! [A: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_612_subset__insertI2,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_613_insert__subsetI,axiom,
! [X: a,A: set_a,X6: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_614_insert__subsetI,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ X6 @ A )
=> ( ord_le7112219575281605754c_fm_a @ ( insert7129737658439086282c_fm_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_615_insert__subsetI,axiom,
! [X: epistemic_fm_a,A: set_Epistemic_fm_a,X6: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_616_subset__UNIV,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ top_to7796028867103199306c_fm_a ) ).
% subset_UNIV
thf(fact_617_Collect__conv__if2,axiom,
! [P4: a > $o,A2: a] :
( ( ( P4 @ A2 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_618_Collect__conv__if2,axiom,
! [P4: set_Epistemic_fm_a > $o,A2: set_Epistemic_fm_a] :
( ( ( P4 @ A2 )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= ( insert7129737658439086282c_fm_a @ A2 @ bot_bo1868452413526482246c_fm_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_conv_if2
thf(fact_619_Collect__conv__if2,axiom,
! [P4: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( ( P4 @ A2 )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( A2 = X3 )
& ( P4 @ X3 ) ) )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_conv_if2
thf(fact_620_Collect__conv__if,axiom,
! [P4: a > $o,A2: a] :
( ( ( P4 @ A2 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_621_Collect__conv__if,axiom,
! [P4: set_Epistemic_fm_a > $o,A2: set_Epistemic_fm_a] :
( ( ( P4 @ A2 )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= ( insert7129737658439086282c_fm_a @ A2 @ bot_bo1868452413526482246c_fm_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= bot_bo1868452413526482246c_fm_a ) ) ) ).
% Collect_conv_if
thf(fact_622_Collect__conv__if,axiom,
! [P4: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( ( P4 @ A2 )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( X3 = A2 )
& ( P4 @ X3 ) ) )
= bot_bo3626323581529592678c_fm_a ) ) ) ).
% Collect_conv_if
thf(fact_623_consistent__disjuncts,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Dis_a @ P @ Q ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
| ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_624_maximal__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( ! [P3: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P3 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ P3 @ S ) ) ) ) ).
% maximal_def
thf(fact_625_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_626_subset__singleton__iff,axiom,
! [X6: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X6 @ ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) )
= ( ( X6 = bot_bo3626323581529592678c_fm_a )
| ( X6
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_627_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_628_subset__singletonD,axiom,
! [A: set_Epistemic_fm_a,X: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X @ bot_bo3626323581529592678c_fm_a ) )
=> ( ( A = bot_bo3626323581529592678c_fm_a )
| ( A
= ( insert7817948963269374698c_fm_a @ X @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singletonD
thf(fact_629_consistent__consequent,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_630_consistent__consequent_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ! [G4: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H4 @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_631_fm_Osimps_I94_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X51 ) @ ( episte9089240958480457552c_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_632_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_633_fm_Osimps_I93_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X41 ) @ ( episte9089240958480457552c_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_634_fm_Osimps_I93_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X41 ) @ ( epistemic_set_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_635_fm_Osimps_I92_J,axiom,
! [X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X31 ) @ ( episte9089240958480457552c_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_636_fm_Osimps_I92_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X31 ) @ ( epistemic_set_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_637_frame_Oupdate__convs_I3_J,axiom,
! [More2: episte1193835314949844379t_unit > episte1193835314949844379t_unit,W6: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte7964699311158398698c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W6 @ K @ More ) )
= ( episte2888590659910966568t_unit @ W6 @ K @ ( More2 @ More ) ) ) ).
% frame.update_convs(3)
thf(fact_638_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ ( set_Epistemic_fm_a2 @ G5 ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% inconsistent_imply
thf(fact_639_le__sup__iff,axiom,
! [X: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_640_le__sup__iff,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X @ Y2 ) @ Z2 )
= ( ( ord_le4043730696559282883fm_a_o @ X @ Z2 )
& ( ord_le4043730696559282883fm_a_o @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_641_le__sup__iff,axiom,
! [X: a > a > $o,Y2: a > a > $o,Z2: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_a_a_o @ X @ Z2 )
& ( ord_less_eq_a_a_o @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_642_le__sup__iff,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o,Z2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ X @ Y2 ) @ Z2 )
= ( ( ord_le3934200179093585166fm_a_o @ X @ Z2 )
& ( ord_le3934200179093585166fm_a_o @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_643_le__sup__iff,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X @ Y2 ) @ Z2 )
= ( ( ord_le3275665582123262618c_fm_a @ X @ Z2 )
& ( ord_le3275665582123262618c_fm_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_644_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_645_sup_Obounded__iff,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 )
= ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
& ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_646_sup_Obounded__iff,axiom,
! [B: a > a > $o,C: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A2 )
= ( ( ord_less_eq_a_a_o @ B @ A2 )
& ( ord_less_eq_a_a_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_647_sup_Obounded__iff,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ B @ C ) @ A2 )
= ( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
& ( ord_le3934200179093585166fm_a_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_648_sup_Obounded__iff,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 )
= ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
& ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_649_extendS__def,axiom,
! [A: epistemic_fm_a > $o,N: epistemic_fm_a,Prev: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S @ N @ Prev )
= ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) ) )
& ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_650_maximal_H__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% maximal'_Extend
thf(fact_651_maximal__maximal_H,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% maximal_maximal'
thf(fact_652_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W4 )
=> ( ( finite3304564945125563331c_fm_a @ W4 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_653_fm_Opred__inject_I2_J,axiom,
! [P4: a > $o,A2: list_char] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Pro_a @ A2 ) )
= ( top_top_list_char_o @ A2 ) ) ).
% fm.pred_inject(2)
thf(fact_654_infinite__UNIV__fm,axiom,
~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% infinite_UNIV_fm
thf(fact_655_sup__set__def,axiom,
( sup_su4305657963326700718c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ( sup_su2585399871592001967fm_a_o
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A3 )
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_656_sup__set__def,axiom,
( sup_su1367922730591523534c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ( sup_su6697315146409554831fm_a_o
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A3 )
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_657_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A3 )
@ ^ [X3: a] : ( member_a @ X3 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_658_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_659_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_660_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_661_inconsistent__finite,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ S )
=> ? [S5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S5 @ S )
& ( finite3304564945125563331c_fm_a @ S5 )
& ~ ( episte2285483198712856226tent_a @ A @ S5 ) ) ) ).
% inconsistent_finite
thf(fact_662_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_663_sup_OcoboundedI2,axiom,
! [C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_664_sup_OcoboundedI2,axiom,
! [C: a > a > $o,B: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ B )
=> ( ord_less_eq_a_a_o @ C @ ( sup_sup_a_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_665_sup_OcoboundedI2,axiom,
! [C: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ C @ B )
=> ( ord_le3934200179093585166fm_a_o @ C @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_666_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_667_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_668_sup_OcoboundedI1,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_669_sup_OcoboundedI1,axiom,
! [C: a > a > $o,A2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ A2 )
=> ( ord_less_eq_a_a_o @ C @ ( sup_sup_a_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_670_sup_OcoboundedI1,axiom,
! [C: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ C @ A2 )
=> ( ord_le3934200179093585166fm_a_o @ C @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_671_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_672_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
( ( sup_sup_set_a @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_673_sup_Oabsorb__iff2,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_674_sup_Oabsorb__iff2,axiom,
( ord_less_eq_a_a_o
= ( ^ [A4: a > a > $o,B2: a > a > $o] :
( ( sup_sup_a_a_o @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_675_sup_Oabsorb__iff2,axiom,
( ord_le3934200179093585166fm_a_o
= ( ^ [A4: epistemic_fm_a > epistemic_fm_a > $o,B2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( sup_su5518344886633089858fm_a_o @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_676_sup_Oabsorb__iff2,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_677_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_678_sup_Oabsorb__iff1,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_679_sup_Oabsorb__iff1,axiom,
( ord_less_eq_a_a_o
= ( ^ [B2: a > a > $o,A4: a > a > $o] :
( ( sup_sup_a_a_o @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_680_sup_Oabsorb__iff1,axiom,
( ord_le3934200179093585166fm_a_o
= ( ^ [B2: epistemic_fm_a > epistemic_fm_a > $o,A4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( sup_su5518344886633089858fm_a_o @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_681_sup_Oabsorb__iff1,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_682_sup_Ocobounded2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_683_sup_Ocobounded2,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ B @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_684_sup_Ocobounded2,axiom,
! [B: a > a > $o,A2: a > a > $o] : ( ord_less_eq_a_a_o @ B @ ( sup_sup_a_a_o @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_685_sup_Ocobounded2,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ B @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_686_sup_Ocobounded2,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_687_sup_Ocobounded1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_688_sup_Ocobounded1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_689_sup_Ocobounded1,axiom,
! [A2: a > a > $o,B: a > a > $o] : ( ord_less_eq_a_a_o @ A2 @ ( sup_sup_a_a_o @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_690_sup_Ocobounded1,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ A2 @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_691_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_692_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_693_sup_Oorder__iff,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( A4
= ( sup_su6697315146409554831fm_a_o @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_694_sup_Oorder__iff,axiom,
( ord_less_eq_a_a_o
= ( ^ [B2: a > a > $o,A4: a > a > $o] :
( A4
= ( sup_sup_a_a_o @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_695_sup_Oorder__iff,axiom,
( ord_le3934200179093585166fm_a_o
= ( ^ [B2: epistemic_fm_a > epistemic_fm_a > $o,A4: epistemic_fm_a > epistemic_fm_a > $o] :
( A4
= ( sup_su5518344886633089858fm_a_o @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_696_sup_Oorder__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( A4
= ( sup_su1367922730591523534c_fm_a @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_697_sup_OboundedI,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_698_sup_OboundedI,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_699_sup_OboundedI,axiom,
! [B: a > a > $o,A2: a > a > $o,C: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( ( ord_less_eq_a_a_o @ C @ A2 )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_700_sup_OboundedI,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( ( ord_le3934200179093585166fm_a_o @ C @ A2 )
=> ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_701_sup_OboundedI,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_702_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_703_sup_OboundedE,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_704_sup_OboundedE,axiom,
! [B: a > a > $o,C: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_a_a_o @ B @ A2 )
=> ~ ( ord_less_eq_a_a_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_705_sup_OboundedE,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ B @ C ) @ A2 )
=> ~ ( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ~ ( ord_le3934200179093585166fm_a_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_706_sup_OboundedE,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ~ ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_707_sup__absorb2,axiom,
! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( sup_sup_set_a @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_708_sup__absorb2,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y2 )
=> ( ( sup_su6697315146409554831fm_a_o @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_709_sup__absorb2,axiom,
! [X: a > a > $o,Y2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ Y2 )
=> ( ( sup_sup_a_a_o @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_710_sup__absorb2,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X @ Y2 )
=> ( ( sup_su5518344886633089858fm_a_o @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_711_sup__absorb2,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y2 )
=> ( ( sup_su1367922730591523534c_fm_a @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_712_sup__absorb1,axiom,
! [Y2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( sup_sup_set_a @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_713_sup__absorb1,axiom,
! [Y2: epistemic_fm_a > $o,X: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X )
=> ( ( sup_su6697315146409554831fm_a_o @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_714_sup__absorb1,axiom,
! [Y2: a > a > $o,X: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y2 @ X )
=> ( ( sup_sup_a_a_o @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_715_sup__absorb1,axiom,
! [Y2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ Y2 @ X )
=> ( ( sup_su5518344886633089858fm_a_o @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_716_sup__absorb1,axiom,
! [Y2: set_Epistemic_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X )
=> ( ( sup_su1367922730591523534c_fm_a @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_717_sup_Oabsorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_718_sup_Oabsorb2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( sup_su6697315146409554831fm_a_o @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_719_sup_Oabsorb2,axiom,
! [A2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ B )
=> ( ( sup_sup_a_a_o @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_720_sup_Oabsorb2,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ B )
=> ( ( sup_su5518344886633089858fm_a_o @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_721_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( sup_su1367922730591523534c_fm_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_722_sup_Oabsorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_723_sup_Oabsorb1,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( sup_su6697315146409554831fm_a_o @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_724_sup_Oabsorb1,axiom,
! [B: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( ( sup_sup_a_a_o @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_725_sup_Oabsorb1,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( ( sup_su5518344886633089858fm_a_o @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_726_sup_Oabsorb1,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( sup_su1367922730591523534c_fm_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_727_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y2: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_728_sup__unique,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y3 @ X2 )
=> ( ( ord_le4043730696559282883fm_a_o @ Z3 @ X2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su6697315146409554831fm_a_o @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_729_sup__unique,axiom,
! [F: ( a > a > $o ) > ( a > a > $o ) > a > a > $o,X: a > a > $o,Y2: a > a > $o] :
( ! [X2: a > a > $o,Y3: a > a > $o] : ( ord_less_eq_a_a_o @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: a > a > $o,Y3: a > a > $o] : ( ord_less_eq_a_a_o @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: a > a > $o,Y3: a > a > $o,Z3: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y3 @ X2 )
=> ( ( ord_less_eq_a_a_o @ Z3 @ X2 )
=> ( ord_less_eq_a_a_o @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_a_a_o @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_730_sup__unique,axiom,
! [F: ( epistemic_fm_a > epistemic_fm_a > $o ) > ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o,Z3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ Y3 @ X2 )
=> ( ( ord_le3934200179093585166fm_a_o @ Z3 @ X2 )
=> ( ord_le3934200179093585166fm_a_o @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su5518344886633089858fm_a_o @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_731_sup__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y3 @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z3 @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1367922730591523534c_fm_a @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_732_sup_OorderI,axiom,
! [A2: set_a,B: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B ) )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_733_sup_OorderI,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( A2
= ( sup_su6697315146409554831fm_a_o @ A2 @ B ) )
=> ( ord_le4043730696559282883fm_a_o @ B @ A2 ) ) ).
% sup.orderI
thf(fact_734_sup_OorderI,axiom,
! [A2: a > a > $o,B: a > a > $o] :
( ( A2
= ( sup_sup_a_a_o @ A2 @ B ) )
=> ( ord_less_eq_a_a_o @ B @ A2 ) ) ).
% sup.orderI
thf(fact_735_sup_OorderI,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A2
= ( sup_su5518344886633089858fm_a_o @ A2 @ B ) )
=> ( ord_le3934200179093585166fm_a_o @ B @ A2 ) ) ).
