TPTP Problem File: SLH0405^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Stalnaker_Logic/0000_Stalnaker_Logic/prob_00235_007799__6483568_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1667 ( 537 unt; 376 typ; 0 def)
% Number of atoms : 3955 (1371 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11487 ( 544 ~; 51 |; 365 &;8653 @)
% ( 0 <=>;1874 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 1256 (1256 >; 0 *; 0 +; 0 <<)
% Number of symbols : 335 ( 332 usr; 38 con; 0-5 aty)
% Number of variables : 3666 ( 235 ^;3202 !; 229 ?;3666 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:07.805
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
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% Explicit typings (332)
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member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1104366573291651755c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member794313724012529841c_fm_i: set_li8269783359719514504c_fm_i > set_se3663484335793657320c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_A,type,
a: epistemic_fm_i > $o ).
thf(sy_v_Sa____,type,
sa: list_Epistemic_fm_i ).
thf(sy_v_V,type,
v: set_Epistemic_fm_i ).
thf(sy_v_a____,type,
a2: epistemic_fm_i ).
% Relevant facts (1275)
thf(fact_0_assms_I1_J,axiom,
episte2285483198712856234tent_i @ a @ v ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ v ).
% assms(2)
thf(fact_2_c1,axiom,
( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ a2 @ sa ) ) @ v )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ a2 @ sa ) ) ) )
=> ( ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ sa ) @ v )
& ( member6642669606046002379c_fm_i @ a2 @ v ) ) ) ).
% c1
thf(fact_3_local_OCons,axiom,
( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ sa ) @ v )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ sa ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ sa ) @ v ) ) ).
% local.Cons
thf(fact_4__092_060open_062set_AS_A_092_060subseteq_062_Aset_A_Ia_A_D_AS_J_092_060close_062,axiom,
ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ sa ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ a2 @ sa ) ) ).
% \<open>set S \<subseteq> set (a # S)\<close>
thf(fact_5_conjunct_Osimps_I2_J,axiom,
! [P: epistemic_fm_i,Ps: list_Epistemic_fm_i] :
( ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ P @ Ps ) )
= ( epistemic_Con_i @ P @ ( stalnaker_conjunct_i @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_6_list_Oinject,axiom,
! [X21: list_Epistemic_fm_i,X22: list_l4360594050116504750c_fm_i,Y21: list_Epistemic_fm_i,Y22: list_l4360594050116504750c_fm_i] :
( ( ( cons_l8703137435948174558c_fm_i @ X21 @ X22 )
= ( cons_l8703137435948174558c_fm_i @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_7_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_8_list_Oinject,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i,Y21: epistemic_fm_i,Y22: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X21 @ X22 )
= ( cons_Epistemic_fm_i @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_9__092_060open_062set_A_091_093_A_092_060subseteq_062_AV_A_092_060and_062_Afinite_A_Iset_A_091_093_J_A_092_060longrightarrow_062_Aconjunct_A_091_093_A_092_060in_062_AV_092_060close_062,axiom,
( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) @ v )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ nil_Epistemic_fm_i ) @ v ) ) ).
% \<open>set [] \<subseteq> V \<and> finite (set []) \<longrightarrow> conjunct [] \<in> V\<close>
thf(fact_10_not__Cons__self2,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( cons_l8703137435948174558c_fm_i @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_11_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_12_not__Cons__self2,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( cons_Epistemic_fm_i @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_13_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i,X21: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ X22 ) )
=> ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_14_list_Oset__intros_I2_J,axiom,
! [Y: set_se3485332733965609186c_fm_i,X22: list_s8635415444435394792c_fm_i,X21: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Y @ ( set_se5076022966804059383c_fm_i @ X22 ) )
=> ( member1461078328125707403c_fm_i @ Y @ ( set_se5076022966804059383c_fm_i @ ( cons_s5806790718070763416c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_15_list_Oset__intros_I2_J,axiom,
! [Y: set_nat,X22: list_set_nat,X21: set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X22 ) )
=> ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_16_list_Oset__intros_I2_J,axiom,
! [Y: list_Epistemic_fm_i,X22: list_l4360594050116504750c_fm_i,X21: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Y @ ( set_li9010496130497932989c_fm_i @ X22 ) )
=> ( member6475149752759350353c_fm_i @ Y @ ( set_li9010496130497932989c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_17_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_18_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_i,X22: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ X22 ) )
=> ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_19_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] : ( member1104366573291651755c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_20_list_Oset__intros_I1_J,axiom,
! [X21: set_se3485332733965609186c_fm_i,X22: list_s8635415444435394792c_fm_i] : ( member1461078328125707403c_fm_i @ X21 @ ( set_se5076022966804059383c_fm_i @ ( cons_s5806790718070763416c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_21_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] : ( member_set_nat2 @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_22_list_Oset__intros_I1_J,axiom,
! [X21: list_Epistemic_fm_i,X22: list_l4360594050116504750c_fm_i] : ( member6475149752759350353c_fm_i @ X21 @ ( set_li9010496130497932989c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_23_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_24_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_25_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ A ) )
=> ( ! [Z2: list_s8081015415394010888c_fm_i] :
( A
!= ( cons_s4962720389763977656c_fm_i @ E @ Z2 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z2: list_s8081015415394010888c_fm_i] :
( ( A
= ( cons_s4962720389763977656c_fm_i @ Z1 @ Z2 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_26_list_Oset__cases,axiom,
! [E: set_se3485332733965609186c_fm_i,A: list_s8635415444435394792c_fm_i] :
( ( member1461078328125707403c_fm_i @ E @ ( set_se5076022966804059383c_fm_i @ A ) )
=> ( ! [Z2: list_s8635415444435394792c_fm_i] :
( A
!= ( cons_s5806790718070763416c_fm_i @ E @ Z2 ) )
=> ~ ! [Z1: set_se3485332733965609186c_fm_i,Z2: list_s8635415444435394792c_fm_i] :
( ( A
= ( cons_s5806790718070763416c_fm_i @ Z1 @ Z2 ) )
=> ~ ( member1461078328125707403c_fm_i @ E @ ( set_se5076022966804059383c_fm_i @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_27_list_Oset__cases,axiom,
! [E: set_nat,A: list_set_nat] :
( ( member_set_nat2 @ E @ ( set_set_nat2 @ A ) )
=> ( ! [Z2: list_set_nat] :
( A
!= ( cons_set_nat @ E @ Z2 ) )
=> ~ ! [Z1: set_nat,Z2: list_set_nat] :
( ( A
= ( cons_set_nat @ Z1 @ Z2 ) )
=> ~ ( member_set_nat2 @ E @ ( set_set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_28_list_Oset__cases,axiom,
! [E: list_Epistemic_fm_i,A: list_l4360594050116504750c_fm_i] :
( ( member6475149752759350353c_fm_i @ E @ ( set_li9010496130497932989c_fm_i @ A ) )
=> ( ! [Z2: list_l4360594050116504750c_fm_i] :