% sup.orderI
thf(fact_736_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_737_sup_OorderE,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_738_sup_OorderE,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( A2
= ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_739_sup_OorderE,axiom,
! [B: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ B @ A2 )
=> ( A2
= ( sup_sup_a_a_o @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_740_sup_OorderE,axiom,
! [B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ B @ A2 )
=> ( A2
= ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_741_sup_OorderE,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( A2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_742_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_743_le__iff__sup,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X3: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_744_le__iff__sup,axiom,
( ord_less_eq_a_a_o
= ( ^ [X3: a > a > $o,Y4: a > a > $o] :
( ( sup_sup_a_a_o @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_745_le__iff__sup,axiom,
( ord_le3934200179093585166fm_a_o
= ( ^ [X3: epistemic_fm_a > epistemic_fm_a > $o,Y4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( sup_su5518344886633089858fm_a_o @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_746_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_747_sup__least,axiom,
! [Y2: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_748_sup__least,axiom,
! [Y2: epistemic_fm_a > $o,X: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ Z2 @ X )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_749_sup__least,axiom,
! [Y2: a > a > $o,X: a > a > $o,Z2: a > a > $o] :
( ( ord_less_eq_a_a_o @ Y2 @ X )
=> ( ( ord_less_eq_a_a_o @ Z2 @ X )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_750_sup__least,axiom,
! [Y2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o,Z2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ Y2 @ X )
=> ( ( ord_le3934200179093585166fm_a_o @ Z2 @ X )
=> ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_751_sup__least,axiom,
! [Y2: set_Epistemic_fm_a,X: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X )
=> ( ( ord_le3275665582123262618c_fm_a @ Z2 @ X )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_752_sup__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_753_sup__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,D2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ D2 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ ( sup_su6697315146409554831fm_a_o @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_754_sup__mono,axiom,
! [A2: a > a > $o,C: a > a > $o,B: a > a > $o,D2: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ C )
=> ( ( ord_less_eq_a_a_o @ B @ D2 )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A2 @ B ) @ ( sup_sup_a_a_o @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_755_sup__mono,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,D2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ C )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ D2 )
=> ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) @ ( sup_su5518344886633089858fm_a_o @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_756_sup__mono,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_757_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_758_sup_Omono,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,D2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ D2 @ B )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ C @ D2 ) @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_759_sup_Omono,axiom,
! [C: a > a > $o,A2: a > a > $o,D2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ C @ A2 )
=> ( ( ord_less_eq_a_a_o @ D2 @ B )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ C @ D2 ) @ ( sup_sup_a_a_o @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_760_sup_Omono,axiom,
! [C: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o,D2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ C @ A2 )
=> ( ( ord_le3934200179093585166fm_a_o @ D2 @ B )
=> ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ C @ D2 ) @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_761_sup_Omono,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,D2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ D2 @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ C @ D2 ) @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_762_le__supI2,axiom,
! [X: set_a,B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_763_le__supI2,axiom,
! [X: epistemic_fm_a > $o,B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ B )
=> ( ord_le4043730696559282883fm_a_o @ X @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_764_le__supI2,axiom,
! [X: a > a > $o,B: a > a > $o,A2: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ B )
=> ( ord_less_eq_a_a_o @ X @ ( sup_sup_a_a_o @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_765_le__supI2,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X @ B )
=> ( ord_le3934200179093585166fm_a_o @ X @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_766_le__supI2,axiom,
! [X: set_Epistemic_fm_a,B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ B )
=> ( ord_le3275665582123262618c_fm_a @ X @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_767_le__supI1,axiom,
! [X: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_768_le__supI1,axiom,
! [X: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ X @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_769_le__supI1,axiom,
! [X: a > a > $o,A2: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ X @ A2 )
=> ( ord_less_eq_a_a_o @ X @ ( sup_sup_a_a_o @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_770_le__supI1,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X @ A2 )
=> ( ord_le3934200179093585166fm_a_o @ X @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_771_le__supI1,axiom,
! [X: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ X @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_772_sup__ge2,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_773_sup__ge2,axiom,
! [Y2: epistemic_fm_a > $o,X: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y2 @ ( sup_su6697315146409554831fm_a_o @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_774_sup__ge2,axiom,
! [Y2: a > a > $o,X: a > a > $o] : ( ord_less_eq_a_a_o @ Y2 @ ( sup_sup_a_a_o @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_775_sup__ge2,axiom,
! [Y2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ Y2 @ ( sup_su5518344886633089858fm_a_o @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_776_sup__ge2,axiom,
! [Y2: set_Epistemic_fm_a,X: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y2 @ ( sup_su1367922730591523534c_fm_a @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_777_sup__ge1,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_778_sup__ge1,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X @ ( sup_su6697315146409554831fm_a_o @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_779_sup__ge1,axiom,
! [X: a > a > $o,Y2: a > a > $o] : ( ord_less_eq_a_a_o @ X @ ( sup_sup_a_a_o @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_780_sup__ge1,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ X @ ( sup_su5518344886633089858fm_a_o @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_781_sup__ge1,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X @ ( sup_su1367922730591523534c_fm_a @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_782_le__supI,axiom,
! [A2: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_783_le__supI,axiom,
! [A2: epistemic_fm_a > $o,X: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ X )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_784_le__supI,axiom,
! [A2: a > a > $o,X: a > a > $o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ A2 @ X )
=> ( ( ord_less_eq_a_a_o @ B @ X )
=> ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_785_le__supI,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ A2 @ X )
=> ( ( ord_le3934200179093585166fm_a_o @ B @ X )
=> ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_786_le__supI,axiom,
! [A2: set_Epistemic_fm_a,X: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ X )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ X )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_787_le__supE,axiom,
! [A2: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_788_le__supE,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,X: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ X )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ A2 @ X )
=> ~ ( ord_le4043730696559282883fm_a_o @ B @ X ) ) ) ).
% le_supE
thf(fact_789_le__supE,axiom,
! [A2: a > a > $o,B: a > a > $o,X: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( sup_sup_a_a_o @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_a_a_o @ A2 @ X )
=> ~ ( ord_less_eq_a_a_o @ B @ X ) ) ) ).
% le_supE
thf(fact_790_le__supE,axiom,
! [A2: epistemic_fm_a > epistemic_fm_a > $o,B: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( sup_su5518344886633089858fm_a_o @ A2 @ B ) @ X )
=> ~ ( ( ord_le3934200179093585166fm_a_o @ A2 @ X )
=> ~ ( ord_le3934200179093585166fm_a_o @ B @ X ) ) ) ).
% le_supE
thf(fact_791_le__supE,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ X )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A2 @ X )
=> ~ ( ord_le3275665582123262618c_fm_a @ B @ X ) ) ) ).
% le_supE
thf(fact_792_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_793_inf__sup__ord_I3_J,axiom,
! [X: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X @ ( sup_su6697315146409554831fm_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_794_inf__sup__ord_I3_J,axiom,
! [X: a > a > $o,Y2: a > a > $o] : ( ord_less_eq_a_a_o @ X @ ( sup_sup_a_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_795_inf__sup__ord_I3_J,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,Y2: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ X @ ( sup_su5518344886633089858fm_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_796_inf__sup__ord_I3_J,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X @ ( sup_su1367922730591523534c_fm_a @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_797_inf__sup__ord_I4_J,axiom,
! [Y2: set_a,X: set_a] : ( ord_less_eq_set_a @ Y2 @ ( sup_sup_set_a @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_798_inf__sup__ord_I4_J,axiom,
! [Y2: epistemic_fm_a > $o,X: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y2 @ ( sup_su6697315146409554831fm_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_799_inf__sup__ord_I4_J,axiom,
! [Y2: a > a > $o,X: a > a > $o] : ( ord_less_eq_a_a_o @ Y2 @ ( sup_sup_a_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_800_inf__sup__ord_I4_J,axiom,
! [Y2: epistemic_fm_a > epistemic_fm_a > $o,X: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ Y2 @ ( sup_su5518344886633089858fm_a_o @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_801_inf__sup__ord_I4_J,axiom,
! [Y2: set_Epistemic_fm_a,X: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y2 @ ( sup_su1367922730591523534c_fm_a @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_802_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [B3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_803_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_804_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_805_finite__subset__induct,axiom,
! [F3: set_a,A: set_a,P4: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ( P4 @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_a @ A6 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_806_finite__subset__induct,axiom,
! [F3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ F3 )
=> ( ( ord_le7112219575281605754c_fm_a @ F3 @ A )
=> ( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a,F4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ F4 )
=> ( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ~ ( member536094252920883875c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7129737658439086282c_fm_a @ A6 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_807_finite__subset__induct,axiom,
! [F3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ F3 )
=> ( ( ord_le3275665582123262618c_fm_a @ F3 @ A )
=> ( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ! [A6: epistemic_fm_a,F4: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ F4 )
=> ( ( member6642669571620171971c_fm_a @ A6 @ A )
=> ( ~ ( member6642669571620171971c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7817948963269374698c_fm_a @ A6 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_808_finite__subset__induct_H,axiom,
! [F3: set_a,A: set_a,P4: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ( P4 @ bot_bot_set_a )
=> ( ! [A6: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A6 @ A )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ~ ( member_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_809_finite__subset__induct_H,axiom,
! [F3: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_se5208064806568342746c_fm_a > $o] :
( ( finite7364866884270929315c_fm_a @ F3 )
=> ( ( ord_le7112219575281605754c_fm_a @ F3 @ A )
=> ( ( P4 @ bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a,F4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ F4 )
=> ( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ F4 @ A )
=> ( ~ ( member536094252920883875c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7129737658439086282c_fm_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_810_finite__subset__induct_H,axiom,
! [F3: set_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a @ F3 )
=> ( ( ord_le3275665582123262618c_fm_a @ F3 @ A )
=> ( ( P4 @ bot_bo3626323581529592678c_fm_a )
=> ( ! [A6: epistemic_fm_a,F4: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ F4 )
=> ( ( member6642669571620171971c_fm_a @ A6 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ F4 @ A )
=> ( ~ ( member6642669571620171971c_fm_a @ A6 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert7817948963269374698c_fm_a @ A6 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_811_finite__Collect__conjI,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
| ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q4 ) ) )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_812_finite__Collect__conjI,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
| ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q4 ) ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_813_finite__Collect__disjI,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) )
= ( ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( finite7364866884270929315c_fm_a @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_814_finite__Collect__disjI,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( P4 @ X3 )
| ( Q4 @ X3 ) ) ) )
= ( ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( finite3304564945125563331c_fm_a @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_815_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_816_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_817_pigeonhole__infinite__rel,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,R: a > epistemic_fm_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ B4 )
& ( R @ X2 @ Xa3 ) ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_818_pigeonhole__infinite__rel,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_Epistemic_fm_a,R: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ~ ( finite7364866884270929315c_fm_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ? [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ B4 )
& ( R @ X2 @ Xa3 ) ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
& ~ ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [A4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_819_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,R: epistemic_fm_a > epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ? [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ B4 )
& ( R @ X2 @ Xa3 ) ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
& ~ ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [A4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_820_finite__has__minimal2,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ( ord_le4043730696559282883fm_a_o @ X2 @ A2 )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_821_finite__has__minimal2,axiom,
! [A: set_a_a_o,A2: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A2 @ A )
=> ? [X2: a > a > $o] :
( ( member_a_a_o @ X2 @ A )
& ( ord_less_eq_a_a_o @ X2 @ A2 )
& ! [Xa3: a > a > $o] :
( ( member_a_a_o @ Xa3 @ A )
=> ( ( ord_less_eq_a_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_822_finite__has__minimal2,axiom,
! [A: set_Ep7530996317732161742fm_a_o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( member8030264063065380503fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X2 @ A )
& ( ord_le3934200179093585166fm_a_o @ X2 @ A2 )
& ! [Xa3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Xa3 @ A )
=> ( ( ord_le3934200179093585166fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_823_finite__has__minimal2,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( ord_le3275665582123262618c_fm_a @ X2 @ A2 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_824_finite__has__maximal2,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ( ord_le4043730696559282883fm_a_o @ A2 @ X2 )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_825_finite__has__maximal2,axiom,
! [A: set_a_a_o,A2: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A2 @ A )
=> ? [X2: a > a > $o] :
( ( member_a_a_o @ X2 @ A )
& ( ord_less_eq_a_a_o @ A2 @ X2 )
& ! [Xa3: a > a > $o] :
( ( member_a_a_o @ Xa3 @ A )
=> ( ( ord_less_eq_a_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_826_finite__has__maximal2,axiom,
! [A: set_Ep7530996317732161742fm_a_o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( member8030264063065380503fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X2 @ A )
& ( ord_le3934200179093585166fm_a_o @ A2 @ X2 )
& ! [Xa3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Xa3 @ A )
=> ( ( ord_le3934200179093585166fm_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_827_finite__has__maximal2,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( ord_le3275665582123262618c_fm_a @ A2 @ X2 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_828_rev__finite__subset,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( finite3304564945125563331c_fm_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_829_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_830_finite__subset,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( finite3304564945125563331c_fm_a @ B4 )
=> ( finite3304564945125563331c_fm_a @ A ) ) ) ).
% finite_subset
thf(fact_831_finite__has__minimal,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_832_finite__has__minimal,axiom,
! [A: set_a_a_o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ? [X2: a > a > $o] :
( ( member_a_a_o @ X2 @ A )
& ! [Xa3: a > a > $o] :
( ( member_a_a_o @ Xa3 @ A )
=> ( ( ord_less_eq_a_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_833_finite__has__minimal,axiom,
! [A: set_Ep7530996317732161742fm_a_o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ? [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X2 @ A )
& ! [Xa3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Xa3 @ A )
=> ( ( ord_le3934200179093585166fm_a_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_834_finite__has__minimal,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_835_finite__has__maximal,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ! [Xa3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_836_finite__has__maximal,axiom,
! [A: set_a_a_o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ? [X2: a > a > $o] :
( ( member_a_a_o @ X2 @ A )
& ! [Xa3: a > a > $o] :
( ( member_a_a_o @ Xa3 @ A )
=> ( ( ord_less_eq_a_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_837_finite__has__maximal,axiom,
! [A: set_Ep7530996317732161742fm_a_o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ? [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X2 @ A )
& ! [Xa3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Xa3 @ A )
=> ( ( ord_le3934200179093585166fm_a_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_838_finite__has__maximal,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_839_maximal_H__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ X3 @ S ) ) ) ) ) ).
% maximal'_def
thf(fact_840_Extend__subset_H,axiom,
! [S: set_Epistemic_fm_a,A: epistemic_fm_a > $o] :
( ( ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) )
!= bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ).
% Extend_subset'
thf(fact_841_set__rec,axiom,
( set_a2
= ( rec_list_set_a_a @ bot_bot_set_a
@ ^ [X3: a,Uu2: list_a] : ( insert_a @ X3 ) ) ) ).
% set_rec
thf(fact_842_set__rec,axiom,
( set_Epistemic_fm_a2
= ( rec_li8401250755714504918c_fm_a @ bot_bo3626323581529592678c_fm_a
@ ^ [X3: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( insert7817948963269374698c_fm_a @ X3 ) ) ) ).
% set_rec
thf(fact_843_subset__singleton__iff__Uniq,axiom,
! [A: set_a] :
( ( ? [A4: a] : ( ord_less_eq_set_a @ A @ ( insert_a @ A4 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_844_subset__singleton__iff__Uniq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( ? [A4: set_Epistemic_fm_a] : ( ord_le7112219575281605754c_fm_a @ A @ ( insert7129737658439086282c_fm_a @ A4 @ bot_bo1868452413526482246c_fm_a ) ) )
= ( uniq_s5323624820373029887c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X3 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_845_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_a] :
( ( ? [A4: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ A4 @ bot_bo3626323581529592678c_fm_a ) ) )
= ( uniq_Epistemic_fm_a
@ ^ [X3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X3 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_846_mono__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,S6: set_Pr8948481099588399239c_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ R2 @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ ( field_Epistemic_fm_a @ S6 ) ) ) ).
% mono_Field
thf(fact_847_Extend__bound,axiom,
! [N: epistemic_fm_a,A: 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 @ A ) @ S @ N ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ).
% Extend_bound
thf(fact_848_extend__subset,axiom,
! [N: epistemic_fm_a,S: set_Epistemic_fm_a,A: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ N ) ) ) ).
% extend_subset
thf(fact_849_consistent__extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( episte2285483198712856226tent_a @ A @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ N ) ) ) ).
% consistent_extend
thf(fact_850_Epistemic__Logic_Oextend__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ N )
= ( wellor7234309849078405374c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ S @ ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S ) @ ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) @ N ) ) ).
% Epistemic_Logic.extend_def
thf(fact_851_extend__under,axiom,
! [M5: epistemic_fm_a,N: epistemic_fm_a,A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ M5 @ ( order_1815523470270482687c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ M5 ) @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ N ) ) ) ).
% extend_under
thf(fact_852_extend__underS,axiom,
! [M5: epistemic_fm_a,N: epistemic_fm_a,A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ M5 @ ( order_1288088311422721504c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ M5 ) @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S @ N ) ) ) ).
% extend_underS
thf(fact_853_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_854_MCS__axioms,axiom,
! [A: epistemic_fm_a > $o] : ( maxima1924290459274269190c_fm_a @ ( episte2285483198712856226tent_a @ A ) ) ).
% MCS_axioms
thf(fact_855_MCS__def,axiom,
( maxima1924290459274269190c_fm_a
= ( ^ [Consistent2: set_Epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a )
& ! [S2: set_Epistemic_fm_a,S7: set_Epistemic_fm_a] :
( ( Consistent2 @ S2 )
=> ( ( ord_le3275665582123262618c_fm_a @ S7 @ S2 )
=> ( Consistent2 @ S7 ) ) )
& ! [S2: set_Epistemic_fm_a] :
( ~ ( Consistent2 @ S2 )
=> ? [S7: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S7 @ S2 )
& ( finite3304564945125563331c_fm_a @ S7 )
& ~ ( Consistent2 @ S7 ) ) ) ) ) ) ).
% MCS_def
thf(fact_856_MCS_Ointro,axiom,
! [Consistent: set_Epistemic_fm_a > $o] :
( ~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a )
=> ( ! [S8: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( Consistent @ S8 )
=> ( ( ord_le3275665582123262618c_fm_a @ S5 @ S8 )
=> ( Consistent @ S5 ) ) )
=> ( ! [S8: set_Epistemic_fm_a] :
( ~ ( Consistent @ S8 )
=> ? [S9: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S9 @ S8 )
& ( finite3304564945125563331c_fm_a @ S9 )
& ~ ( Consistent @ S9 ) ) )
=> ( maxima1924290459274269190c_fm_a @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_857_underS__subset__under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_1288088311422721504c_fm_a @ R2 @ A2 ) @ ( order_1815523470270482687c_fm_a @ R2 @ A2 ) ) ).
% underS_subset_under
thf(fact_858_under__incr2,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,R3: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ R2 @ R3 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1815523470270482687c_fm_a @ R2 @ A2 ) @ ( order_1815523470270482687c_fm_a @ R3 @ A2 ) ) ) ).
% under_incr2
thf(fact_859_underS__incr2,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,R3: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ R2 @ R3 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1288088311422721504c_fm_a @ R2 @ A2 ) @ ( order_1288088311422721504c_fm_a @ R3 @ A2 ) ) ) ).
% underS_incr2
thf(fact_860_under__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_1815523470270482687c_fm_a @ R2 @ A2 ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% under_Field
thf(fact_861_Order__Relation_OunderS__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_1288088311422721504c_fm_a @ R2 @ A2 ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% Order_Relation.underS_Field
thf(fact_862_adm__woL__extendL,axiom,
wellor8531372491315211020c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) ).
% adm_woL_extendL
thf(fact_863_MCS__Lim__Ord_OExtend__bound,axiom,
! [R2: set_Product_prod_a_a,Consistent: set_a > $o,N: a,S: set_a] :
( ( maxima3603147619815099129_Ord_a @ R2 @ Consistent )
=> ( ( member_a @ N @ ( field_a @ R2 ) )
=> ( ord_less_eq_set_a @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ N ) @ ( maxima2486273825106742127tend_a @ R2 @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_864_MCS__Lim__Ord_OExtend__bound,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,Consistent: set_se5208064806568342746c_fm_a > $o,N: set_Epistemic_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( maxima8505690773950114933c_fm_a @ R2 @ Consistent )
=> ( ( member536094252920883875c_fm_a @ N @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ N ) @ ( maxima2136919731711262699c_fm_a @ R2 @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_865_MCS__Lim__Ord_OExtend__bound,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,N: epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ N ) @ ( maxima2580775624958445067c_fm_a @ R2 @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_866_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_867_wo__rel_Ofinite__underS__bound,axiom,
! [R2: set_Product_prod_a_a,X6: set_a,N: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( finite_finite_a @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( order_underS_a @ R2 @ N ) )
=> ( ( X6 != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ X6 )
& ! [Xa3: a] :
( ( member_a @ Xa3 @ X6 )
=> ( member_a @ Xa3 @ ( order_under_a @ R2 @ X2 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_868_wo__rel_Ofinite__underS__bound,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,X6: set_se5208064806568342746c_fm_a,N: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( finite7364866884270929315c_fm_a @ X6 )
=> ( ( ord_le7112219575281605754c_fm_a @ X6 @ ( order_7871341057202914240c_fm_a @ R2 @ N ) )
=> ( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
& ! [Xa3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa3 @ X6 )
=> ( member536094252920883875c_fm_a @ Xa3 @ ( order_4320674923736412895c_fm_a @ R2 @ X2 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_869_wo__rel_Ofinite__underS__bound,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,X6: set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( finite3304564945125563331c_fm_a @ X6 )
=> ( ( ord_le3275665582123262618c_fm_a @ X6 @ ( order_1288088311422721504c_fm_a @ R2 @ N ) )
=> ( ( X6 != bot_bo3626323581529592678c_fm_a )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ X6 )
& ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ X6 )
=> ( member6642669571620171971c_fm_a @ Xa3 @ ( order_1815523470270482687c_fm_a @ R2 @ X2 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_870_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S )
=> ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_871_wo__rel__r,axiom,
bNF_We8255954830765735686c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% wo_rel_r
thf(fact_872_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 )
=> ? [S5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S5 @ S )
& ( finite3304564945125563331c_fm_a @ S5 )
& ~ ( Consistent @ S5 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_873_MCS__Lim__Ord_Oextend__subset,axiom,
! [R2: set_Product_prod_a_a,Consistent: set_a > $o,N: a,S: set_a] :
( ( maxima3603147619815099129_Ord_a @ R2 @ Consistent )
=> ( ( member_a @ N @ ( field_a @ R2 ) )
=> ( ord_less_eq_set_a @ S @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_874_MCS__Lim__Ord_Oextend__subset,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,Consistent: set_se5208064806568342746c_fm_a > $o,N: set_Epistemic_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( maxima8505690773950114933c_fm_a @ R2 @ Consistent )
=> ( ( member536094252920883875c_fm_a @ N @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ord_le7112219575281605754c_fm_a @ S @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_875_MCS__Lim__Ord_Oextend__subset,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,N: epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ S @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_876_MCS__Lim__Ord_Oextend__underS,axiom,
! [R2: set_Product_prod_a_a,Consistent: set_a > $o,M5: a,N: a,S: set_a] :
( ( maxima3603147619815099129_Ord_a @ R2 @ Consistent )
=> ( ( member_a @ M5 @ ( order_underS_a @ R2 @ N ) )
=> ( ord_less_eq_set_a @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_877_MCS__Lim__Ord_Oextend__underS,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,Consistent: set_se5208064806568342746c_fm_a > $o,M5: set_Epistemic_fm_a,N: set_Epistemic_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( maxima8505690773950114933c_fm_a @ R2 @ Consistent )
=> ( ( member536094252920883875c_fm_a @ M5 @ ( order_7871341057202914240c_fm_a @ R2 @ N ) )
=> ( ord_le7112219575281605754c_fm_a @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_878_MCS__Lim__Ord_Oextend__underS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,M5: epistemic_fm_a,N: epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( member6642669571620171971c_fm_a @ M5 @ ( order_1288088311422721504c_fm_a @ R2 @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_879_MCS__Lim__Ord_Oextend__under,axiom,
! [R2: set_Product_prod_a_a,Consistent: set_a > $o,M5: a,N: a,S: set_a] :
( ( maxima3603147619815099129_Ord_a @ R2 @ Consistent )
=> ( ( member_a @ M5 @ ( order_under_a @ R2 @ N ) )
=> ( ord_less_eq_set_a @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima2937105246822620047tend_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_880_MCS__Lim__Ord_Oextend__under,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,Consistent: set_se5208064806568342746c_fm_a > $o,M5: set_Epistemic_fm_a,N: set_Epistemic_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( maxima8505690773950114933c_fm_a @ R2 @ Consistent )
=> ( ( member536094252920883875c_fm_a @ M5 @ ( order_4320674923736412895c_fm_a @ R2 @ N ) )
=> ( ord_le7112219575281605754c_fm_a @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima2942051681900060683c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_881_MCS__Lim__Ord_Oextend__under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,M5: epistemic_fm_a,N: epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( member6642669571620171971c_fm_a @ M5 @ ( order_1815523470270482687c_fm_a @ R2 @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ M5 ) @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_882_MCS__Lim__Ord__axioms,axiom,
! [A: epistemic_fm_a > $o] : ( maxima8116622252184345237c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) ) ).
% MCS_Lim_Ord_axioms
thf(fact_883_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ? [X_1: epistemic_fm_a] : ( bNF_We541765430379730896c_fm_a @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_884_wo__rel_OunderS__suc,axiom,
! [R2: set_Product_prod_a_a,B: a,A: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ B @ ( order_underS_a @ R2 @ ( bNF_We6154283375207884895_suc_a @ R2 @ A ) ) )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( member_a @ B @ ( order_under_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_885_wo__rel_OunderS__suc,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( member536094252920883875c_fm_a @ B @ ( order_7871341057202914240c_fm_a @ R2 @ ( bNF_We1342240761504683227c_fm_a @ R2 @ A ) ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( member536094252920883875c_fm_a @ B @ ( order_4320674923736412895c_fm_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_886_wo__rel_OunderS__suc,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( member6642669571620171971c_fm_a @ B @ ( order_1288088311422721504c_fm_a @ R2 @ ( bNF_We6750397625931619067c_fm_a @ R2 @ A ) ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ( member6642669571620171971c_fm_a @ B @ ( order_1815523470270482687c_fm_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_887_Sup__fin_Osubset,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ( sup_su1367922730591523534c_fm_a @ ( lattic3176627891459829297c_fm_a @ B4 ) @ ( lattic3176627891459829297c_fm_a @ A ) )
= ( lattic3176627891459829297c_fm_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_888_Sup__fin_Osubset,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B4 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B4 ) @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_889_Inf__fin_Osubset__imp,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( finite5407549452671652090fm_a_o @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ B4 ) @ ( lattic8370817070757793286fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_890_Inf__fin_Osubset__imp,axiom,
! [A: set_a_a_o,B4: set_a_a_o] :
( ( ord_le7145157750676031086_a_a_o @ A @ B4 )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( finite_finite_a_a_o @ B4 )
=> ( ord_less_eq_a_a_o @ ( lattic6163869305583905163_a_a_o @ B4 ) @ ( lattic6163869305583905163_a_a_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_891_Inf__fin_Osubset__imp,axiom,
! [A: set_Ep7530996317732161742fm_a_o,B4: set_Ep7530996317732161742fm_a_o] :
( ( ord_le4099779637921079406fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( finite3930751176993980823fm_a_o @ B4 )
=> ( ord_le3934200179093585166fm_a_o @ ( lattic2865504167514852235fm_a_o @ B4 ) @ ( lattic2865504167514852235fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_892_Inf__fin_Osubset__imp,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ B4 ) @ ( lattic7927669795667837335c_fm_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_893_Inf__fin__le__Sup__fin,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ A ) @ ( lattic18181228415053292fm_a_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_894_Inf__fin__le__Sup__fin,axiom,
! [A: set_a_a_o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ord_less_eq_a_a_o @ ( lattic6163869305583905163_a_a_o @ A ) @ ( lattic8845106930119437093_a_a_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_895_Inf__fin__le__Sup__fin,axiom,
! [A: set_Ep7530996317732161742fm_a_o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ord_le3934200179093585166fm_a_o @ ( lattic2865504167514852235fm_a_o @ A ) @ ( lattic1265468659591431973fm_a_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_896_Inf__fin__le__Sup__fin,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A ) @ ( lattic3176627891459829297c_fm_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_897_Sup__fin_OcoboundedI,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( lattic18181228415053292fm_a_o @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_898_Sup__fin_OcoboundedI,axiom,
! [A: set_a_a_o,A2: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A2 @ A )
=> ( ord_less_eq_a_a_o @ A2 @ ( lattic8845106930119437093_a_a_o @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_899_Sup__fin_OcoboundedI,axiom,
! [A: set_Ep7530996317732161742fm_a_o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( member8030264063065380503fm_a_o @ A2 @ A )
=> ( ord_le3934200179093585166fm_a_o @ A2 @ ( lattic1265468659591431973fm_a_o @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_900_Sup__fin_OcoboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( lattic3176627891459829297c_fm_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_901_Inf__fin_OcoboundedI,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_902_Inf__fin_OcoboundedI,axiom,
! [A: set_a_a_o,A2: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A2 @ A )
=> ( ord_less_eq_a_a_o @ ( lattic6163869305583905163_a_a_o @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_903_Inf__fin_OcoboundedI,axiom,
! [A: set_Ep7530996317732161742fm_a_o,A2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( member8030264063065380503fm_a_o @ A2 @ A )
=> ( ord_le3934200179093585166fm_a_o @ ( lattic2865504167514852235fm_a_o @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_904_Inf__fin_OcoboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_905_Sup__fin_Obounded__iff,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X )
= ( ! [X3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_906_Sup__fin_Obounded__iff,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( ord_less_eq_a_a_o @ ( lattic8845106930119437093_a_a_o @ A ) @ X )
= ( ! [X3: a > a > $o] :
( ( member_a_a_o @ X3 @ A )
=> ( ord_less_eq_a_a_o @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_907_Sup__fin_Obounded__iff,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( ord_le3934200179093585166fm_a_o @ ( lattic1265468659591431973fm_a_o @ A ) @ X )
= ( ! [X3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X3 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_908_Sup__fin_Obounded__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_909_Inf__fin_Obounded__iff,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ X @ ( lattic8370817070757793286fm_a_o @ A ) )
= ( ! [X3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_910_Inf__fin_Obounded__iff,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( ord_less_eq_a_a_o @ X @ ( lattic6163869305583905163_a_a_o @ A ) )
= ( ! [X3: a > a > $o] :
( ( member_a_a_o @ X3 @ A )
=> ( ord_less_eq_a_a_o @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_911_Inf__fin_Obounded__iff,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( ord_le3934200179093585166fm_a_o @ X @ ( lattic2865504167514852235fm_a_o @ A ) )
= ( ! [X3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X3 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_912_Inf__fin_Obounded__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ ( lattic7927669795667837335c_fm_a @ A ) )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_913_Sup__fin_OboundedI,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ! [A6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A6 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A6 @ X ) )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_914_Sup__fin_OboundedI,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ! [A6: a > a > $o] :
( ( member_a_a_o @ A6 @ A )
=> ( ord_less_eq_a_a_o @ A6 @ X ) )
=> ( ord_less_eq_a_a_o @ ( lattic8845106930119437093_a_a_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_915_Sup__fin_OboundedI,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ! [A6: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ A6 @ A )
=> ( ord_le3934200179093585166fm_a_o @ A6 @ X ) )
=> ( ord_le3934200179093585166fm_a_o @ ( lattic1265468659591431973fm_a_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_916_Sup__fin_OboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A6 @ X ) )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_917_Sup__fin_OboundedE,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X )
=> ! [A7: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A7 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_918_Sup__fin_OboundedE,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( ord_less_eq_a_a_o @ ( lattic8845106930119437093_a_a_o @ A ) @ X )
=> ! [A7: a > a > $o] :
( ( member_a_a_o @ A7 @ A )
=> ( ord_less_eq_a_a_o @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_919_Sup__fin_OboundedE,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( ord_le3934200179093585166fm_a_o @ ( lattic1265468659591431973fm_a_o @ A ) @ X )
=> ! [A7: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ A7 @ A )
=> ( ord_le3934200179093585166fm_a_o @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_920_Sup__fin_OboundedE,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X )
=> ! [A7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A7 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_921_Inf__fin_OboundedI,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ! [A6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A6 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ A6 ) )
=> ( ord_le4043730696559282883fm_a_o @ X @ ( lattic8370817070757793286fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_922_Inf__fin_OboundedI,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ! [A6: a > a > $o] :
( ( member_a_a_o @ A6 @ A )
=> ( ord_less_eq_a_a_o @ X @ A6 ) )
=> ( ord_less_eq_a_a_o @ X @ ( lattic6163869305583905163_a_a_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_923_Inf__fin_OboundedI,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ! [A6: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ A6 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X @ A6 ) )
=> ( ord_le3934200179093585166fm_a_o @ X @ ( lattic2865504167514852235fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_924_Inf__fin_OboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ A6 ) )
=> ( ord_le3275665582123262618c_fm_a @ X @ ( lattic7927669795667837335c_fm_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_925_Inf__fin_OboundedE,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ X @ ( lattic8370817070757793286fm_a_o @ A ) )
=> ! [A7: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A7 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_926_Inf__fin_OboundedE,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( ord_less_eq_a_a_o @ X @ ( lattic6163869305583905163_a_a_o @ A ) )
=> ! [A7: a > a > $o] :
( ( member_a_a_o @ A7 @ A )
=> ( ord_less_eq_a_a_o @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_927_Inf__fin_OboundedE,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ( finite3930751176993980823fm_a_o @ A )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( ord_le3934200179093585166fm_a_o @ X @ ( lattic2865504167514852235fm_a_o @ A ) )
=> ! [A7: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ A7 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_928_Inf__fin_OboundedE,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ ( lattic7927669795667837335c_fm_a @ A ) )
=> ! [A7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A7 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_929_Sup__fin_Osubset__imp,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( finite5407549452671652090fm_a_o @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ ( lattic18181228415053292fm_a_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_930_Sup__fin_Osubset__imp,axiom,
! [A: set_a_a_o,B4: set_a_a_o] :
( ( ord_le7145157750676031086_a_a_o @ A @ B4 )
=> ( ( A != bot_bot_set_a_a_o )
=> ( ( finite_finite_a_a_o @ B4 )
=> ( ord_less_eq_a_a_o @ ( lattic8845106930119437093_a_a_o @ A ) @ ( lattic8845106930119437093_a_a_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_931_Sup__fin_Osubset__imp,axiom,
! [A: set_Ep7530996317732161742fm_a_o,B4: set_Ep7530996317732161742fm_a_o] :
( ( ord_le4099779637921079406fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4126678323887997754fm_a_o )
=> ( ( finite3930751176993980823fm_a_o @ B4 )
=> ( ord_le3934200179093585166fm_a_o @ ( lattic1265468659591431973fm_a_o @ A ) @ ( lattic1265468659591431973fm_a_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_932_Sup__fin_Osubset__imp,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ ( lattic3176627891459829297c_fm_a @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_933_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( bNF_We541765430379730896c_fm_a @ R2 @ B4 @ ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_934_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_AboveS_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( ( member_a @ B @ B4 )
=> ( member_a @ B @ ( order_underS_a @ R2 @ ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_935_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_6674880131641743173c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( ( member536094252920883875c_fm_a @ B @ B4 )
=> ( member536094252920883875c_fm_a @ B @ ( order_7871341057202914240c_fm_a @ R2 @ ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_936_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ( member6642669571620171971c_fm_a @ B @ B4 )
=> ( member6642669571620171971c_fm_a @ B @ ( order_1288088311422721504c_fm_a @ R2 @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_937_wo__rel_Osuc__inField,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_AboveS_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) @ ( field_a @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_938_wo__rel_Osuc__inField,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_6674880131641743173c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) @ ( field_3462791820427926727c_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_939_wo__rel_Osuc__inField,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) @ ( field_Epistemic_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_940_wo__rel_Osuc__AboveS,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_AboveS_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) @ ( order_AboveS_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_941_wo__rel_Osuc__AboveS,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_6674880131641743173c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) @ ( order_6674880131641743173c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_942_wo__rel_Osuc__AboveS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) @ ( order_225095655763363173c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_943_AboveS__decr,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_225095655763363173c_fm_a @ R2 @ B4 ) @ ( order_225095655763363173c_fm_a @ R2 @ A ) ) ) ).