( A
!= ( cons_l8703137435948174558c_fm_i @ E @ Z2 ) )
=> ~ ! [Z1: list_Epistemic_fm_i,Z2: list_l4360594050116504750c_fm_i] :
( ( A
= ( cons_l8703137435948174558c_fm_i @ Z1 @ Z2 ) )
=> ~ ( member6475149752759350353c_fm_i @ E @ ( set_li9010496130497932989c_fm_i @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_29_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_30_list_Oset__cases,axiom,
! [E: epistemic_fm_i,A: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ A ) )
=> ( ! [Z2: list_Epistemic_fm_i] :
( A
!= ( cons_Epistemic_fm_i @ E @ Z2 ) )
=> ~ ! [Z1: epistemic_fm_i,Z2: list_Epistemic_fm_i] :
( ( A
= ( cons_Epistemic_fm_i @ Z1 @ Z2 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_31_set__ConsD,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_32_set__ConsD,axiom,
! [Y: set_se3485332733965609186c_fm_i,X: set_se3485332733965609186c_fm_i,Xs: list_s8635415444435394792c_fm_i] :
( ( member1461078328125707403c_fm_i @ Y @ ( set_se5076022966804059383c_fm_i @ ( cons_s5806790718070763416c_fm_i @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member1461078328125707403c_fm_i @ Y @ ( set_se5076022966804059383c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_33_set__ConsD,axiom,
! [Y: set_nat,X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_34_set__ConsD,axiom,
! [Y: list_Epistemic_fm_i,X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( member6475149752759350353c_fm_i @ Y @ ( set_li9010496130497932989c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member6475149752759350353c_fm_i @ Y @ ( set_li9010496130497932989c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_35_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_36_set__ConsD,axiom,
! [Y: epistemic_fm_i,X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_37_member__rec_I1_J,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Y: list_Epistemic_fm_i] :
( ( member1162511396847685827c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member1162511396847685827c_fm_i @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_38_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_39_member__rec_I1_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ( member6038508299535739453c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member6038508299535739453c_fm_i @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_40_ShiftD,axiom,
! [Kl: list_l4360594050116504750c_fm_i,Kl2: set_li9045521773454279566c_fm_i,K: list_Epistemic_fm_i] :
( ( member5742582314450595671c_fm_i @ Kl @ ( bNF_Gr2889578550766211527c_fm_i @ Kl2 @ K ) )
=> ( member5742582314450595671c_fm_i @ ( cons_l8703137435948174558c_fm_i @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_41_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_42_ShiftD,axiom,
! [Kl: list_Epistemic_fm_i,Kl2: set_li8269783359719514504c_fm_i,K: epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Kl @ ( bNF_Gr8437504169225076033c_fm_i @ Kl2 @ K ) )
=> ( member6475149752759350353c_fm_i @ ( cons_Epistemic_fm_i @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_43_null__rec_I1_J,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
~ ( null_l7332364675292880694c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_44_null__rec_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_45_null__rec_I1_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
~ ( null_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_46_mcs__conjunction,axiom,
! [A2: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ V )
=> ( ( ( member6642669606046002379c_fm_i @ P @ V )
& ( member6642669606046002379c_fm_i @ Q @ V ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_47_List_Ofinite__set,axiom,
! [Xs: list_l4360594050116504750c_fm_i] : ( finite3106398504192302929c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_48_List_Ofinite__set,axiom,
! [Xs: list_Sum_sum_nat_nat] : ( finite6187706683773761046at_nat @ ( set_Sum_sum_nat_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_49_List_Ofinite__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i] : ( finite7933139204641697195c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_50_List_Ofinite__set,axiom,
! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_51_List_Ofinite__set,axiom,
! [Xs: list_E6081740836645470325c_fm_i] : ( finite4369126853184676732c_fm_i @ ( set_Ep2053541840733318416c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_52_List_Ofinite__set,axiom,
! [Xs: list_n5500797864759111951c_fm_i] : ( finite2015836717251302962c_fm_i @ ( set_na1706856518922123678c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_53_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_i] : ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% List.finite_set
thf(fact_54_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_55_null__rec_I2_J,axiom,
null_l7332364675292880694c_fm_i @ nil_li3019469239499002158c_fm_i ).
% null_rec(2)
thf(fact_56_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_57_null__rec_I2_J,axiom,
null_Epistemic_fm_i @ nil_Epistemic_fm_i ).
% null_rec(2)
thf(fact_58_member__rec_I2_J,axiom,
! [Y: list_Epistemic_fm_i] :
~ ( member1162511396847685827c_fm_i @ nil_li3019469239499002158c_fm_i @ Y ) ).
% member_rec(2)
thf(fact_59_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_60_member__rec_I2_J,axiom,
! [Y: epistemic_fm_i] :
~ ( member6038508299535739453c_fm_i @ nil_Epistemic_fm_i @ Y ) ).
% member_rec(2)
thf(fact_61_subset__code_I1_J,axiom,
! [Xs: list_s8635415444435394792c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ ( set_se5076022966804059383c_fm_i @ Xs ) @ B )
= ( ! [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ ( set_se5076022966804059383c_fm_i @ Xs ) )
=> ( member1461078328125707403c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_62_subset__code_I1_J,axiom,
! [Xs: list_l4360594050116504750c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) @ B )
= ( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( member6475149752759350353c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_63_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_64_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_65_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_66_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_67_transpose_Ocases,axiom,
! [X: list_l2400788496623510452c_fm_i] :
( ( X != nil_li3456212153916897716c_fm_i )
=> ( ! [Xss: list_l2400788496623510452c_fm_i] :
( X
!= ( cons_l756769862478126436c_fm_i @ nil_li3019469239499002158c_fm_i @ Xss ) )
=> ~ ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i,Xss: list_l2400788496623510452c_fm_i] :
( X
!= ( cons_l756769862478126436c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_68_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_69_transpose_Ocases,axiom,
! [X: list_l4360594050116504750c_fm_i] :
( ( X != nil_li3019469239499002158c_fm_i )
=> ( ! [Xss: list_l4360594050116504750c_fm_i] :
( X
!= ( cons_l8703137435948174558c_fm_i @ nil_Epistemic_fm_i @ Xss ) )
=> ~ ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i,Xss: list_l4360594050116504750c_fm_i] :
( X
!= ( cons_l8703137435948174558c_fm_i @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_70_eq__Nil__null,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( Xs = nil_li3019469239499002158c_fm_i )
= ( null_l7332364675292880694c_fm_i @ Xs ) ) ).
% eq_Nil_null
thf(fact_71_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_72_eq__Nil__null,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( Xs = nil_Epistemic_fm_i )
= ( null_Epistemic_fm_i @ Xs ) ) ).
% eq_Nil_null
thf(fact_73_finite__list,axiom,
! [A2: set_li8269783359719514504c_fm_i] :
( ( finite3106398504192302929c_fm_i @ A2 )
=> ? [Xs2: list_l4360594050116504750c_fm_i] :
( ( set_li9010496130497932989c_fm_i @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_74_finite__list,axiom,
! [A2: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ A2 )
=> ? [Xs2: list_Sum_sum_nat_nat] :
( ( set_Sum_sum_nat_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_75_finite__list,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ? [Xs2: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_76_finite__list,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ? [Xs2: list_set_nat] :
( ( set_set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_77_finite__list,axiom,
! [A2: set_Ep2190104726579546907c_fm_i] :
( ( finite4369126853184676732c_fm_i @ A2 )
=> ? [Xs2: list_E6081740836645470325c_fm_i] :
( ( set_Ep2053541840733318416c_fm_i @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_78_finite__list,axiom,
! [A2: set_na2176394682799156201c_fm_i] :
( ( finite2015836717251302962c_fm_i @ A2 )
=> ? [Xs2: list_n5500797864759111951c_fm_i] :
( ( set_na1706856518922123678c_fm_i @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_79_finite__list,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ? [Xs2: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_80_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_81_in__set__member,axiom,
! [X: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( member946333751272142877c_fm_i @ Xs @ X ) ) ).
% in_set_member
thf(fact_82_in__set__member,axiom,
! [X: set_se3485332733965609186c_fm_i,Xs: list_s8635415444435394792c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( set_se5076022966804059383c_fm_i @ Xs ) )
= ( member5074515665864507901c_fm_i @ Xs @ X ) ) ).
% in_set_member
thf(fact_83_in__set__member,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
= ( member_set_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_84_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_85_in__set__member,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
= ( member1162511396847685827c_fm_i @ Xs @ X ) ) ).
% in_set_member
thf(fact_86_in__set__member,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( member6038508299535739453c_fm_i @ Xs @ X ) ) ).
% in_set_member
thf(fact_87_list__nonempty__induct,axiom,
! [Xs: list_l4360594050116504750c_fm_i,P2: list_l4360594050116504750c_fm_i > $o] :
( ( Xs != nil_li3019469239499002158c_fm_i )
=> ( ! [X3: list_Epistemic_fm_i] : ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ nil_li3019469239499002158c_fm_i ) )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] :
( ( Xs2 != nil_li3019469239499002158c_fm_i )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_88_list__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_89_list__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( Xs != nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i] : ( P2 @ ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( Xs2 != nil_Epistemic_fm_i )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_90_list__induct2_H,axiom,
! [P2: list_Epistemic_fm_i > list_l4360594050116504750c_fm_i > $o,Xs: list_Epistemic_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( P2 @ nil_Epistemic_fm_i @ nil_li3019469239499002158c_fm_i )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] : ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ nil_li3019469239499002158c_fm_i )
=> ( ! [Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] : ( P2 @ nil_Epistemic_fm_i @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_91_list__induct2_H,axiom,
! [P2: list_Epistemic_fm_i > list_nat > $o,Xs: list_Epistemic_fm_i,Ys: list_nat] :
( ( P2 @ nil_Epistemic_fm_i @ nil_nat )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] : ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_Epistemic_fm_i @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i,Y2: nat,Ys2: list_nat] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_92_list__induct2_H,axiom,
! [P2: list_l4360594050116504750c_fm_i > list_Epistemic_fm_i > $o,Xs: list_l4360594050116504750c_fm_i,Ys: list_Epistemic_fm_i] :
( ( P2 @ nil_li3019469239499002158c_fm_i @ nil_Epistemic_fm_i )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] : ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ nil_Epistemic_fm_i )
=> ( ! [Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] : ( P2 @ nil_li3019469239499002158c_fm_i @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i,Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_93_list__induct2_H,axiom,
! [P2: list_l4360594050116504750c_fm_i > list_l4360594050116504750c_fm_i > $o,Xs: list_l4360594050116504750c_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( P2 @ nil_li3019469239499002158c_fm_i @ nil_li3019469239499002158c_fm_i )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] : ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ nil_li3019469239499002158c_fm_i )
=> ( ! [Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] : ( P2 @ nil_li3019469239499002158c_fm_i @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i,Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_94_list__induct2_H,axiom,
! [P2: list_l4360594050116504750c_fm_i > list_nat > $o,Xs: list_l4360594050116504750c_fm_i,Ys: list_nat] :
( ( P2 @ nil_li3019469239499002158c_fm_i @ nil_nat )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] : ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_li3019469239499002158c_fm_i @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i,Y2: nat,Ys2: list_nat] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_95_list__induct2_H,axiom,
! [P2: list_nat > list_Epistemic_fm_i > $o,Xs: list_nat,Ys: list_Epistemic_fm_i] :
( ( P2 @ nil_nat @ nil_Epistemic_fm_i )
=> ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_Epistemic_fm_i )
=> ( ! [Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] : ( P2 @ nil_nat @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_96_list__induct2_H,axiom,
! [P2: list_nat > list_l4360594050116504750c_fm_i > $o,Xs: list_nat,Ys: list_l4360594050116504750c_fm_i] :
( ( P2 @ nil_nat @ nil_li3019469239499002158c_fm_i )
=> ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_li3019469239499002158c_fm_i )
=> ( ! [Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] : ( P2 @ nil_nat @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: list_Epistemic_fm_i,Ys2: list_l4360594050116504750c_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_97_list__induct2_H,axiom,
! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_98_list__induct2_H,axiom,
! [P2: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( P2 @ nil_Epistemic_fm_i @ nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] : ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ nil_Epistemic_fm_i )
=> ( ! [Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] : ( P2 @ nil_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i,Y2: epistemic_fm_i,Ys2: list_Epistemic_fm_i] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_99_neq__Nil__conv,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( Xs != nil_li3019469239499002158c_fm_i )
= ( ? [Y3: list_Epistemic_fm_i,Ys3: list_l4360594050116504750c_fm_i] :
( Xs
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_100_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_101_neq__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( Xs != nil_Epistemic_fm_i )
= ( ? [Y3: epistemic_fm_i,Ys3: list_Epistemic_fm_i] :
( Xs
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_102_remdups__adj_Ocases,axiom,
! [X: list_l4360594050116504750c_fm_i] :
( ( X != nil_li3019469239499002158c_fm_i )
=> ( ! [X3: list_Epistemic_fm_i] :
( X
!= ( cons_l8703137435948174558c_fm_i @ X3 @ nil_li3019469239499002158c_fm_i ) )
=> ~ ! [X3: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] :
( X
!= ( cons_l8703137435948174558c_fm_i @ X3 @ ( cons_l8703137435948174558c_fm_i @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_103_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_104_remdups__adj_Ocases,axiom,
! [X: list_Epistemic_fm_i] :
( ( X != nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i] :
( X
!= ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) )
=> ~ ! [X3: epistemic_fm_i,Y2: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( X
!= ( cons_Epistemic_fm_i @ X3 @ ( cons_Epistemic_fm_i @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_105_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_106_list_Oexhaust,axiom,
! [Y: list_l4360594050116504750c_fm_i] :
( ( Y != nil_li3019469239499002158c_fm_i )
=> ~ ! [X212: list_Epistemic_fm_i,X222: list_l4360594050116504750c_fm_i] :
( Y
!= ( cons_l8703137435948174558c_fm_i @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_107_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_108_list_Oexhaust,axiom,
! [Y: list_Epistemic_fm_i] :
( ( Y != nil_Epistemic_fm_i )
=> ~ ! [X212: epistemic_fm_i,X222: list_Epistemic_fm_i] :
( Y
!= ( cons_Epistemic_fm_i @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_109_list_OdiscI,axiom,
! [List: list_l4360594050116504750c_fm_i,X21: list_Epistemic_fm_i,X22: list_l4360594050116504750c_fm_i] :
( ( List
= ( cons_l8703137435948174558c_fm_i @ X21 @ X22 ) )
=> ( List != nil_li3019469239499002158c_fm_i ) ) ).