% AboveS_decr
thf(fact_944_AboveS__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_225095655763363173c_fm_a @ R2 @ A ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% AboveS_Field
thf(fact_945_wo__rel_Ominim__in,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_946_wo__rel_Ominim__in,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_947_wo__rel_Ominim__in,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_948_wo__rel_Ominim__inField,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) @ ( field_a @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_949_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) @ ( field_3462791820427926727c_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_950_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) @ ( field_Epistemic_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_951_wo__rel_Osuc__greater,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_AboveS_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( ( member_a @ B @ B4 )
=> ( ( ( bNF_We6154283375207884895_suc_a @ R2 @ B4 )
!= B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_952_wo__rel_Osuc__greater,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_6674880131641743173c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( ( member536094252920883875c_fm_a @ B @ B4 )
=> ( ( ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 )
!= B )
& ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B @ ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_953_wo__rel_Osuc__greater,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ( member6642669571620171971c_fm_a @ B @ B4 )
=> ( ( ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 )
!= B )
& ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_954_wo__rel_OunderS__supr,axiom,
! [R2: set_Product_prod_a_a,B: a,A: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ B @ ( order_underS_a @ R2 @ ( bNF_We1039243157376381174supr_a @ R2 @ A ) ) )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( member_a @ B @ ( order_underS_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_supr
thf(fact_955_wo__rel_OunderS__supr,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( member536094252920883875c_fm_a @ B @ ( order_7871341057202914240c_fm_a @ R2 @ ( bNF_We2920625255559816434c_fm_a @ R2 @ A ) ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( member536094252920883875c_fm_a @ B @ ( order_7871341057202914240c_fm_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_supr
thf(fact_956_wo__rel_OunderS__supr,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( member6642669571620171971c_fm_a @ B @ ( order_1288088311422721504c_fm_a @ R2 @ ( bNF_We3666122547472320274c_fm_a @ R2 @ A ) ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ( member6642669571620171971c_fm_a @ B @ ( order_1288088311422721504c_fm_a @ R2 @ X2 ) ) ) ) ) ) ).
% wo_rel.underS_supr
thf(fact_957_wo__rel_Ominim__UnderS__underS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( order_2700047599516024256c_fm_a @ R2 @ A )
= ( order_1288088311422721504c_fm_a @ R2 @ ( bNF_We6247850539267499238c_fm_a @ R2 @ A ) ) ) ) ) ) ).
% wo_rel.minim_UnderS_underS
thf(fact_958_wo__rel_Ominim__Under__under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( order_2859517270982914335c_fm_a @ R2 @ A )
= ( order_1815523470270482687c_fm_a @ R2 @ ( bNF_We6247850539267499238c_fm_a @ R2 @ A ) ) ) ) ) ) ).
% wo_rel.minim_Under_under
thf(fact_959_UnderS__subset__Under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_2700047599516024256c_fm_a @ R2 @ A ) @ ( order_2859517270982914335c_fm_a @ R2 @ A ) ) ).
% UnderS_subset_Under
thf(fact_960_wo__rel_Oequals__supr,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( field_a @ R2 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B6 @ A2 ) @ R2 ) )
=> ( ! [A8: a] :
( ( member_a @ A8 @ ( field_a @ R2 ) )
=> ( ! [B7: a] :
( ( member_a @ B7 @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B7 @ A8 ) @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_supr
thf(fact_961_wo__rel_Oequals__supr,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B6 @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B6 @ A2 ) @ R2 ) )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A8 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B7 @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B7 @ A8 ) @ R2 ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_supr
thf(fact_962_wo__rel_Oequals__supr,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B6 @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B6 @ A2 ) @ R2 ) )
=> ( ! [A8: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A8 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B7: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B7 @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B7 @ A8 ) @ R2 ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_supr
thf(fact_963_wo__rel_Osupr__least,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( field_a @ R2 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B6 @ A2 ) @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_least
thf(fact_964_wo__rel_Osupr__least,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B6 @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B6 @ A2 ) @ R2 ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_least
thf(fact_965_wo__rel_Osupr__least,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B6 @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B6 @ A2 ) @ R2 ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_least
thf(fact_966_Under__decr,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) @ ( order_2859517270982914335c_fm_a @ R2 @ A ) ) ) ).
% Under_decr
thf(fact_967_underS__def,axiom,
( order_7871341057202914240c_fm_a
= ( ^ [R4: set_Pr4537461906582779015c_fm_a,A4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] :
( ( B2 != A4 )
& ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B2 @ A4 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_968_underS__def,axiom,
( order_1288088311422721504c_fm_a
= ( ^ [R4: set_Pr8948481099588399239c_fm_a,A4: epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [B2: epistemic_fm_a] :
( ( B2 != A4 )
& ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B2 @ A4 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_969_under__def,axiom,
( order_4320674923736412895c_fm_a
= ( ^ [R4: set_Pr4537461906582779015c_fm_a,A4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] : ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B2 @ A4 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_970_under__def,axiom,
( order_1815523470270482687c_fm_a
= ( ^ [R4: set_Pr8948481099588399239c_fm_a,A4: epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [B2: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B2 @ A4 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_971_UnderS__decr,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_2700047599516024256c_fm_a @ R2 @ B4 ) @ ( order_2700047599516024256c_fm_a @ R2 @ A ) ) ) ).
% UnderS_decr
thf(fact_972_pred__subset__eq2,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X3: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R )
@ ^ [X3: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ S ) )
= ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_973_pred__subset__eq2,axiom,
! [R: set_Pr8948481099588399239c_fm_a,S: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3934200179093585166fm_a_o
@ ^ [X3: epistemic_fm_a,Y4: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X3 @ Y4 ) @ R )
@ ^ [X3: epistemic_fm_a,Y4: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X3 @ Y4 ) @ S ) )
= ( ord_le8222437384497874983c_fm_a @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_974_Under__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_2859517270982914335c_fm_a @ R2 @ A ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% Under_Field
thf(fact_975_UnderS__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_2700047599516024256c_fm_a @ R2 @ A ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% UnderS_Field
thf(fact_976_wo__rel_Osuc__least,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( field_a @ R2 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B6 @ A2 ) @ R2 ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_977_wo__rel_Osuc__least,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B6 @ A2 ) @ R2 ) ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_978_wo__rel_Osuc__least,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B6 @ A2 ) @ R2 ) ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_979_wo__rel_Oequals__suc,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( field_a @ R2 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B6 @ A2 ) @ R2 ) ) )
=> ( ! [A8: a] :
( ( member_a @ A8 @ ( field_a @ R2 ) )
=> ( ! [B7: a] :
( ( member_a @ B7 @ B4 )
=> ( ( A8 != B7 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B7 @ A8 ) @ R2 ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_980_wo__rel_Oequals__suc,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B6 @ A2 ) @ R2 ) ) )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A8 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ! [B7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B7 @ B4 )
=> ( ( A8 != B7 )
& ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B7 @ A8 ) @ R2 ) ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_981_wo__rel_Oequals__suc,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B6 @ B4 )
=> ( ( A2 != B6 )
& ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B6 @ A2 ) @ R2 ) ) )
=> ( ! [A8: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A8 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ! [B7: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B7 @ B4 )
=> ( ( A8 != B7 )
& ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B7 @ A8 ) @ R2 ) ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ A8 ) @ R2 ) ) )
=> ( A2
= ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_982_wo__rel_OTrans__Under__regressive,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( order_2859517270982914335c_fm_a @ R2 @ ( order_2859517270982914335c_fm_a @ R2 @ A ) ) @ ( order_2859517270982914335c_fm_a @ R2 @ A ) ) ) ) ) ).
% wo_rel.Trans_Under_regressive
thf(fact_983_wo__rel_Oequals__minim__Under,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ B4 )
=> ( ( member_a @ A2 @ ( order_Under_a @ R2 @ B4 ) )
=> ( A2
= ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_984_wo__rel_Oequals__minim__Under,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ B4 )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( order_2458666036161633023c_fm_a @ R2 @ B4 ) )
=> ( A2
= ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_985_wo__rel_Oequals__minim__Under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ B4 )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) )
=> ( A2
= ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_986_wo__rel_Ominim__least,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_987_wo__rel_Ominim__least,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ B @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_988_wo__rel_Ominim__least,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ B @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_989_wo__rel_Oequals__minim,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ B4 )
=> ( ! [B6: a] :
( ( member_a @ B6 @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B6 ) @ R2 ) )
=> ( A2
= ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_990_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ B4 )
=> ( ! [B6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B6 @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ B6 ) @ R2 ) )
=> ( A2
= ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_991_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ B4 )
=> ( ! [B6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B6 @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ B6 ) @ R2 ) )
=> ( A2
= ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_992_UnderS__AboveS__Galois,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ C2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_225095655763363173c_fm_a @ R2 @ C2 ) )
= ( ord_le3275665582123262618c_fm_a @ C2 @ ( order_2700047599516024256c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% UnderS_AboveS_Galois
thf(fact_993_subset__AboveS__UnderS,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_225095655763363173c_fm_a @ R2 @ ( order_2700047599516024256c_fm_a @ R2 @ B4 ) ) ) ) ).
% subset_AboveS_UnderS
thf(fact_994_subset__UnderS__AboveS,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_2700047599516024256c_fm_a @ R2 @ ( order_225095655763363173c_fm_a @ R2 @ B4 ) ) ) ) ).
% subset_UnderS_AboveS
thf(fact_995_wo__rel_Ominim__iff__In__Under,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( ( A2
= ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) )
= ( ( member_a @ A2 @ B4 )
& ( member_a @ A2 @ ( order_Under_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_996_wo__rel_Ominim__iff__In__Under,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( A2
= ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) )
= ( ( member536094252920883875c_fm_a @ A2 @ B4 )
& ( member536094252920883875c_fm_a @ A2 @ ( order_2458666036161633023c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_997_wo__rel_Ominim__iff__In__Under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( ( A2
= ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) )
= ( ( member6642669571620171971c_fm_a @ A2 @ B4 )
& ( member6642669571620171971c_fm_a @ A2 @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_998_wo__rel_Ominim__Under,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B4 ) @ ( order_Under_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_999_wo__rel_Ominim__Under,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We5423793456056005318c_fm_a @ R2 @ B4 ) @ ( order_2458666036161633023c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_1000_wo__rel_Ominim__Under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We6247850539267499238c_fm_a @ R2 @ B4 ) @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_1001_wo__rel_Ocases__Total,axiom,
! [R2: set_Product_prod_a_a,A2: a,B: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1002_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a,B: epistemic_fm_a,Phi: epistemic_fm_a > epistemic_fm_a > $o] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ A2 @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1003_wo__rel_Oequals__suc__AboveS,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( order_AboveS_a @ R2 @ B4 ) )
=> ( ! [A8: a] :
( ( member_a @ A8 @ ( order_AboveS_a @ R2 @ B4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We6154283375207884895_suc_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1004_wo__rel_Oequals__suc__AboveS,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( order_6674880131641743173c_fm_a @ R2 @ B4 ) )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A8 @ ( order_6674880131641743173c_fm_a @ R2 @ B4 ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We1342240761504683227c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1005_wo__rel_Oequals__suc__AboveS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( order_225095655763363173c_fm_a @ R2 @ B4 ) )
=> ( ! [A8: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A8 @ ( order_225095655763363173c_fm_a @ R2 @ B4 ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1006_wo__rel_Osuc__above__AboveS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ( order_225095655763363173c_fm_a @ R2 @ B4 )
= ( order_1596567918915635226c_fm_a @ R2 @ ( bNF_We6750397625931619067c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.suc_above_AboveS
thf(fact_1007_wo__rel_Oin__underS__supr,axiom,
! [R2: set_Product_prod_a_a,J: a,I3: a,A: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ J @ ( order_underS_a @ R2 @ I3 ) )
=> ( ( member_a @ I3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( field_a @ R2 ) )
=> ( ( ( order_Above_a @ R2 @ A )
!= bot_bot_set_a )
=> ( member_a @ J @ ( order_underS_a @ R2 @ ( bNF_We1039243157376381174supr_a @ R2 @ A ) ) ) ) ) ) ) ) ).
% wo_rel.in_underS_supr
thf(fact_1008_wo__rel_Oin__underS__supr,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,J: set_Epistemic_fm_a,I3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( member536094252920883875c_fm_a @ J @ ( order_7871341057202914240c_fm_a @ R2 @ I3 ) )
=> ( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_606786824185998874c_fm_a @ R2 @ A )
!= bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ J @ ( order_7871341057202914240c_fm_a @ R2 @ ( bNF_We2920625255559816434c_fm_a @ R2 @ A ) ) ) ) ) ) ) ) ).
% wo_rel.in_underS_supr
thf(fact_1009_wo__rel_Oin__underS__supr,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,J: epistemic_fm_a,I3: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( member6642669571620171971c_fm_a @ J @ ( order_1288088311422721504c_fm_a @ R2 @ I3 ) )
=> ( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_2640561719628066874c_fm_a @ R2 @ A )
!= bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ J @ ( order_1288088311422721504c_fm_a @ R2 @ ( bNF_We3666122547472320274c_fm_a @ R2 @ A ) ) ) ) ) ) ) ) ).
% wo_rel.in_underS_supr
thf(fact_1010_wo__rel_Osupr__greater,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_Above_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( ( member_a @ B @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_greater
thf(fact_1011_wo__rel_Osupr__greater,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_606786824185998874c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( ( member536094252920883875c_fm_a @ B @ B4 )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ B @ ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_greater
thf(fact_1012_wo__rel_Osupr__greater,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_2640561719628066874c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ( member6642669571620171971c_fm_a @ B @ B4 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ B @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.supr_greater
thf(fact_1013_wo__rel_Osupr__inField,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_Above_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) @ ( field_a @ R2 ) ) ) ) ) ).
% wo_rel.supr_inField
thf(fact_1014_wo__rel_Osupr__inField,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_606786824185998874c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) @ ( field_3462791820427926727c_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.supr_inField
thf(fact_1015_wo__rel_Osupr__inField,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_2640561719628066874c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) @ ( field_Epistemic_fm_a @ R2 ) ) ) ) ) ).
% wo_rel.supr_inField
thf(fact_1016_Above__decr,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) @ ( order_2640561719628066874c_fm_a @ R2 @ A ) ) ) ).