% list.discI
thf(fact_110_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_111_list_OdiscI,axiom,
! [List: list_Epistemic_fm_i,X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( List
= ( cons_Epistemic_fm_i @ X21 @ X22 ) )
=> ( List != nil_Epistemic_fm_i ) ) ).
% list.discI
thf(fact_112_list_Odistinct_I1_J,axiom,
! [X21: list_Epistemic_fm_i,X22: list_l4360594050116504750c_fm_i] :
( nil_li3019469239499002158c_fm_i
!= ( cons_l8703137435948174558c_fm_i @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_113_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_114_list_Odistinct_I1_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( nil_Epistemic_fm_i
!= ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_115_mem__Collect__eq,axiom,
! [A: list_Epistemic_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( member6475149752759350353c_fm_i @ A @ ( collec5759350117362652307c_fm_i @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( member1461078328125707403c_fm_i @ A @ ( collec2761885132556217805c_fm_i @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( member_set_nat2 @ A @ ( collect_set_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
! [A: epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A2: set_li8269783359719514504c_fm_i] :
( ( collec5759350117362652307c_fm_i
@ ^ [X2: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A2: set_se7339729205154126530c_fm_i] :
( ( collec2761885132556217805c_fm_i
@ ^ [X2: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_127_Collect__cong,axiom,
! [P2: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec4904205187116291597c_fm_i @ P2 )
= ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_cong
thf(fact_128_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_129_set__subset__Cons,axiom,
! [Xs: list_l4360594050116504750c_fm_i,X: list_Epistemic_fm_i] : ( ord_le1209247765174941992c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) @ ( set_li9010496130497932989c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_130_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_131_set__subset__Cons,axiom,
! [Xs: list_s8081015415394010888c_fm_i,X: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_132_set__subset__Cons,axiom,
! [Xs: list_set_nat,X: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_133_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_134_inconsistent__finite,axiom,
! [A2: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A2 @ S )
=> ? [S2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
& ( finite3304564979551393739c_fm_i @ S2 )
& ~ ( episte2285483198712856234tent_i @ A2 @ S2 ) ) ) ).
% inconsistent_finite
thf(fact_135_maximal__extension,axiom,
! [A2: epistemic_fm_i > $o,V: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ V )
=> ~ ! [W: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V @ W )
=> ( ( episte2285483198712856234tent_i @ A2 @ W )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ W ) ) ) ) ).
% maximal_extension
thf(fact_136_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_137_is__empty__set,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( is_emp6097853100520738298c_fm_i @ ( set_li9010496130497932989c_fm_i @ Xs ) )
= ( null_l7332364675292880694c_fm_i @ Xs ) ) ).
% is_empty_set
thf(fact_138_is__empty__set,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( is_emp7643256682846039796c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( null_Epistemic_fm_i @ Xs ) ) ).
% is_empty_set
thf(fact_139_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( ( epistemic_Con_i @ X41 @ X42 )
= ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_140_subsetI,axiom,
! [A2: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ! [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ A2 )
=> ( member6475149752759350353c_fm_i @ X3 @ B ) )
=> ( ord_le1209247765174941992c_fm_i @ A2 @ B ) ) ).
% subsetI
thf(fact_141_subsetI,axiom,
! [A2: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A2 )
=> ( member1461078328125707403c_fm_i @ X3 @ B ) )
=> ( ord_le2772500130360476258c_fm_i @ A2 @ B ) ) ).
% subsetI
thf(fact_142_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( member_nat2 @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_143_subsetI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
=> ( member1104366573291651755c_fm_i @ X3 @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ B ) ) ).
% subsetI
thf(fact_144_subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
=> ( member_set_nat2 @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_145_subsetI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( member6642669606046002379c_fm_i @ X3 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% subsetI
thf(fact_146_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_147_subset__antisym,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_148_subset__antisym,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_149_subset__antisym,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_150_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_151_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_152_order__refl,axiom,
! [X: $o > set_Epistemic_fm_i] : ( ord_le8883210156954716975c_fm_i @ X @ X ) ).
% order_refl
thf(fact_153_order__refl,axiom,
! [X: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X @ X ) ).
% order_refl
thf(fact_154_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_155_order__refl,axiom,
! [X: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X @ X ) ).
% order_refl
thf(fact_156_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_157_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_158_dual__order_Orefl,axiom,
! [A: $o > set_Epistemic_fm_i] : ( ord_le8883210156954716975c_fm_i @ A @ A ) ).
% dual_order.refl
thf(fact_159_dual__order_Orefl,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ A ) ).
% dual_order.refl
thf(fact_160_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_161_dual__order_Orefl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% dual_order.refl
thf(fact_162_ax__in__maximal,axiom,
! [A2: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ V )
=> ( ( A2 @ P )
=> ( member6642669606046002379c_fm_i @ P @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_163_consistent__hereditary,axiom,
! [A2: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( episte2285483198712856234tent_i @ A2 @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_164_the__elem__set,axiom,
! [X: list_Epistemic_fm_i] :
( ( the_el9152279969184108317c_fm_i @ ( set_li9010496130497932989c_fm_i @ ( cons_l8703137435948174558c_fm_i @ X @ nil_li3019469239499002158c_fm_i ) ) )
= X ) ).
% the_elem_set
thf(fact_165_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_166_the__elem__set,axiom,
! [X: epistemic_fm_i] :
( ( the_el2173195912186371479c_fm_i @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) )
= X ) ).
% the_elem_set
thf(fact_167_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_168_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_169_order__antisym__conv,axiom,
! [Y: $o > set_Epistemic_fm_i,X: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ Y @ X )
=> ( ( ord_le8883210156954716975c_fm_i @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_170_order__antisym__conv,axiom,
! [Y: set_se3485332733965609186c_fm_i,X: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X )
=> ( ( ord_le5389487502678872194c_fm_i @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_171_order__antisym__conv,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_172_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_173_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_174_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_175_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_176_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_177_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_178_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_ord__le__eq__subst,axiom,
! [A: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_183_ord__le__eq__subst,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_nat,C: set_nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_184_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_185_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_186_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_187_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_188_ord__eq__le__subst,axiom,
! [A: nat,F: set_Epistemic_fm_i > nat,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_193_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_Epistemic_fm_i > set_nat,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_194_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_195_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_196_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_197_order__eq__refl,axiom,
! [X: $o > set_Epistemic_fm_i,Y: $o > set_Epistemic_fm_i] :
( ( X = Y )
=> ( ord_le8883210156954716975c_fm_i @ X @ Y ) ) ).
% order_eq_refl
thf(fact_198_order__eq__refl,axiom,
! [X: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( X = Y )
=> ( ord_le5389487502678872194c_fm_i @ X @ Y ) ) ).
% order_eq_refl
thf(fact_199_order__eq__refl,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( X = Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_200_order__eq__refl,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( X = Y )
=> ( ord_le3843937902494030498c_fm_i @ X @ Y ) ) ).
% order_eq_refl
thf(fact_201_order__subst2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_202_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_203_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_204_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_205_order__subst2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_206_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_207_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_208_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_209_order__subst2,axiom,
! [A: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_210_order__subst2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_nat,C: set_nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_211_order__subst1,axiom,
! [A: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_212_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_213_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_214_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_215_order__subst1,axiom,
! [A: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B2: nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_216_order__subst1,axiom,
! [A: nat,F: set_Epistemic_fm_i > nat,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_217_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_218_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_219_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_220_order__subst1,axiom,
! [A: set_Epistemic_fm_i,F: set_nat > set_Epistemic_fm_i,B2: set_nat,C: set_nat] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_221_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_222_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_223_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_Epistemic_fm_i,Z: $o > set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: $o > set_Epistemic_fm_i,B3: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ A3 @ B3 )
& ( ord_le8883210156954716975c_fm_i @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_224_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
& ( ord_le5389487502678872194c_fm_i @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_225_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_226_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_227_le__fun__def,axiom,
( ord_le8883210156954716975c_fm_i
= ( ^ [F2: $o > set_Epistemic_fm_i,G: $o > set_Epistemic_fm_i] :
! [X2: $o] : ( ord_le3843937902494030498c_fm_i @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_228_le__funI,axiom,
! [F: $o > set_Epistemic_fm_i,G2: $o > set_Epistemic_fm_i] :
( ! [X3: $o] : ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le8883210156954716975c_fm_i @ F @ G2 ) ) ).
% le_funI
thf(fact_229_le__funE,axiom,
! [F: $o > set_Epistemic_fm_i,G2: $o > set_Epistemic_fm_i,X: $o] :
( ( ord_le8883210156954716975c_fm_i @ F @ G2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_230_le__funD,axiom,
! [F: $o > set_Epistemic_fm_i,G2: $o > set_Epistemic_fm_i,X: $o] :
( ( ord_le8883210156954716975c_fm_i @ F @ G2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_231_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_232_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_233_antisym,axiom,
! [A: $o > set_Epistemic_fm_i,B2: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ A @ B2 )
=> ( ( ord_le8883210156954716975c_fm_i @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_234_antisym,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_235_antisym,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_236_antisym,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_237_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_238_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_239_dual__order_Otrans,axiom,
! [B2: $o > set_Epistemic_fm_i,A: $o > set_Epistemic_fm_i,C: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ B2 @ A )
=> ( ( ord_le8883210156954716975c_fm_i @ C @ B2 )
=> ( ord_le8883210156954716975c_fm_i @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_240_dual__order_Otrans,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ C @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_241_dual__order_Otrans,axiom,
! [B2: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_242_dual__order_Otrans,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_243_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_244_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_245_dual__order_Oantisym,axiom,
! [B2: $o > set_Epistemic_fm_i,A: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ B2 @ A )
=> ( ( ord_le8883210156954716975c_fm_i @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oantisym,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_248_dual__order_Oantisym,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_249_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_250_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_251_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > set_Epistemic_fm_i,Z: $o > set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: $o > set_Epistemic_fm_i,B3: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ B3 @ A3 )
& ( ord_le8883210156954716975c_fm_i @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_252_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B3 @ A3 )
& ( ord_le5389487502678872194c_fm_i @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_253_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
& ( ord_le6893508408891458716et_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_254_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
& ( ord_le3843937902494030498c_fm_i @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_255_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_256_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X: $o > set_Epistemic_fm_i,Y: $o > set_Epistemic_fm_i,Z3: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ X @ Y )
=> ( ( ord_le8883210156954716975c_fm_i @ Y @ Z3 )
=> ( ord_le8883210156954716975c_fm_i @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_259_order__trans,axiom,
! [X: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ Y @ Z3 )
=> ( ord_le5389487502678872194c_fm_i @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_260_order__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_261_order__trans,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_262_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_263_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_264_order_Otrans,axiom,
! [A: $o > set_Epistemic_fm_i,B2: $o > set_Epistemic_fm_i,C: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ A @ B2 )
=> ( ( ord_le8883210156954716975c_fm_i @ B2 @ C )
=> ( ord_le8883210156954716975c_fm_i @ A @ C ) ) ) ).
% order.trans
thf(fact_265_order_Otrans,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A @ C ) ) ) ).
% order.trans
thf(fact_266_order_Otrans,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_267_order_Otrans,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% order.trans
thf(fact_268_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_269_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_270_order__antisym,axiom,
! [X: $o > set_Epistemic_fm_i,Y: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ X @ Y )
=> ( ( ord_le8883210156954716975c_fm_i @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_271_order__antisym,axiom,
! [X: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_272_order__antisym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_273_order__antisym,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_274_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_275_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_276_ord__le__eq__trans,axiom,
! [A: $o > set_Epistemic_fm_i,B2: $o > set_Epistemic_fm_i,C: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le8883210156954716975c_fm_i @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_277_ord__le__eq__trans,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le5389487502678872194c_fm_i @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_278_ord__le__eq__trans,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_279_ord__le__eq__trans,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_280_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_281_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_282_ord__eq__le__trans,axiom,
! [A: $o > set_Epistemic_fm_i,B2: $o > set_Epistemic_fm_i,C: $o > set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ( ord_le8883210156954716975c_fm_i @ B2 @ C )
=> ( ord_le8883210156954716975c_fm_i @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_283_ord__eq__le__trans,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( A = B2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_284_ord__eq__le__trans,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( A = B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_285_ord__eq__le__trans,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_286_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_287_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_288_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_Epistemic_fm_i,Z: $o > set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [X2: $o > set_Epistemic_fm_i,Y3: $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ X2 @ Y3 )
& ( ord_le8883210156954716975c_fm_i @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_289_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y4 = Z ) )
= ( ^ [X2: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y3 )
& ( ord_le5389487502678872194c_fm_i @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_290_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
& ( ord_le6893508408891458716et_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_291_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y3 )
& ( ord_le3843937902494030498c_fm_i @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_292_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_293_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_294_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_295_Collect__mono__iff,axiom,
! [P2: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_296_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) )
= ( ! [X2: set_nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_297_Collect__mono__iff,axiom,
! [P2: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) )
= ( ! [X2: epistemic_fm_i] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_298_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_299_set__eq__subset,axiom,
( ( ^ [Y4: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y4 = Z ) )
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A5 @ B5 )
& ( ord_le5389487502678872194c_fm_i @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_300_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_301_set__eq__subset,axiom,
( ( ^ [Y4: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y4 = Z ) )
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_302_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_303_subset__trans,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_304_subset__trans,axiom,
! [A2: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_305_subset__trans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_306_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_307_Collect__mono,axiom,
! [P2: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ! [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_308_Collect__mono,axiom,
! [P2: set_nat > $o,Q2: set_nat > $o] :
( ! [X3: set_nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_309_Collect__mono,axiom,
! [P2: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_310_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_311_subset__refl,axiom,
! [A2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A2 @ A2 ) ).