% Above_decr
thf(fact_1017_above__def,axiom,
( order_2468795711760778746c_fm_a
= ( ^ [R4: set_Pr4537461906582779015c_fm_a,A4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] : ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A4 @ B2 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1018_above__def,axiom,
( order_1596567918915635226c_fm_a
= ( ^ [R4: set_Pr8948481099588399239c_fm_a,A4: epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [B2: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A4 @ B2 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1019_Above__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_2640561719628066874c_fm_a @ R2 @ A ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% Above_Field
thf(fact_1020_above__Field,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_1596567918915635226c_fm_a @ R2 @ A2 ) @ ( field_Epistemic_fm_a @ R2 ) ) ).
% above_Field
thf(fact_1021_above__incr2,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,R3: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a] :
( ( ord_le8222437384497874983c_fm_a @ R2 @ R3 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1596567918915635226c_fm_a @ R2 @ A2 ) @ ( order_1596567918915635226c_fm_a @ R3 @ A2 ) ) ) ).
% above_incr2
thf(fact_1022_AboveS__subset__Above,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( order_225095655763363173c_fm_a @ R2 @ A ) @ ( order_2640561719628066874c_fm_a @ R2 @ A ) ) ).
% AboveS_subset_Above
thf(fact_1023_wo__rel_Osupr__above__Above,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_2640561719628066874c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( ( order_2640561719628066874c_fm_a @ R2 @ B4 )
= ( order_1596567918915635226c_fm_a @ R2 @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.supr_above_Above
thf(fact_1024_subset__Under__Above,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_2859517270982914335c_fm_a @ R2 @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) ) ) ) ).
% subset_Under_Above
thf(fact_1025_subset__Above__Under,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_2640561719628066874c_fm_a @ R2 @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) ) ) ) ).
% subset_Above_Under
thf(fact_1026_Under__Above__Galois,axiom,
! [B4: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ C2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( order_2640561719628066874c_fm_a @ R2 @ C2 ) )
= ( ord_le3275665582123262618c_fm_a @ C2 @ ( order_2859517270982914335c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% Under_Above_Galois
thf(fact_1027_wo__rel_Osupr__least__Above,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( order_Above_a @ R2 @ B4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ).
% wo_rel.supr_least_Above
thf(fact_1028_wo__rel_Osupr__least__Above,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( order_606786824185998874c_fm_a @ R2 @ B4 ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ).
% wo_rel.supr_least_Above
thf(fact_1029_wo__rel_Osupr__least__Above,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) @ A2 ) @ R2 ) ) ) ) ).
% wo_rel.supr_least_Above
thf(fact_1030_wo__rel_Oequals__supr__Above,axiom,
! [R2: set_Product_prod_a_a,B4: set_a,A2: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( member_a @ A2 @ ( order_Above_a @ R2 @ B4 ) )
=> ( ! [A8: a] :
( ( member_a @ A8 @ ( order_Above_a @ R2 @ B4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_supr_Above
thf(fact_1031_wo__rel_Oequals__supr__Above,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( order_606786824185998874c_fm_a @ R2 @ B4 ) )
=> ( ! [A8: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A8 @ ( order_606786824185998874c_fm_a @ R2 @ B4 ) )
=> ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_supr_Above
thf(fact_1032_wo__rel_Oequals__supr__Above,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) )
=> ( ! [A8: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A8 @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ A8 ) @ R2 ) )
=> ( A2
= ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_supr_Above
thf(fact_1033_wo__rel_Osupr__Above,axiom,
! [R2: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R2 ) )
=> ( ( ( order_Above_a @ R2 @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We1039243157376381174supr_a @ R2 @ B4 ) @ ( order_Above_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.supr_Above
thf(fact_1034_wo__rel_Osupr__Above,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ( order_606786824185998874c_fm_a @ R2 @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( bNF_We2920625255559816434c_fm_a @ R2 @ B4 ) @ ( order_606786824185998874c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.supr_Above
thf(fact_1035_wo__rel_Osupr__Above,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ( order_2640561719628066874c_fm_a @ R2 @ B4 )
!= bot_bo3626323581529592678c_fm_a )
=> ( member6642669571620171971c_fm_a @ ( bNF_We3666122547472320274c_fm_a @ R2 @ B4 ) @ ( order_2640561719628066874c_fm_a @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.supr_Above
thf(fact_1036_underS__incl__iff,axiom,
! [R2: set_Product_prod_a_a,A2: a,B: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A2 @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A2 ) @ ( order_underS_a @ R2 @ B ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1037_underS__incl__iff,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( order_5511511401533808303c_fm_a @ ( field_3462791820427926727c_fm_a @ R2 ) @ R2 )
=> ( ( member536094252920883875c_fm_a @ A2 @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( member536094252920883875c_fm_a @ B @ ( field_3462791820427926727c_fm_a @ R2 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ ( order_7871341057202914240c_fm_a @ R2 @ A2 ) @ ( order_7871341057202914240c_fm_a @ R2 @ B ) )
= ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1038_underS__incl__iff,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( order_5410817246541083855c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ R2 )
=> ( ( member6642669571620171971c_fm_a @ A2 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( member6642669571620171971c_fm_a @ B @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( order_1288088311422721504c_fm_a @ R2 @ A2 ) @ ( order_1288088311422721504c_fm_a @ R2 @ B ) )
= ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1039_the__elem__def,axiom,
( the_elem_a
= ( ^ [X4: set_a] :
( the_a
@ ^ [X3: a] :
( X4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1040_the__elem__def,axiom,
( the_el2173195877760541071c_fm_a
= ( ^ [X4: set_Epistemic_fm_a] :
( the_Epistemic_fm_a
@ ^ [X3: epistemic_fm_a] :
( X4
= ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1041_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr8948481099588399239c_fm_a] :
( ( order_5410817246541083855c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ R2 )
=> ( ( order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ R2 )
= ( ! [A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( field_Epistemic_fm_a @ R2 ) )
=> ( ( A3 != bot_bo3626323581529592678c_fm_a )
=> ? [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ A3 )
=> ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X3 @ Y4 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1042_wo__rel_Oofilter__UnderS,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( order_7157697347518734512c_fm_a @ R2 @ ( order_2700047599516024256c_fm_a @ R2 @ A ) ) ) ) ).
% wo_rel.ofilter_UnderS
thf(fact_1043_wo__rel_Oofilter__Under,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
=> ( order_7157697347518734512c_fm_a @ R2 @ ( order_2859517270982914335c_fm_a @ R2 @ A ) ) ) ) ).
% wo_rel.ofilter_Under
thf(fact_1044_wo__rel_Oofilter__linord,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( order_7157697347518734512c_fm_a @ R2 @ A )
=> ( ( order_7157697347518734512c_fm_a @ R2 @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
| ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_1045_ofilter__def,axiom,
( order_7157697347518734512c_fm_a
= ( ^ [R4: set_Pr8948481099588399239c_fm_a,A3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ ( field_Epistemic_fm_a @ R4 ) )
& ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1815523470270482687c_fm_a @ R4 @ X3 ) @ A3 ) ) ) ) ) ).
% ofilter_def
thf(fact_1046_WELL,axiom,
order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% WELL
thf(fact_1047_wo__rel_Oofilter__def,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( order_7157697347518734512c_fm_a @ R2 @ A )
= ( ( ord_le3275665582123262618c_fm_a @ A @ ( field_Epistemic_fm_a @ R2 ) )
& ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1815523470270482687c_fm_a @ R2 @ X3 ) @ A ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_1048_under__mono,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,I3: epistemic_fm_a,J: epistemic_fm_a] :
( ( order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ R2 )
=> ( ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ I3 @ J ) @ R2 )
=> ( ord_le3275665582123262618c_fm_a @ ( order_1815523470270482687c_fm_a @ R2 @ I3 ) @ ( order_1815523470270482687c_fm_a @ R2 @ J ) ) ) ) ).
% under_mono
thf(fact_1049_MCS__Lim__Ord__def,axiom,
( maxima8116622252184345237c_fm_a
= ( ^ [R4: set_Pr8948481099588399239c_fm_a,Consistent2: set_Epistemic_fm_a > $o] :
( ( order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ R4 ) @ R4 )
& ( wellor5744531135217078139c_fm_a @ R4 )
& ! [S2: set_Epistemic_fm_a,S7: set_Epistemic_fm_a] :
( ( Consistent2 @ S2 )
=> ( ( ord_le3275665582123262618c_fm_a @ S7 @ S2 )
=> ( Consistent2 @ S7 ) ) )
& ! [S2: set_Epistemic_fm_a] :
( ~ ( Consistent2 @ S2 )
=> ? [S7: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S7 @ S2 )
& ( finite3304564945125563331c_fm_a @ S7 )
& ~ ( Consistent2 @ S7 ) ) ) ) ) ) ).
% MCS_Lim_Ord_def
thf(fact_1050_MCS__Lim__Ord_Ointro,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o] :
( ( order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ R2 ) @ R2 )
=> ( ( wellor5744531135217078139c_fm_a @ R2 )
=> ( ! [S8: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( Consistent @ S8 )
=> ( ( ord_le3275665582123262618c_fm_a @ S5 @ S8 )
=> ( Consistent @ S5 ) ) )
=> ( ! [S8: set_Epistemic_fm_a] :
( ~ ( Consistent @ S8 )
=> ? [S9: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S9 @ S8 )
& ( finite3304564945125563331c_fm_a @ S9 )
& ~ ( Consistent @ S9 ) ) )
=> ( maxima8116622252184345237c_fm_a @ R2 @ Consistent ) ) ) ) ) ).
% MCS_Lim_Ord.intro
thf(fact_1051_isLimOrd__r,axiom,
wellor5744531135217078139c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% isLimOrd_r
thf(fact_1052_infinite__Func,axiom,
! [A: set_Epistemic_fm_a,B1: a,B23: a,B4: set_a] :
( ~ ( finite3304564945125563331c_fm_a @ A )
=> ( ( ord_less_eq_set_a @ ( insert_a @ B1 @ ( insert_a @ B23 @ bot_bot_set_a ) ) @ B4 )
=> ( ( B1 != B23 )
=> ~ ( finite1016825363819911648fm_a_a @ ( bNF_We1622471989791679736fm_a_a @ A @ B4 ) ) ) ) ) ).
% infinite_Func
thf(fact_1053_infinite__Func,axiom,
! [A: set_Epistemic_fm_a,B1: epistemic_fm_a,B23: epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ~ ( finite3304564945125563331c_fm_a @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ B1 @ ( insert7817948963269374698c_fm_a @ B23 @ bot_bo3626323581529592678c_fm_a ) ) @ B4 )
=> ( ( B1 != B23 )
=> ~ ( finite8130236415186311548c_fm_a @ ( bNF_We4818972062080617492c_fm_a @ A @ B4 ) ) ) ) ) ).
% infinite_Func
thf(fact_1054_sup__Inf2__distrib,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B4 )
=> ( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( sup_su1367922730591523534c_fm_a @ ( lattic7927669795667837335c_fm_a @ A ) @ ( lattic7927669795667837335c_fm_a @ B4 ) )
= ( lattic7927669795667837335c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( Uu2
= ( sup_su1367922730591523534c_fm_a @ A4 @ B2 ) )
& ( member536094252920883875c_fm_a @ A4 @ A )
& ( member536094252920883875c_fm_a @ B2 @ B4 ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_1055_sup__Inf2__distrib,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B4 )
=> ( ( B4 != bot_bot_set_set_a )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic8209813465164889211_set_a @ B4 ) )
= ( lattic8209813465164889211_set_a
@ ( collect_set_a
@ ^ [Uu2: set_a] :
? [A4: set_a,B2: set_a] :
( ( Uu2
= ( sup_sup_set_a @ A4 @ B2 ) )
& ( member_set_a @ A4 @ A )
& ( member_set_a @ B2 @ B4 ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_1056_finite__Collect__bounded__ex,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 )
& ( Q4 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( Q4 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1057_finite__Collect__bounded__ex,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 )
& ( Q4 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: set_Epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( Q4 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1058_finite__Collect__bounded__ex,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 )
& ( Q4 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : ( Q4 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1059_finite__Collect__bounded__ex,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 )
& ( Q4 @ X3 @ Y4 ) ) ) )
= ( ! [Y4: epistemic_fm_a] :
( ( P4 @ Y4 )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] : ( Q4 @ X3 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1060_Collect__insert,axiom,
! [R: a > a,J1: a,J2: set_a] :
( ( collect_a
@ ^ [Uu2: a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ ( insert_a @ J1 @ J2 ) ) ) )
= ( insert_a @ ( R @ J1 )
@ ( collect_a
@ ^ [Uu2: a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1061_Collect__insert,axiom,
! [R: set_Epistemic_fm_a > a,J1: set_Epistemic_fm_a,J2: set_se5208064806568342746c_fm_a] :
( ( collect_a
@ ^ [Uu2: a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ ( insert7129737658439086282c_fm_a @ J1 @ J2 ) ) ) )
= ( insert_a @ ( R @ J1 )
@ ( collect_a
@ ^ [Uu2: a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1062_Collect__insert,axiom,
! [R: epistemic_fm_a > a,J1: epistemic_fm_a,J2: set_Epistemic_fm_a] :
( ( collect_a
@ ^ [Uu2: a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ ( insert7817948963269374698c_fm_a @ J1 @ J2 ) ) ) )
= ( insert_a @ ( R @ J1 )
@ ( collect_a
@ ^ [Uu2: a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1063_Collect__insert,axiom,
! [R: a > set_Epistemic_fm_a,J1: a,J2: set_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ ( insert_a @ J1 @ J2 ) ) ) )
= ( insert7129737658439086282c_fm_a @ ( R @ J1 )
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1064_Collect__insert,axiom,
! [R: set_Epistemic_fm_a > set_Epistemic_fm_a,J1: set_Epistemic_fm_a,J2: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ ( insert7129737658439086282c_fm_a @ J1 @ J2 ) ) ) )
= ( insert7129737658439086282c_fm_a @ ( R @ J1 )
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1065_Collect__insert,axiom,
! [R: epistemic_fm_a > set_Epistemic_fm_a,J1: epistemic_fm_a,J2: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ ( insert7817948963269374698c_fm_a @ J1 @ J2 ) ) ) )
= ( insert7129737658439086282c_fm_a @ ( R @ J1 )
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1066_Collect__insert,axiom,
! [R: a > epistemic_fm_a,J1: a,J2: set_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ ( insert_a @ J1 @ J2 ) ) ) )
= ( insert7817948963269374698c_fm_a @ ( R @ J1 )
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: a] :
( ( Uu2
= ( R @ J3 ) )
& ( member_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1067_Collect__insert,axiom,
! [R: set_Epistemic_fm_a > epistemic_fm_a,J1: set_Epistemic_fm_a,J2: set_se5208064806568342746c_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ ( insert7129737658439086282c_fm_a @ J1 @ J2 ) ) ) )
= ( insert7817948963269374698c_fm_a @ ( R @ J1 )
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: set_Epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member536094252920883875c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1068_Collect__insert,axiom,
! [R: epistemic_fm_a > epistemic_fm_a,J1: epistemic_fm_a,J2: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ ( insert7817948963269374698c_fm_a @ J1 @ J2 ) ) ) )
= ( insert7817948963269374698c_fm_a @ ( R @ J1 )
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [J3: epistemic_fm_a] :
( ( Uu2
= ( R @ J3 ) )
& ( member6642669571620171971c_fm_a @ J3 @ J2 ) ) ) ) ) ).