% subset_refl
thf(fact_312_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_313_subset__refl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% subset_refl
thf(fact_314_subset__iff,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A5: set_li8269783359719514504c_fm_i,B5: set_li8269783359719514504c_fm_i] :
! [T: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ T @ A5 )
=> ( member6475149752759350353c_fm_i @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_315_subset__iff,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A5: set_se7339729205154126530c_fm_i,B5: set_se7339729205154126530c_fm_i] :
! [T: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ T @ A5 )
=> ( member1461078328125707403c_fm_i @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_316_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat2 @ T @ A5 )
=> ( member_nat2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_317_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A5 )
=> ( member1104366573291651755c_fm_i @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_318_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat2 @ T @ A5 )
=> ( member_set_nat2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_319_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A5 )
=> ( member6642669606046002379c_fm_i @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_320_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_321_equalityD2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A2 = B )
=> ( ord_le5389487502678872194c_fm_i @ B @ A2 ) ) ).
% equalityD2
thf(fact_322_equalityD2,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_323_equalityD2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2 = B )
=> ( ord_le3843937902494030498c_fm_i @ B @ A2 ) ) ).
% equalityD2
thf(fact_324_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_325_equalityD1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A2 = B )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ B ) ) ).
% equalityD1
thf(fact_326_equalityD1,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_327_equalityD1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2 = B )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% equalityD1
thf(fact_328_subset__eq,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A5: set_li8269783359719514504c_fm_i,B5: set_li8269783359719514504c_fm_i] :
! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ A5 )
=> ( member6475149752759350353c_fm_i @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_329_subset__eq,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A5: set_se7339729205154126530c_fm_i,B5: set_se7339729205154126530c_fm_i] :
! [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A5 )
=> ( member1461078328125707403c_fm_i @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_330_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A5 )
=> ( member_nat2 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_331_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A5 )
=> ( member1104366573291651755c_fm_i @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_332_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A5 )
=> ( member_set_nat2 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_333_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A5 )
=> ( member6642669606046002379c_fm_i @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_334_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_335_equalityE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A2 = B )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ~ ( ord_le5389487502678872194c_fm_i @ B @ A2 ) ) ) ).
% equalityE
thf(fact_336_equalityE,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_337_equalityE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2 = B )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ A2 ) ) ) ).
% equalityE
thf(fact_338_subsetD,axiom,
! [A2: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,C: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A2 @ B )
=> ( ( member6475149752759350353c_fm_i @ C @ A2 )
=> ( member6475149752759350353c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_339_subsetD,axiom,
! [A2: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ B )
=> ( ( member1461078328125707403c_fm_i @ C @ A2 )
=> ( member1461078328125707403c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_340_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_341_subsetD,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( member1104366573291651755c_fm_i @ C @ A2 )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_342_subsetD,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat2 @ C @ A2 )
=> ( member_set_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_343_subsetD,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_344_in__mono,axiom,
! [A2: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,X: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A2 @ B )
=> ( ( member6475149752759350353c_fm_i @ X @ A2 )
=> ( member6475149752759350353c_fm_i @ X @ B ) ) ) ).
% in_mono
thf(fact_345_in__mono,axiom,
! [A2: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,X: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ B )
=> ( ( member1461078328125707403c_fm_i @ X @ A2 )
=> ( member1461078328125707403c_fm_i @ X @ B ) ) ) ).
% in_mono
thf(fact_346_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ X @ B ) ) ) ).
% in_mono
thf(fact_347_in__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( member1104366573291651755c_fm_i @ X @ A2 )
=> ( member1104366573291651755c_fm_i @ X @ B ) ) ) ).
% in_mono
thf(fact_348_in__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat2 @ X @ A2 )
=> ( member_set_nat2 @ X @ B ) ) ) ).
% in_mono
thf(fact_349_in__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( member6642669606046002379c_fm_i @ X @ B ) ) ) ).
% in_mono
thf(fact_350_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat2 @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_351_finite__has__minimal2,axiom,
! [A2: set_o_4689017541235349509c_fm_i,A: $o > set_Epistemic_fm_i] :
( ( finite785680710116111718c_fm_i @ A2 )
=> ( ( member8251784763030151270c_fm_i @ A @ A2 )
=> ? [X3: $o > set_Epistemic_fm_i] :
( ( member8251784763030151270c_fm_i @ X3 @ A2 )
& ( ord_le8883210156954716975c_fm_i @ X3 @ A )
& ! [Xa: $o > set_Epistemic_fm_i] :
( ( member8251784763030151270c_fm_i @ Xa @ A2 )
=> ( ( ord_le8883210156954716975c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_352_finite__has__minimal2,axiom,
! [A2: set_se7339729205154126530c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A2 )
=> ( ( member1461078328125707403c_fm_i @ A @ A2 )
=> ? [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A2 )
& ( ord_le5389487502678872194c_fm_i @ X3 @ A )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_353_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( ord_le6893508408891458716et_nat @ X3 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_354_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_355_finite__has__minimal2,axiom,
! [A2: set_se3485332733965609186c_fm_i,A: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( member1104366573291651755c_fm_i @ A @ A2 )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ X3 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_356_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat2 @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_357_finite__has__maximal2,axiom,
! [A2: set_o_4689017541235349509c_fm_i,A: $o > set_Epistemic_fm_i] :
( ( finite785680710116111718c_fm_i @ A2 )
=> ( ( member8251784763030151270c_fm_i @ A @ A2 )
=> ? [X3: $o > set_Epistemic_fm_i] :
( ( member8251784763030151270c_fm_i @ X3 @ A2 )
& ( ord_le8883210156954716975c_fm_i @ A @ X3 )
& ! [Xa: $o > set_Epistemic_fm_i] :
( ( member8251784763030151270c_fm_i @ Xa @ A2 )
=> ( ( ord_le8883210156954716975c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_358_finite__has__maximal2,axiom,