% Collect_insert
thf(fact_1069_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1070_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1071_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1072_finite__image__set2,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1073_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1074_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1075_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( 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 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1076_finite__image__set2,axiom,
! [P4: epistemic_fm_a > $o,Q4: 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 @ Q4 ) )
=> ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a,Y4: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 @ Y4 ) )
& ( P4 @ X3 )
& ( Q4 @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1077_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_1078_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_1079_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_1080_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_1081_sup__Inf1__distrib,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( sup_su1367922730591523534c_fm_a @ X @ ( lattic7927669795667837335c_fm_a @ A ) )
= ( lattic7927669795667837335c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A4: set_Epistemic_fm_a] :
( ( Uu2
= ( sup_su1367922730591523534c_fm_a @ X @ A4 ) )
& ( member536094252920883875c_fm_a @ A4 @ A ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_1082_sup__Inf1__distrib,axiom,
! [A: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( sup_sup_set_a @ X @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a
@ ( collect_set_a
@ ^ [Uu2: set_a] :
? [A4: set_a] :
( ( Uu2
= ( sup_sup_set_a @ X @ A4 ) )
& ( member_set_a @ A4 @ A ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_1083_set__Cons__def,axiom,
( set_Cons_a
= ( ^ [A3: set_a,XS: set_list_a] :
( collect_list_a
@ ^ [Z5: list_a] :
? [X3: a,Xs2: list_a] :
( ( Z5
= ( cons_a @ X3 @ Xs2 ) )
& ( member_a @ X3 @ A3 )
& ( member_list_a @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_1084_set__Cons__def,axiom,
( set_Co7170608787392608944c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,XS: set_li4204741992506657632c_fm_a] :
( collec3923057941238076011c_fm_a
@ ^ [Z5: list_s580375451141968640c_fm_a] :
? [X3: set_Epistemic_fm_a,Xs2: list_s580375451141968640c_fm_a] :
( ( Z5
= ( cons_s4394448069393209776c_fm_a @ X3 @ Xs2 ) )
& ( member536094252920883875c_fm_a @ X3 @ A3 )
& ( member2328277852289759785c_fm_a @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_1085_set__Cons__def,axiom,
( set_Co4868574198862222544c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,XS: set_li769143395467472256c_fm_a] :
( collec5191077796991884427c_fm_a
@ ^ [Z5: list_Epistemic_fm_a] :
? [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Z5
= ( cons_Epistemic_fm_a @ X3 @ Xs2 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A3 )
& ( member5906877432388582473c_fm_a @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_1086_finite__def,axiom,
( finite_finite_a
= ( comple3243854344658494246et_a_o
@ ^ [P3: set_a > $o,X3: set_a] :
( ( X3 = bot_bot_set_a )
| ? [A3: set_a,A4: a] :
( ( X3
= ( insert_a @ A4 @ A3 ) )
& ( P3 @ A3 ) ) ) ) ) ).
% finite_def
thf(fact_1087_finite__def,axiom,
( finite3304564945125563331c_fm_a
= ( comple2002745723559171210fm_a_o
@ ^ [P3: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a] :
( ( X3 = bot_bo3626323581529592678c_fm_a )
| ? [A3: set_Epistemic_fm_a,A4: epistemic_fm_a] :
( ( X3
= ( insert7817948963269374698c_fm_a @ A4 @ A3 ) )
& ( P3 @ A3 ) ) ) ) ) ).
% finite_def
thf(fact_1088_lfp__greatest,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o] :
( ! [U3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( F @ U3 ) @ U3 )
=> ( ord_le4043730696559282883fm_a_o @ A @ U3 ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( comple3933800597814182506fm_a_o @ F ) ) ) ).
% lfp_greatest
thf(fact_1089_lfp__greatest,axiom,
! [F: ( a > a > $o ) > a > a > $o,A: a > a > $o] :
( ! [U3: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( F @ U3 ) @ U3 )
=> ( ord_less_eq_a_a_o @ A @ U3 ) )
=> ( ord_less_eq_a_a_o @ A @ ( comple6427898171932868135_a_a_o @ F ) ) ) ).
% lfp_greatest
thf(fact_1090_lfp__greatest,axiom,
! [F: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o,A: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [U3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( F @ U3 ) @ U3 )
=> ( ord_le3934200179093585166fm_a_o @ A @ U3 ) )
=> ( ord_le3934200179093585166fm_a_o @ A @ ( comple846538940498443303fm_a_o @ F ) ) ) ).
% lfp_greatest
thf(fact_1091_lfp__greatest,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ! [U3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ U3 ) @ U3 )
=> ( ord_le3275665582123262618c_fm_a @ A @ U3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( comple693844135356717107c_fm_a @ F ) ) ) ).
% lfp_greatest
thf(fact_1092_lfp__lowerbound,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1093_lfp__lowerbound,axiom,
! [F: ( a > a > $o ) > a > a > $o,A: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( F @ A ) @ A )
=> ( ord_less_eq_a_a_o @ ( comple6427898171932868135_a_a_o @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1094_lfp__lowerbound,axiom,
! [F: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o,A: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( F @ A ) @ A )
=> ( ord_le3934200179093585166fm_a_o @ ( comple846538940498443303fm_a_o @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1095_lfp__lowerbound,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1096_lfp__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,G: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [Z6: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ ( comple3933800597814182506fm_a_o @ G ) ) ) ).
% lfp_mono
thf(fact_1097_lfp__mono,axiom,
! [F: ( a > a > $o ) > a > a > $o,G: ( a > a > $o ) > a > a > $o] :
( ! [Z6: a > a > $o] : ( ord_less_eq_a_a_o @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_less_eq_a_a_o @ ( comple6427898171932868135_a_a_o @ F ) @ ( comple6427898171932868135_a_a_o @ G ) ) ) ).
% lfp_mono
thf(fact_1098_lfp__mono,axiom,
! [F: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o,G: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z6: epistemic_fm_a > epistemic_fm_a > $o] : ( ord_le3934200179093585166fm_a_o @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_le3934200179093585166fm_a_o @ ( comple846538940498443303fm_a_o @ F ) @ ( comple846538940498443303fm_a_o @ G ) ) ) ).
% lfp_mono
thf(fact_1099_lfp__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,G: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z6: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ Z6 ) @ ( G @ Z6 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ ( comple693844135356717107c_fm_a @ G ) ) ) ).
% lfp_mono
thf(fact_1100_lfp__lfp,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o,W2: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
=> ( ( ord_le4043730696559282883fm_a_o @ W2 @ Z3 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ W2 ) @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( comple3933800597814182506fm_a_o
@ ^ [X3: epistemic_fm_a > $o] : ( comple3933800597814182506fm_a_o @ ( F @ X3 ) ) )
= ( comple3933800597814182506fm_a_o
@ ^ [X3: epistemic_fm_a > $o] : ( F @ X3 @ X3 ) ) ) ) ).
% lfp_lfp
thf(fact_1101_lfp__lfp,axiom,
! [F: ( a > a > $o ) > ( a > a > $o ) > a > a > $o] :
( ! [X2: a > a > $o,Y3: a > a > $o,W2: a > a > $o,Z3: a > a > $o] :
( ( ord_less_eq_a_a_o @ X2 @ Y3 )
=> ( ( ord_less_eq_a_a_o @ W2 @ Z3 )
=> ( ord_less_eq_a_a_o @ ( F @ X2 @ W2 ) @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( comple6427898171932868135_a_a_o
@ ^ [X3: a > a > $o] : ( comple6427898171932868135_a_a_o @ ( F @ X3 ) ) )
= ( comple6427898171932868135_a_a_o
@ ^ [X3: a > a > $o] : ( F @ X3 @ X3 ) ) ) ) ).
% lfp_lfp
thf(fact_1102_lfp__lfp,axiom,
! [F: ( epistemic_fm_a > epistemic_fm_a > $o ) > ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > epistemic_fm_a > $o,Y3: epistemic_fm_a > epistemic_fm_a > $o,W2: epistemic_fm_a > epistemic_fm_a > $o,Z3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ X2 @ Y3 )
=> ( ( ord_le3934200179093585166fm_a_o @ W2 @ Z3 )
=> ( ord_le3934200179093585166fm_a_o @ ( F @ X2 @ W2 ) @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( comple846538940498443303fm_a_o
@ ^ [X3: epistemic_fm_a > epistemic_fm_a > $o] : ( comple846538940498443303fm_a_o @ ( F @ X3 ) ) )
= ( comple846538940498443303fm_a_o
@ ^ [X3: epistemic_fm_a > epistemic_fm_a > $o] : ( F @ X3 @ X3 ) ) ) ) ).
% lfp_lfp
thf(fact_1103_lfp__lfp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a,W2: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
=> ( ( ord_le3275665582123262618c_fm_a @ W2 @ Z3 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ W2 ) @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( 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_1104_ordLeq__Func,axiom,
! [B1: a,B23: a,B4: set_a,A: set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ ( insert_a @ B1 @ ( insert_a @ B23 @ bot_bot_set_a ) ) @ B4 )
=> ( ( B1 != B23 )
=> ( member2939515678471440220fm_a_a @ ( produc8772851772689315491fm_a_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca4839443212964495725fm_a_a @ ( bNF_We1622471989791679736fm_a_a @ A @ B4 ) ) ) @ bNF_We7379407385213540796fm_a_a ) ) ) ).
% ordLeq_Func
thf(fact_1105_ordLeq__Func,axiom,
! [B1: epistemic_fm_a,B23: epistemic_fm_a,B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ B1 @ ( insert7817948963269374698c_fm_a @ B23 @ bot_bo3626323581529592678c_fm_a ) ) @ B4 )
=> ( ( B1 != B23 )
=> ( member5250668776978656404c_fm_a @ ( produc707058088808417755c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca7196388322009124745c_fm_a @ ( bNF_We4818972062080617492c_fm_a @ A @ B4 ) ) ) @ bNF_We5212170316383168344c_fm_a ) ) ) ).
% ordLeq_Func
thf(fact_1106_card__of__mono1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca1305897159876240246c_fm_a @ B4 ) ) @ bNF_We8976657667165594343c_fm_a ) ) ).
% card_of_mono1
thf(fact_1107_card__of__Func__mono,axiom,
! [A1: set_Epistemic_fm_a,A23: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A1 @ A23 )
=> ( ( B4 != bot_bo3626323581529592678c_fm_a )
=> ( member5994151988976616464c_fm_a @ ( produc4714088577910911575c_fm_a @ ( bNF_Ca7196388322009124745c_fm_a @ ( bNF_We4818972062080617492c_fm_a @ A1 @ B4 ) ) @ ( bNF_Ca7196388322009124745c_fm_a @ ( bNF_We4818972062080617492c_fm_a @ A23 @ B4 ) ) ) @ bNF_We8812984459868641459c_fm_a ) ) ) ).
% card_of_Func_mono
thf(fact_1108_card__of__inj__rel,axiom,
! [R: set_Pr4537461906582779015c_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a,Y7: set_Epistemic_fm_a] :
( ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ X2 @ Y3 ) @ R )
=> ( ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ X2 @ Y7 ) @ R )
=> ( Y3 = Y7 ) ) )
=> ( member2407610774516601616c_fm_a
@ ( produc5569453956477096791c_fm_a
@ ( bNF_Ca2234214912197128534c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] : ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ X3 @ Y4 ) @ R ) ) )
@ ( bNF_Ca2234214912197128534c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
? [Y4: set_Epistemic_fm_a] : ( member6846535391323571920c_fm_a @ ( produc7744651086898197655c_fm_a @ X3 @ Y4 ) @ R ) ) ) )
@ bNF_We1521092302956861159c_fm_a ) ) ).
% card_of_inj_rel
thf(fact_1109_card__of__inj__rel,axiom,
! [R: set_Pr8704351555129029223c_fm_a] :
( ! [X2: epistemic_fm_a,Y3: set_Epistemic_fm_a,Y7: set_Epistemic_fm_a] :
( ( member4818202340005250224c_fm_a @ ( produc4556709532593425015c_fm_a @ X2 @ Y3 ) @ R )
=> ( ( member4818202340005250224c_fm_a @ ( produc4556709532593425015c_fm_a @ X2 @ Y7 ) @ R )
=> ( Y3 = Y7 ) ) )
=> ( member2483015056062791440c_fm_a
@ ( produc9004553225164312407c_fm_a
@ ( bNF_Ca2234214912197128534c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] : ( member4818202340005250224c_fm_a @ ( produc4556709532593425015c_fm_a @ X3 @ Y4 ) @ R ) ) )
@ ( bNF_Ca1305897159876240246c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
? [Y4: set_Epistemic_fm_a] : ( member4818202340005250224c_fm_a @ ( produc4556709532593425015c_fm_a @ X3 @ Y4 ) @ R ) ) ) )
@ bNF_We3240852806290328839c_fm_a ) ) ).
% card_of_inj_rel
thf(fact_1110_card__of__inj__rel,axiom,
! [R: set_Pr8826056995237917351c_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y3: epistemic_fm_a,Y7: epistemic_fm_a] :
( ( member1928953963652080880c_fm_a @ ( produc6669457843103984311c_fm_a @ X2 @ Y3 ) @ R )
=> ( ( member1928953963652080880c_fm_a @ ( produc6669457843103984311c_fm_a @ X2 @ Y7 ) @ R )
=> ( Y3 = Y7 ) ) )
=> ( member2298073897539299088c_fm_a
@ ( produc8794081022805088087c_fm_a
@ ( bNF_Ca1305897159876240246c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Y4: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] : ( member1928953963652080880c_fm_a @ ( produc6669457843103984311c_fm_a @ X3 @ Y4 ) @ R ) ) )
@ ( bNF_Ca2234214912197128534c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
? [Y4: epistemic_fm_a] : ( member1928953963652080880c_fm_a @ ( produc6669457843103984311c_fm_a @ X3 @ Y4 ) @ R ) ) ) )
@ bNF_We1128104495779769543c_fm_a ) ) ).
% card_of_inj_rel
thf(fact_1111_card__of__inj__rel,axiom,
! [R: set_Pr8948481099588399239c_fm_a] :
( ! [X2: epistemic_fm_a,Y3: epistemic_fm_a,Y7: epistemic_fm_a] :
( ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X2 @ Y3 ) @ R )
=> ( ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X2 @ Y7 ) @ R )
=> ( Y3 = Y7 ) ) )
=> ( member1688387778307953424c_fm_a
@ ( produc7431400533060703063c_fm_a
@ ( bNF_Ca1305897159876240246c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [Y4: epistemic_fm_a] :
? [X3: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X3 @ Y4 ) @ R ) ) )
@ ( bNF_Ca1305897159876240246c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
? [Y4: epistemic_fm_a] : ( member5029746173879296720c_fm_a @ ( produc8582950737844404375c_fm_a @ X3 @ Y4 ) @ R ) ) ) )
@ bNF_We8976657667165594343c_fm_a ) ) ).
% card_of_inj_rel
thf(fact_1112_range__constant,axiom,
! [X: a] :
( ( image_7228943142519265129fm_a_a
@ ^ [Uu2: epistemic_fm_a] : X
@ top_to7796028867103199306c_fm_a )
= ( insert_a @ X @ bot_bot_set_a ) ) ).
% range_constant
thf(fact_1113_range__constant,axiom,
! [X: epistemic_fm_a] :
( ( image_4449434806354059013c_fm_a
@ ^ [Uu2: epistemic_fm_a] : X
@ top_to7796028867103199306c_fm_a )
= ( insert7817948963269374698c_fm_a @ X @ bot_bo3626323581529592678c_fm_a ) ) ).
% range_constant
thf(fact_1114_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > a,B4: set_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_4497856574827002249fm_a_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1115_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1116_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > a,B4: set_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1117_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1118_image__Collect__subsetI,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ ( collec2519470961442302949c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1119_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_1120_subset__image__iff,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B4
= ( image_4449434806354059013c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1121_subset__imageE,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B4
!= ( image_4449434806354059013c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1122_image__subsetI,axiom,
! [A: set_a,F: a > a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1123_image__subsetI,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1124_image__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > a,B4: set_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_4497856574827002249fm_a_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1125_image__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1126_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > a,B4: set_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1127_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member536094252920883875c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1128_image__subsetI,axiom,
! [A: set_a,F: a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_1898462159489238305c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1129_image__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1130_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1131_image__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ ( image_4449434806354059013c_fm_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1132_all__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ( P4 @ B3 ) ) )
= ( ! [B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1133_imageE,axiom,
! [B: a,F: a > a,A: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1134_imageE,axiom,
! [B: a,F: set_Epistemic_fm_a > a,A: set_se5208064806568342746c_fm_a] :
( ( member_a @ B @ ( image_4497856574827002249fm_a_a @ F @ A ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member536094252920883875c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1135_imageE,axiom,
! [B: a,F: epistemic_fm_a > a,A: set_Epistemic_fm_a] :
( ( member_a @ B @ ( image_7228943142519265129fm_a_a @ F @ A ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member6642669571620171971c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1136_imageE,axiom,
! [B: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a,A: set_a] :
( ( member536094252920883875c_fm_a @ B @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1137_imageE,axiom,
! [B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ B @ ( image_8582343580727714565c_fm_a @ F @ A ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member536094252920883875c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1138_imageE,axiom,
! [B: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member6642669571620171971c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1139_imageE,axiom,
! [B: epistemic_fm_a,F: a > epistemic_fm_a,A: set_a] :
( ( member6642669571620171971c_fm_a @ B @ ( image_1898462159489238305c_fm_a @ F @ A ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1140_imageE,axiom,
! [B: epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ B @ ( image_3376480847214465317c_fm_a @ F @ A ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member536094252920883875c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1141_imageE,axiom,
! [B: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member6642669571620171971c_fm_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1142_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P4: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
& ( P4 @ X3 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1143_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_a > a,A: set_se5208064806568342746c_fm_a,P4: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_4497856574827002249fm_a_a @ F @ A ) )
& ( P4 @ X3 ) ) )
= ( image_4497856574827002249fm_a_a @ F
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1144_Compr__image__eq,axiom,
! [F: epistemic_fm_a > a,A: set_Epistemic_fm_a,P4: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_7228943142519265129fm_a_a @ F @ A ) )
& ( P4 @ X3 ) ) )
= ( image_7228943142519265129fm_a_a @ F
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1145_Compr__image__eq,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a,P4: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( image_3480842985853347585c_fm_a @ F @ A ) )
& ( P4 @ X3 ) ) )
= ( image_3480842985853347585c_fm_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1146_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: 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 @ A ) )
& ( P4 @ X3 ) ) )
= ( image_8582343580727714565c_fm_a @ F
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1147_Compr__image__eq,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: 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 @ A ) )
& ( P4 @ X3 ) ) )
= ( image_1263732536703906021c_fm_a @ F
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1148_Compr__image__eq,axiom,
! [F: a > epistemic_fm_a,A: set_a,P4: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( image_1898462159489238305c_fm_a @ F @ A ) )
& ( P4 @ X3 ) ) )
= ( image_1898462159489238305c_fm_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1149_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A: 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 @ A ) )
& ( P4 @ X3 ) ) )
= ( image_3376480847214465317c_fm_a @ F
@ ( collec2519470961442302949c_fm_a
@ ^ [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1150_Compr__image__eq,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: 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 @ A ) )
& ( P4 @ X3 ) ) )
= ( image_4449434806354059013c_fm_a @ F
@ ( collec4904205152690461189c_fm_a
@ ^ [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1151_rangeE,axiom,
! [B: a,F: epistemic_fm_a > a] :
( ( member_a @ B @ ( image_7228943142519265129fm_a_a @ F @ top_to7796028867103199306c_fm_a ) )
=> ~ ! [X2: epistemic_fm_a] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1152_rangeE,axiom,
! [B: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B @ ( image_1263732536703906021c_fm_a @ F @ top_to7796028867103199306c_fm_a ) )
=> ~ ! [X2: epistemic_fm_a] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1153_rangeE,axiom,
! [B: epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B @ ( image_4449434806354059013c_fm_a @ F @ top_to7796028867103199306c_fm_a ) )
=> ~ ! [X2: epistemic_fm_a] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_1154_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_1155_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_1156_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_1157_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_1158_Setcompr__eq__image,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: a] :
( ( Uu2
= ( F @ X3 ) )
& ( member_a @ X3 @ A ) ) )
= ( image_3480842985853347585c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1159_Setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member536094252920883875c_fm_a @ X3 @ A ) ) )
= ( image_8582343580727714565c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1160_Setcompr__eq__image,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A ) ) )
= ( image_1263732536703906021c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1161_Setcompr__eq__image,axiom,
! [F: a > epistemic_fm_a,A: set_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: a] :
( ( Uu2
= ( F @ X3 ) )
& ( member_a @ X3 @ A ) ) )
= ( image_1898462159489238305c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1162_Setcompr__eq__image,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: set_Epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member536094252920883875c_fm_a @ X3 @ A ) ) )
= ( image_3376480847214465317c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1163_Setcompr__eq__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [Uu2: epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( ( Uu2
= ( F @ X3 ) )
& ( member6642669571620171971c_fm_a @ X3 @ A ) ) )
= ( image_4449434806354059013c_fm_a @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_1164_pigeonhole__infinite,axiom,
! [A: set_a,F: a > epistemic_fm_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ ( image_1898462159489238305c_fm_a @ F @ A ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1165_pigeonhole__infinite,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a] :
( ~ ( finite7364866884270929315c_fm_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ ( image_3376480847214465317c_fm_a @ F @ A ) )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ~ ( finite7364866884270929315c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [A4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1166_pigeonhole__infinite,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ~ ( finite3304564945125563331c_fm_a @ A )
=> ( ( finite3304564945125563331c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ~ ( finite3304564945125563331c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [A4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1167_range__subsetD,axiom,
! [F: epistemic_fm_a > a,B4: set_a,I3: epistemic_fm_a] :
( ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ top_to7796028867103199306c_fm_a ) @ B4 )
=> ( member_a @ ( F @ I3 ) @ B4 ) ) ).
% range_subsetD
thf(fact_1168_range__subsetD,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a,I3: epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ top_to7796028867103199306c_fm_a ) @ B4 )
=> ( member536094252920883875c_fm_a @ ( F @ I3 ) @ B4 ) ) ).
% range_subsetD
thf(fact_1169_range__subsetD,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,B4: set_Epistemic_fm_a,I3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ top_to7796028867103199306c_fm_a ) @ B4 )
=> ( member6642669571620171971c_fm_a @ ( F @ I3 ) @ B4 ) ) ).
% range_subsetD
thf(fact_1170_all__finite__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B3: set_Epistemic_fm_a] :
( ( ( finite3304564945125563331c_fm_a @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A ) ) )
=> ( P4 @ B3 ) ) )
= ( ! [B3: set_Epistemic_fm_a] :
( ( ( finite3304564945125563331c_fm_a @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ A ) )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1171_ex__finite__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ? [B3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
& ( P4 @ B3 ) ) )
= ( ? [B3: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ A )
& ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1172_finite__subset__image,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ? [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
& ( finite3304564945125563331c_fm_a @ C3 )
& ( B4
= ( image_4449434806354059013c_fm_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1173_finite__surj,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ( finite3304564945125563331c_fm_a @ B4 ) ) ) ).
% finite_surj
thf(fact_1174_finite__range__imageI,axiom,
! [G: epistemic_fm_a > epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ ( image_4449434806354059013c_fm_a @ G @ top_to7796028867103199306c_fm_a ) )
=> ( finite3304564945125563331c_fm_a
@ ( image_4449434806354059013c_fm_a
@ ^ [X3: epistemic_fm_a] : ( F @ ( G @ X3 ) )
@ top_to7796028867103199306c_fm_a ) ) ) ).
% finite_range_imageI
thf(fact_1175_full__SetCompr__eq,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [U2: set_Epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( U2
= ( F @ X3 ) ) )
= ( image_1263732536703906021c_fm_a @ F @ top_to7796028867103199306c_fm_a ) ) ).
% full_SetCompr_eq
thf(fact_1176_full__SetCompr__eq,axiom,
! [F: epistemic_fm_a > epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [U2: epistemic_fm_a] :
? [X3: epistemic_fm_a] :
( U2
= ( F @ X3 ) ) )
= ( image_4449434806354059013c_fm_a @ F @ top_to7796028867103199306c_fm_a ) ) ).
% full_SetCompr_eq
thf(fact_1177_image__constant__conv,axiom,
! [A: set_Epistemic_fm_a,C: a] :
( ( ( A = bot_bo3626323581529592678c_fm_a )
=> ( ( image_7228943142519265129fm_a_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= bot_bot_set_a ) )
& ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( image_7228943142519265129fm_a_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1178_image__constant__conv,axiom,
! [A: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( ( A = bot_bo3626323581529592678c_fm_a )
=> ( ( image_4449434806354059013c_fm_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= bot_bo3626323581529592678c_fm_a ) )
& ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( image_4449434806354059013c_fm_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= ( insert7817948963269374698c_fm_a @ C @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% image_constant_conv
thf(fact_1179_image__constant,axiom,
! [X: a,A: set_a,C: a] :
( ( member_a @ X @ A )
=> ( ( image_a_a
@ ^ [X3: a] : C
@ A )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1180_image__constant,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,C: a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ( image_4497856574827002249fm_a_a
@ ^ [X3: set_Epistemic_fm_a] : C
@ A )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1181_image__constant,axiom,
! [X: epistemic_fm_a,A: set_Epistemic_fm_a,C: a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ( image_7228943142519265129fm_a_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1182_image__constant,axiom,
! [X: a,A: set_a,C: epistemic_fm_a] :
( ( member_a @ X @ A )
=> ( ( image_1898462159489238305c_fm_a
@ ^ [X3: a] : C
@ A )
= ( insert7817948963269374698c_fm_a @ C @ bot_bo3626323581529592678c_fm_a ) ) ) ).
% image_constant
thf(fact_1183_image__constant,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,C: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ( image_3376480847214465317c_fm_a
@ ^ [X3: set_Epistemic_fm_a] : C
@ A )
= ( insert7817948963269374698c_fm_a @ C @ bot_bo3626323581529592678c_fm_a ) ) ) ).
% image_constant
thf(fact_1184_image__constant,axiom,
! [X: epistemic_fm_a,A: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ( image_4449434806354059013c_fm_a
@ ^ [X3: epistemic_fm_a] : C
@ A )
= ( insert7817948963269374698c_fm_a @ C @ bot_bo3626323581529592678c_fm_a ) ) ) ).
% image_constant
thf(fact_1185_surj__imp__ordLeq,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ B4 ) @ ( bNF_Ca1305897159876240246c_fm_a @ A ) ) @ bNF_We8976657667165594343c_fm_a ) ) ).
% surj_imp_ordLeq
thf(fact_1186_Extend__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) ) ) ) ).
% Extend_def
thf(fact_1187_extendL__def,axiom,
! [Rec: epistemic_fm_a > set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ Rec @ N )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ Rec @ ( order_1288088311422721504c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) ) ) ) ).
% extendL_def
thf(fact_1188_card__of__Func__UNIV,axiom,
! [B4: set_Epistemic_fm_a] :
( member5994151988976616464c_fm_a
@ ( produc4714088577910911575c_fm_a @ ( bNF_Ca7196388322009124745c_fm_a @ ( bNF_We4818972062080617492c_fm_a @ top_to7796028867103199306c_fm_a @ B4 ) )
@ ( bNF_Ca7196388322009124745c_fm_a
@ ( collec8583107250894946746c_fm_a
@ ^ [F2: epistemic_fm_a > epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F2 @ top_to7796028867103199306c_fm_a ) @ B4 ) ) ) )
@ bNF_We8075338327528360864c_fm_a ) ).
% card_of_Func_UNIV
thf(fact_1189_finite__UN__I,axiom,
! [A: set_a,B4: a > set_Epistemic_fm_a] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( finite3304564945125563331c_fm_a @ ( B4 @ A6 ) ) )
=> ( finite3304564945125563331c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B4 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1190_finite__UN__I,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( finite3304564945125563331c_fm_a @ ( B4 @ A6 ) ) )
=> ( finite3304564945125563331c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ B4 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1191_finite__UN__I,axiom,
! [A: set_Epistemic_fm_a,B4: epistemic_fm_a > set_Epistemic_fm_a] :
( ( finite3304564945125563331c_fm_a @ A )
=> ( ! [A6: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A6 @ A )
=> ( finite3304564945125563331c_fm_a @ ( B4 @ A6 ) ) )
=> ( finite3304564945125563331c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1192_internalize__card__of__ordLeq2,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca1305897159876240246c_fm_a @ C2 ) ) @ bNF_We8976657667165594343c_fm_a )
= ( ? [B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ C2 )
& ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca1305897159876240246c_fm_a @ B3 ) ) @ bNF_We2357358836176181844c_fm_a )
& ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ B3 ) @ ( bNF_Ca1305897159876240246c_fm_a @ C2 ) ) @ bNF_We8976657667165594343c_fm_a ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_1193_card__of__UNION__ordLeq__infinite,axiom,
! [B4: set_Epistemic_fm_a,I4: set_Epistemic_fm_a,A: epistemic_fm_a > set_Epistemic_fm_a] :
( ~ ( finite3304564945125563331c_fm_a @ B4 )
=> ( ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ I4 ) @ ( bNF_Ca1305897159876240246c_fm_a @ B4 ) ) @ bNF_We8976657667165594343c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ I4 )
=> ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ ( A @ X2 ) ) @ ( bNF_Ca1305897159876240246c_fm_a @ B4 ) ) @ bNF_We8976657667165594343c_fm_a ) )
=> ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) ) ) @ ( bNF_Ca1305897159876240246c_fm_a @ B4 ) ) @ bNF_We8976657667165594343c_fm_a ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_1194_wo__rel_Oofilter__under__Union,axiom,
! [R2: set_Product_prod_a_a,A: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( order_ofilter_a @ R2 @ A )
=> ( A
= ( comple2307003609928055243_set_a
@ ( collect_set_a
@ ^ [Uu2: set_a] :
? [A4: a] :
( ( Uu2
= ( order_under_a @ R2 @ A4 ) )
& ( member_a @ A4 @ A ) ) ) ) ) ) ) ).
% wo_rel.ofilter_under_Union
thf(fact_1195_wo__rel_Oofilter__under__Union,axiom,
! [R2: set_Pr4537461906582779015c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( bNF_We8880040591803976934c_fm_a @ R2 )
=> ( ( order_7372089894593676944c_fm_a @ R2 @ A )
=> ( A
= ( comple7261557488043011271c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [Uu2: set_se5208064806568342746c_fm_a] :
? [A4: set_Epistemic_fm_a] :
( ( Uu2
= ( order_4320674923736412895c_fm_a @ R2 @ A4 ) )
& ( member536094252920883875c_fm_a @ A4 @ A ) ) ) ) ) ) ) ).
% wo_rel.ofilter_under_Union
thf(fact_1196_wo__rel_Oofilter__under__Union,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,A: set_Epistemic_fm_a] :
( ( bNF_We8255954830765735686c_fm_a @ R2 )
=> ( ( order_7157697347518734512c_fm_a @ R2 @ A )
=> ( A
= ( comple7868773486375872231c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [Uu2: set_Epistemic_fm_a] :
? [A4: epistemic_fm_a] :
( ( Uu2
= ( order_1815523470270482687c_fm_a @ R2 @ A4 ) )
& ( member6642669571620171971c_fm_a @ A4 @ A ) ) ) ) ) ) ) ).
% wo_rel.ofilter_under_Union
thf(fact_1197_internalize__card__of__ordLeq,axiom,
! [A: set_Epistemic_fm_a,R2: set_Pr8948481099588399239c_fm_a] :
( ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ R2 ) @ bNF_We8976657667165594343c_fm_a )
= ( ? [B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ ( field_Epistemic_fm_a @ R2 ) )
& ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ A ) @ ( bNF_Ca1305897159876240246c_fm_a @ B3 ) ) @ bNF_We2357358836176181844c_fm_a )
& ( member1688387778307953424c_fm_a @ ( produc7431400533060703063c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ B3 ) @ R2 ) @ bNF_We8976657667165594343c_fm_a ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_1198_MCS__Lim__Ord_OExtend__def,axiom,
! [R2: set_Pr8948481099588399239c_fm_a,Consistent: set_Epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima8116622252184345237c_fm_a @ R2 @ Consistent )
=> ( ( maxima2580775624958445067c_fm_a @ R2 @ Consistent @ S )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ ( maxima1168816336865142315c_fm_a @ R2 @ Consistent @ S ) @ ( field_Epistemic_fm_a @ R2 ) ) ) ) ) ).
% MCS_Lim_Ord.Extend_def
thf(fact_1199_UN__insert,axiom,
! [B4: epistemic_fm_a > set_Epistemic_fm_a,A2: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ ( insert7817948963269374698c_fm_a @ A2 @ A ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( B4 @ A2 ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ A ) ) ) ) ).
% UN_insert
thf(fact_1200_UN__insert,axiom,
! [B4: a > set_Epistemic_fm_a,A2: a,A: set_a] :
( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B4 @ ( insert_a @ A2 @ A ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( B4 @ A2 ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B4 @ A ) ) ) ) ).
% UN_insert
thf(fact_1201_UN__insert,axiom,
! [B4: epistemic_fm_a > set_a,A2: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ ( insert7817948963269374698c_fm_a @ A2 @ A ) ) )
= ( sup_sup_set_a @ ( B4 @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ A ) ) ) ) ).
% UN_insert
thf(fact_1202_UN__insert,axiom,
! [B4: a > set_a,A2: a,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ ( insert_a @ A2 @ A ) ) )
= ( sup_sup_set_a @ ( B4 @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ A ) ) ) ) ).