! [A2: set_se7339729205154126530c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A2 )
=> ( ( member1461078328125707403c_fm_i @ A @ A2 )
=> ? [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A2 )
& ( ord_le5389487502678872194c_fm_i @ A @ X3 )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_359_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X3 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_360_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_361_finite__has__maximal2,axiom,
! [A2: set_se3485332733965609186c_fm_i,A: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( member1104366573291651755c_fm_i @ A @ A2 )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ A @ X3 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_362_rev__finite__subset,axiom,
! [B: set_Sum_sum_nat_nat,A2: set_Sum_sum_nat_nat] :
( ( finite6187706683773761046at_nat @ B )
=> ( ( ord_le5967974642961909525at_nat @ A2 @ B )
=> ( finite6187706683773761046at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_363_rev__finite__subset,axiom,
! [B: set_Ep2190104726579546907c_fm_i,A2: set_Ep2190104726579546907c_fm_i] :
( ( finite4369126853184676732c_fm_i @ B )
=> ( ( ord_le4359956957808536443c_fm_i @ A2 @ B )
=> ( finite4369126853184676732c_fm_i @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_364_rev__finite__subset,axiom,
! [B: set_na2176394682799156201c_fm_i,A2: set_na2176394682799156201c_fm_i] :
( ( finite2015836717251302962c_fm_i @ B )
=> ( ( ord_le7659027508473013129c_fm_i @ A2 @ B )
=> ( finite2015836717251302962c_fm_i @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_365_rev__finite__subset,axiom,
! [B: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( finite7933139204641697195c_fm_i @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_366_rev__finite__subset,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_367_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_368_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_369_infinite__super,axiom,
! [S: set_Sum_sum_nat_nat,T2: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ S @ T2 )
=> ( ~ ( finite6187706683773761046at_nat @ S )
=> ~ ( finite6187706683773761046at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_370_infinite__super,axiom,
! [S: set_Ep2190104726579546907c_fm_i,T2: set_Ep2190104726579546907c_fm_i] :
( ( ord_le4359956957808536443c_fm_i @ S @ T2 )
=> ( ~ ( finite4369126853184676732c_fm_i @ S )
=> ~ ( finite4369126853184676732c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_371_infinite__super,axiom,
! [S: set_na2176394682799156201c_fm_i,T2: set_na2176394682799156201c_fm_i] :
( ( ord_le7659027508473013129c_fm_i @ S @ T2 )
=> ( ~ ( finite2015836717251302962c_fm_i @ S )
=> ~ ( finite2015836717251302962c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_372_infinite__super,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ S @ T2 )
=> ( ~ ( finite7933139204641697195c_fm_i @ S )
=> ~ ( finite7933139204641697195c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_373_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_374_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_375_infinite__super,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_376_finite__subset,axiom,
! [A2: set_Sum_sum_nat_nat,B: set_Sum_sum_nat_nat] :
( ( ord_le5967974642961909525at_nat @ A2 @ B )
=> ( ( finite6187706683773761046at_nat @ B )
=> ( finite6187706683773761046at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_377_finite__subset,axiom,
! [A2: set_Ep2190104726579546907c_fm_i,B: set_Ep2190104726579546907c_fm_i] :
( ( ord_le4359956957808536443c_fm_i @ A2 @ B )
=> ( ( finite4369126853184676732c_fm_i @ B )
=> ( finite4369126853184676732c_fm_i @ A2 ) ) ) ).
% finite_subset
thf(fact_378_finite__subset,axiom,
! [A2: set_na2176394682799156201c_fm_i,B: set_na2176394682799156201c_fm_i] :
( ( ord_le7659027508473013129c_fm_i @ A2 @ B )
=> ( ( finite2015836717251302962c_fm_i @ B )
=> ( finite2015836717251302962c_fm_i @ A2 ) ) ) ).
% finite_subset
thf(fact_379_finite__subset,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( finite7933139204641697195c_fm_i @ A2 ) ) ) ).
% finite_subset
thf(fact_380_finite__subset,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_381_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_382_finite__subset,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_subset
thf(fact_383_product__lists_Osimps_I1_J,axiom,
( ( produc5002056178352495895c_fm_i @ nil_li3456212153916897716c_fm_i )
= ( cons_l756769862478126436c_fm_i @ nil_li3019469239499002158c_fm_i @ nil_li3456212153916897716c_fm_i ) ) ).
% product_lists.simps(1)
thf(fact_384_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_385_product__lists_Osimps_I1_J,axiom,
( ( produc1391074722258694289c_fm_i @ nil_li3019469239499002158c_fm_i )
= ( cons_l8703137435948174558c_fm_i @ nil_Epistemic_fm_i @ nil_li3019469239499002158c_fm_i ) ) ).
% product_lists.simps(1)
thf(fact_386_empty__Shift,axiom,
! [Kl2: set_li2934481837907835752c_fm_i,K: set_Epistemic_fm_i] :
( ( member605545779687026225c_fm_i @ nil_se670848885773549064c_fm_i @ Kl2 )
=> ( ( member1104366573291651755c_fm_i @ K @ ( bNF_Gr2376118584863155869c_fm_i @ Kl2 @ nil_se670848885773549064c_fm_i ) )
=> ( member605545779687026225c_fm_i @ nil_se670848885773549064c_fm_i @ ( bNF_Gr8275819806307007265c_fm_i @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_387_empty__Shift,axiom,
! [Kl2: set_li1928893501965799752c_fm_i,K: set_se3485332733965609186c_fm_i] :
( ( member1400835821191078929c_fm_i @ nil_se1681839564182969320c_fm_i @ Kl2 )
=> ( ( member1461078328125707403c_fm_i @ K @ ( bNF_Gr8592748930649239677c_fm_i @ Kl2 @ nil_se1681839564182969320c_fm_i ) )
=> ( member1400835821191078929c_fm_i @ nil_se1681839564182969320c_fm_i @ ( bNF_Gr5737545205521609985c_fm_i @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_388_empty__Shift,axiom,
! [Kl2: set_list_set_nat,K: set_nat] :
( ( member_list_set_nat @ nil_set_nat @ Kl2 )
=> ( ( member_set_nat2 @ K @ ( bNF_Gr3282828795834814635et_nat @ Kl2 @ nil_set_nat ) )
=> ( member_list_set_nat @ nil_set_nat @ ( bNF_Gr2891354507007493415et_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_389_empty__Shift,axiom,
! [Kl2: set_li9045521773454279566c_fm_i,K: list_Epistemic_fm_i] :
( ( member5742582314450595671c_fm_i @ nil_li3019469239499002158c_fm_i @ Kl2 )
=> ( ( member6475149752759350353c_fm_i @ K @ ( bNF_Gr7749281761897660227c_fm_i @ Kl2 @ nil_li3019469239499002158c_fm_i ) )
=> ( member5742582314450595671c_fm_i @ nil_li3019469239499002158c_fm_i @ ( bNF_Gr2889578550766211527c_fm_i @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_390_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl2 )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_391_empty__Shift,axiom,
! [Kl2: set_li8269783359719514504c_fm_i,K: epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ Kl2 )
=> ( ( member6642669606046002379c_fm_i @ K @ ( bNF_Gr1093487169986386109c_fm_i @ Kl2 @ nil_Epistemic_fm_i ) )
=> ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ ( bNF_Gr8437504169225076033c_fm_i @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_392_Succ__Shift,axiom,
! [Kl2: set_li9045521773454279566c_fm_i,K: list_Epistemic_fm_i,Kl: list_l4360594050116504750c_fm_i] :
( ( bNF_Gr7749281761897660227c_fm_i @ ( bNF_Gr2889578550766211527c_fm_i @ Kl2 @ K ) @ Kl )
= ( bNF_Gr7749281761897660227c_fm_i @ Kl2 @ ( cons_l8703137435948174558c_fm_i @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_393_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_394_Succ__Shift,axiom,
! [Kl2: set_li8269783359719514504c_fm_i,K: epistemic_fm_i,Kl: list_Epistemic_fm_i] :
( ( bNF_Gr1093487169986386109c_fm_i @ ( bNF_Gr8437504169225076033c_fm_i @ Kl2 @ K ) @ Kl )
= ( bNF_Gr1093487169986386109c_fm_i @ Kl2 @ ( cons_Epistemic_fm_i @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_395_subseqs_Osimps_I1_J,axiom,
( ( subseq3738671058005954685c_fm_i @ nil_li3019469239499002158c_fm_i )
= ( cons_l756769862478126436c_fm_i @ nil_li3019469239499002158c_fm_i @ nil_li3456212153916897716c_fm_i ) ) ).