% UN_insert
thf(fact_1203_UN__simps_I3_J,axiom,
! [C2: set_Epistemic_fm_a,A: set_a,B4: epistemic_fm_a > set_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( sup_sup_set_a @ A @ ( B4 @ X3 ) )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( sup_sup_set_a @ A @ ( B4 @ X3 ) )
@ C2 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1204_UN__simps_I3_J,axiom,
! [C2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( sup_su1367922730591523534c_fm_a @ A @ ( B4 @ X3 ) )
@ C2 ) )
= bot_bo3626323581529592678c_fm_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( sup_su1367922730591523534c_fm_a @ A @ ( B4 @ X3 ) )
@ C2 ) )
= ( sup_su1367922730591523534c_fm_a @ A @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1205_UN__simps_I2_J,axiom,
! [C2: set_Epistemic_fm_a,A: epistemic_fm_a > set_a,B4: set_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( sup_sup_set_a @ ( A @ X3 ) @ B4 )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( sup_sup_set_a @ ( A @ X3 ) @ B4 )
@ C2 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ A @ C2 ) ) @ B4 ) ) ) ) ).
% UN_simps(2)
thf(fact_1206_UN__simps_I2_J,axiom,
! [C2: set_Epistemic_fm_a,A: epistemic_fm_a > set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( sup_su1367922730591523534c_fm_a @ ( A @ X3 ) @ B4 )
@ C2 ) )
= bot_bo3626323581529592678c_fm_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( sup_su1367922730591523534c_fm_a @ ( A @ X3 ) @ B4 )
@ C2 ) )
= ( sup_su1367922730591523534c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ C2 ) ) @ B4 ) ) ) ) ).
% UN_simps(2)
thf(fact_1207_UN__I,axiom,
! [A2: a,A: set_a,B: a,B4: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1208_UN__I,axiom,
! [A2: a,A: set_a,B: set_Epistemic_fm_a,B4: a > set_se5208064806568342746c_fm_a] :
( ( member_a @ A2 @ A )
=> ( ( member536094252920883875c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_4864492491237557473c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1209_UN__I,axiom,
! [A2: a,A: set_a,B: epistemic_fm_a,B4: a > set_Epistemic_fm_a] :
( ( member_a @ A2 @ A )
=> ( ( member6642669571620171971c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1210_UN__I,axiom,
! [A2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: a,B4: set_Epistemic_fm_a > set_a] :
( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_7006806330250622697_set_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1211_UN__I,axiom,
! [A2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a,B4: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ( member536094252920883875c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_3447524139502481637c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1212_UN__I,axiom,
! [A2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: epistemic_fm_a,B4: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ( member6642669571620171971c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1213_UN__I,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: a,B4: epistemic_fm_a > set_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ( member_a @ B @ ( B4 @ A2 ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1214_UN__I,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: set_Epistemic_fm_a,B4: epistemic_fm_a > set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ( member536094252920883875c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_2544827684617158341c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1215_UN__I,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: epistemic_fm_a,B4: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ( member6642669571620171971c_fm_a @ B @ ( B4 @ A2 ) )
=> ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1216_UN__constant,axiom,
! [A: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ( A = bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [Y4: epistemic_fm_a] : C
@ A ) )
= bot_bo3626323581529592678c_fm_a ) )
& ( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [Y4: epistemic_fm_a] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1217_UN__Un,axiom,
! [M: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ M @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ M @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ M @ B4 ) ) ) ) ).
% UN_Un
thf(fact_1218_UN__Un,axiom,
! [M: epistemic_fm_a > set_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ M @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ M @ B4 ) ) ) ) ).
% UN_Un
thf(fact_1219_UN__Un,axiom,
! [M: a > set_Epistemic_fm_a,A: set_a,B4: set_a] :
( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ M @ ( sup_sup_set_a @ A @ B4 ) ) )
= ( sup_su1367922730591523534c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ M @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ M @ B4 ) ) ) ) ).
% UN_Un
thf(fact_1220_UN__Un,axiom,
! [M: a > set_a,A: set_a,B4: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ ( sup_sup_set_a @ A @ B4 ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ M @ B4 ) ) ) ) ).
% UN_Un
thf(fact_1221_UN__simps_I1_J,axiom,
! [C2: set_Epistemic_fm_a,A2: a,B4: epistemic_fm_a > set_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( insert_a @ A2 @ ( B4 @ X3 ) )
@ C2 ) )
= bot_bot_set_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple2307003609928055243_set_a
@ ( image_7937322393147004105_set_a
@ ^ [X3: epistemic_fm_a] : ( insert_a @ A2 @ ( B4 @ X3 ) )
@ C2 ) )
= ( insert_a @ A2 @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1222_UN__simps_I1_J,axiom,
! [C2: set_Epistemic_fm_a,A2: epistemic_fm_a,B4: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ( C2 = bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( insert7817948963269374698c_fm_a @ A2 @ ( B4 @ X3 ) )
@ C2 ) )
= bot_bo3626323581529592678c_fm_a ) )
& ( ( C2 != bot_bo3626323581529592678c_fm_a )
=> ( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( insert7817948963269374698c_fm_a @ A2 @ ( B4 @ X3 ) )
@ C2 ) )
= ( insert7817948963269374698c_fm_a @ A2 @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1223_UN__singleton,axiom,
! [A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X3: a] : ( insert_a @ X3 @ bot_bot_set_a )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1224_UN__singleton,axiom,
! [A: set_Epistemic_fm_a] :
( ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X3: epistemic_fm_a] : ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1225_Sup__eqI,axiom,
! [A: set_Epistemic_fm_a_o,X: epistemic_fm_a > $o] :
( ! [Y3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y3 @ X ) )
=> ( ! [Y3: epistemic_fm_a > $o] :
( ! [Z7: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Z7 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Z7 @ Y3 ) )
=> ( ord_le4043730696559282883fm_a_o @ X @ Y3 ) )
=> ( ( comple310706794042174646fm_a_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_1226_Sup__eqI,axiom,
! [A: set_a_a_o,X: a > a > $o] :
( ! [Y3: a > a > $o] :
( ( member_a_a_o @ Y3 @ A )
=> ( ord_less_eq_a_a_o @ Y3 @ X ) )
=> ( ! [Y3: a > a > $o] :
( ! [Z7: a > a > $o] :
( ( member_a_a_o @ Z7 @ A )
=> ( ord_less_eq_a_a_o @ Z7 @ Y3 ) )
=> ( ord_less_eq_a_a_o @ X @ Y3 ) )
=> ( ( comple5693171575053104859_a_a_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_1227_Sup__eqI,axiom,
! [A: set_Ep7530996317732161742fm_a_o,X: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Y3 @ A )
=> ( ord_le3934200179093585166fm_a_o @ Y3 @ X ) )
=> ( ! [Y3: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z7: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ Z7 @ A )
=> ( ord_le3934200179093585166fm_a_o @ Z7 @ Y3 ) )
=> ( ord_le3934200179093585166fm_a_o @ X @ Y3 ) )
=> ( ( comple6797666690712414427fm_a_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_1228_Sup__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ! [Y3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y3 @ X ) )
=> ( ! [Y3: set_Epistemic_fm_a] :
( ! [Z7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z7 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Z7 @ Y3 ) )
=> ( ord_le3275665582123262618c_fm_a @ X @ Y3 ) )
=> ( ( comple7868773486375872231c_fm_a @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_1229_Sup__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ! [A6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A6 @ A )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ A6 @ X5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_1230_Sup__mono,axiom,
! [A: set_a_a_o,B4: set_a_a_o] :
( ! [A6: a > a > $o] :
( ( member_a_a_o @ A6 @ A )
=> ? [X5: a > a > $o] :
( ( member_a_a_o @ X5 @ B4 )
& ( ord_less_eq_a_a_o @ A6 @ X5 ) ) )
=> ( ord_less_eq_a_a_o @ ( comple5693171575053104859_a_a_o @ A ) @ ( comple5693171575053104859_a_a_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_1231_Sup__mono,axiom,
! [A: set_Ep7530996317732161742fm_a_o,B4: set_Ep7530996317732161742fm_a_o] :
( ! [A6: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ A6 @ A )
=> ? [X5: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X5 @ B4 )
& ( ord_le3934200179093585166fm_a_o @ A6 @ X5 ) ) )
=> ( ord_le3934200179093585166fm_a_o @ ( comple6797666690712414427fm_a_o @ A ) @ ( comple6797666690712414427fm_a_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_1232_Sup__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ A6 @ X5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_mono
thf(fact_1233_Sup__least,axiom,
! [A: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_1234_Sup__least,axiom,
! [A: set_a_a_o,Z2: a > a > $o] :
( ! [X2: a > a > $o] :
( ( member_a_a_o @ X2 @ A )
=> ( ord_less_eq_a_a_o @ X2 @ Z2 ) )
=> ( ord_less_eq_a_a_o @ ( comple5693171575053104859_a_a_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_1235_Sup__least,axiom,
! [A: set_Ep7530996317732161742fm_a_o,Z2: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X2 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X2 @ Z2 ) )
=> ( ord_le3934200179093585166fm_a_o @ ( comple6797666690712414427fm_a_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_1236_Sup__least,axiom,
! [A: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_1237_Sup__upper,axiom,
! [X: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ ( comple310706794042174646fm_a_o @ A ) ) ) ).
% Sup_upper
thf(fact_1238_Sup__upper,axiom,
! [X: a > a > $o,A: set_a_a_o] :
( ( member_a_a_o @ X @ A )
=> ( ord_less_eq_a_a_o @ X @ ( comple5693171575053104859_a_a_o @ A ) ) ) ).
% Sup_upper
thf(fact_1239_Sup__upper,axiom,
! [X: epistemic_fm_a > epistemic_fm_a > $o,A: set_Ep7530996317732161742fm_a_o] :
( ( member8030264063065380503fm_a_o @ X @ A )
=> ( ord_le3934200179093585166fm_a_o @ X @ ( comple6797666690712414427fm_a_o @ A ) ) ) ).
% Sup_upper
thf(fact_1240_Sup__upper,axiom,
! [X: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Sup_upper
thf(fact_1241_Sup__le__iff,axiom,
! [A: set_Epistemic_fm_a_o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ B )
= ( ! [X3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1242_Sup__le__iff,axiom,
! [A: set_a_a_o,B: a > a > $o] :
( ( ord_less_eq_a_a_o @ ( comple5693171575053104859_a_a_o @ A ) @ B )
= ( ! [X3: a > a > $o] :
( ( member_a_a_o @ X3 @ A )
=> ( ord_less_eq_a_a_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1243_Sup__le__iff,axiom,
! [A: set_Ep7530996317732161742fm_a_o,B: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ ( comple6797666690712414427fm_a_o @ A ) @ B )
= ( ! [X3: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ X3 @ A )
=> ( ord_le3934200179093585166fm_a_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1244_Sup__le__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ B )
= ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1245_Sup__upper2,axiom,
! [U4: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o,V4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ U4 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ V4 @ U4 )
=> ( ord_le4043730696559282883fm_a_o @ V4 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1246_Sup__upper2,axiom,
! [U4: a > a > $o,A: set_a_a_o,V4: a > a > $o] :
( ( member_a_a_o @ U4 @ A )
=> ( ( ord_less_eq_a_a_o @ V4 @ U4 )
=> ( ord_less_eq_a_a_o @ V4 @ ( comple5693171575053104859_a_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1247_Sup__upper2,axiom,
! [U4: epistemic_fm_a > epistemic_fm_a > $o,A: set_Ep7530996317732161742fm_a_o,V4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member8030264063065380503fm_a_o @ U4 @ A )
=> ( ( ord_le3934200179093585166fm_a_o @ V4 @ U4 )
=> ( ord_le3934200179093585166fm_a_o @ V4 @ ( comple6797666690712414427fm_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1248_Sup__upper2,axiom,
! [U4: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,V4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ U4 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ V4 @ U4 )
=> ( ord_le3275665582123262618c_fm_a @ V4 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1249_Union__subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ? [Y5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ X2 @ Y5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Union_subsetI
thf(fact_1250_Union__upper,axiom,
! [B4: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ B4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Union_upper
thf(fact_1251_Union__least,axiom,
! [A: set_se5208064806568342746c_fm_a,C2: set_Epistemic_fm_a] :
( ! [X7: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X7 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X7 @ C2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1252_Union__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Union_mono
thf(fact_1253_SUP__UNIV__bool__expand,axiom,
! [A: $o > set_Epistemic_fm_a] :
( ( comple7868773486375872231c_fm_a @ ( image_2964458278485386983c_fm_a @ A @ top_top_set_o ) )
= ( sup_su1367922730591523534c_fm_a @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_1254_SUP__UNIV__bool__expand,axiom,
! [A: $o > set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ top_top_set_o ) )
= ( sup_sup_set_a @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_1255_UN__bool__eq,axiom,
! [A: $o > set_Epistemic_fm_a] :
( ( comple7868773486375872231c_fm_a @ ( image_2964458278485386983c_fm_a @ A @ top_top_set_o ) )
= ( sup_su1367922730591523534c_fm_a @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% UN_bool_eq
thf(fact_1256_UN__bool__eq,axiom,
! [A: $o > set_a] :
( ( comple2307003609928055243_set_a @ ( image_o_set_a @ A @ top_top_set_o ) )
= ( sup_sup_set_a @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% UN_bool_eq
thf(fact_1257_Un__eq__UN,axiom,
( sup_su1367922730591523534c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( comple7868773486375872231c_fm_a
@ ( image_2964458278485386983c_fm_a
@ ^ [B2: $o] : ( if_set2375185401397246656c_fm_a @ B2 @ A3 @ B3 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_1258_Un__eq__UN,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( comple2307003609928055243_set_a
@ ( image_o_set_a
@ ^ [B2: $o] : ( if_set_a @ B2 @ A3 @ B3 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_1259_UN__E,axiom,
! [B: a,B4: a > set_a,A: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B4 @ A ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1260_UN__E,axiom,
! [B: a,B4: set_Epistemic_fm_a > set_a,A: set_se5208064806568342746c_fm_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_7006806330250622697_set_a @ B4 @ A ) ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ~ ( member_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1261_UN__E,axiom,
! [B: a,B4: epistemic_fm_a > set_a,A: set_Epistemic_fm_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_7937322393147004105_set_a @ B4 @ A ) ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ~ ( member_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1262_UN__E,axiom,
! [B: set_Epistemic_fm_a,B4: a > set_se5208064806568342746c_fm_a,A: set_a] :
( ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_4864492491237557473c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member536094252920883875c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1263_UN__E,axiom,
! [B: set_Epistemic_fm_a,B4: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_3447524139502481637c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ~ ( member536094252920883875c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1264_UN__E,axiom,
! [B: set_Epistemic_fm_a,B4: epistemic_fm_a > set_se5208064806568342746c_fm_a,A: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B @ ( comple7261557488043011271c_fm_a @ ( image_2544827684617158341c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ~ ( member536094252920883875c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1265_UN__E,axiom,
! [B: epistemic_fm_a,B4: a > set_Epistemic_fm_a,A: set_a] :
( ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member6642669571620171971c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1266_UN__E,axiom,
! [B: epistemic_fm_a,B4: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ~ ( member6642669571620171971c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1267_UN__E,axiom,
! [B: epistemic_fm_a,B4: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ B @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ B4 @ A ) ) )
=> ~ ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ~ ( member6642669571620171971c_fm_a @ B @ ( B4 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1268_SUP__eq,axiom,
! [A: set_a,B4: set_se5208064806568342746c_fm_a,F: a > set_Epistemic_fm_a,G: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J4 @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1269_SUP__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a,G: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: a] :
( ( member_a @ I @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J4 @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1270_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G: a > set_Epistemic_fm_a] :
( ! [I: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: a] :
( ( member_a @ J4 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1271_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J4 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1272_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J4 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1273_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,F: epistemic_fm_a > set_Epistemic_fm_a,G: a > set_Epistemic_fm_a] :
( ! [I: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: a] :
( ( member_a @ J4 @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1274_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J4 @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_1275_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( G @ X5 ) ) ) )
=> ( ! [J4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J4 @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G @ J4 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G @ B4 ) ) ) ) ) ).
% SUP_eq
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y2: set_a] :
( ( if_set_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y2: set_a] :
( ( if_set_a @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [G4: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H4 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ a3 @ b ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ a3 @ c ) @ ( epistemic_Imp_a @ a3 @ ( epistemic_Con_a @ b @ c ) ) ) ) ) ).
%------------------------------------------------------------------------------