% subseqs.simps(1)
thf(fact_396_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_397_subseqs_Osimps_I1_J,axiom,
( ( subseq859285874047815415c_fm_i @ nil_Epistemic_fm_i )
= ( cons_l8703137435948174558c_fm_i @ nil_Epistemic_fm_i @ nil_li3019469239499002158c_fm_i ) ) ).
% subseqs.simps(1)
thf(fact_398_subset__code_I3_J,axiom,
~ ( ord_le1209247765174941992c_fm_i @ ( coset_1955608913136020891c_fm_i @ nil_li3019469239499002158c_fm_i ) @ ( set_li9010496130497932989c_fm_i @ nil_li3019469239499002158c_fm_i ) ) ).
% subset_code(3)
thf(fact_399_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_400_subset__code_I3_J,axiom,
~ ( ord_le5389487502678872194c_fm_i @ ( coset_1342939534705115381c_fm_i @ nil_se670848885773549064c_fm_i ) @ ( set_se200842218512397079c_fm_i @ nil_se670848885773549064c_fm_i ) ) ).
% subset_code(3)
thf(fact_401_subset__code_I3_J,axiom,
~ ( ord_le6893508408891458716et_nat @ ( coset_set_nat @ nil_set_nat ) @ ( set_set_nat2 @ nil_set_nat ) ) ).
% subset_code(3)
thf(fact_402_subset__code_I3_J,axiom,
~ ( ord_le3843937902494030498c_fm_i @ ( coset_Epistemic_fm_i @ nil_Epistemic_fm_i ) @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% subset_code(3)
thf(fact_403_not__in__set__insert,axiom,
! [X: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X @ Xs )
= ( cons_s4962720389763977656c_fm_i @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_404_not__in__set__insert,axiom,
! [X: set_se3485332733965609186c_fm_i,Xs: list_s8635415444435394792c_fm_i] :
( ~ ( member1461078328125707403c_fm_i @ X @ ( set_se5076022966804059383c_fm_i @ Xs ) )
=> ( ( insert6197727299958409214c_fm_i @ X @ Xs )
= ( cons_s5806790718070763416c_fm_i @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_405_not__in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= ( cons_set_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_406_not__in__set__insert,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ~ ( member6475149752759350353c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( insert903435116114438724c_fm_i @ X @ Xs )
= ( cons_l8703137435948174558c_fm_i @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_407_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_408_not__in__set__insert,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X @ Xs )
= ( cons_Epistemic_fm_i @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_409_insert__Nil,axiom,
! [X: list_Epistemic_fm_i] :
( ( insert903435116114438724c_fm_i @ X @ nil_li3019469239499002158c_fm_i )
= ( cons_l8703137435948174558c_fm_i @ X @ nil_li3019469239499002158c_fm_i ) ) ).
% insert_Nil
thf(fact_410_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_411_insert__Nil,axiom,
! [X: epistemic_fm_i] :
( ( insert177310195918387262c_fm_i @ X @ nil_Epistemic_fm_i )
= ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) ).
% insert_Nil
thf(fact_412_list__ex1__simps_I1_J,axiom,
! [P2: list_Epistemic_fm_i > $o] :
~ ( list_e1940110243566727793c_fm_i @ P2 @ nil_li3019469239499002158c_fm_i ) ).
% list_ex1_simps(1)
thf(fact_413_list__ex1__simps_I1_J,axiom,
! [P2: nat > $o] :
~ ( list_ex1_nat @ P2 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_414_list__ex1__simps_I1_J,axiom,
! [P2: epistemic_fm_i > $o] :
~ ( list_e2031426328022727403c_fm_i @ P2 @ nil_Epistemic_fm_i ) ).
% list_ex1_simps(1)
thf(fact_415_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_416_map__tailrec__rev_Oelims,axiom,
! [X: epistemic_fm_i > list_Epistemic_fm_i,Xa2: list_Epistemic_fm_i,Xb: list_l4360594050116504750c_fm_i,Y: list_l4360594050116504750c_fm_i] :
( ( ( map_ta7280051635357636987c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_Epistemic_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: epistemic_fm_i,As: list_Epistemic_fm_i] :
( ( Xa2
= ( cons_Epistemic_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta7280051635357636987c_fm_i @ X @ As @ ( cons_l8703137435948174558c_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_417_map__tailrec__rev_Oelims,axiom,
! [X: epistemic_fm_i > nat,Xa2: list_Epistemic_fm_i,Xb: list_nat,Y: list_nat] :
( ( ( map_ta2419475119591317693_i_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_Epistemic_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: epistemic_fm_i,As: list_Epistemic_fm_i] :
( ( Xa2
= ( cons_Epistemic_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta2419475119591317693_i_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_418_map__tailrec__rev_Oelims,axiom,
! [X: list_Epistemic_fm_i > epistemic_fm_i,Xa2: list_l4360594050116504750c_fm_i,Xb: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( ( map_ta4902254756410701423c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_li3019469239499002158c_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: list_Epistemic_fm_i,As: list_l4360594050116504750c_fm_i] :
( ( Xa2
= ( cons_l8703137435948174558c_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta4902254756410701423c_fm_i @ X @ As @ ( cons_Epistemic_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_419_map__tailrec__rev_Oelims,axiom,
! [X: list_Epistemic_fm_i > list_Epistemic_fm_i,Xa2: list_l4360594050116504750c_fm_i,Xb: list_l4360594050116504750c_fm_i,Y: list_l4360594050116504750c_fm_i] :
( ( ( map_ta6471692776667290613c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_li3019469239499002158c_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: list_Epistemic_fm_i,As: list_l4360594050116504750c_fm_i] :
( ( Xa2
= ( cons_l8703137435948174558c_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta6471692776667290613c_fm_i @ X @ As @ ( cons_l8703137435948174558c_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_420_map__tailrec__rev_Oelims,axiom,
! [X: list_Epistemic_fm_i > nat,Xa2: list_l4360594050116504750c_fm_i,Xb: list_nat,Y: list_nat] :
( ( ( map_ta8021280163625540547_i_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_li3019469239499002158c_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: list_Epistemic_fm_i,As: list_l4360594050116504750c_fm_i] :
( ( Xa2
= ( cons_l8703137435948174558c_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta8021280163625540547_i_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_421_map__tailrec__rev_Oelims,axiom,
! [X: nat > epistemic_fm_i,Xa2: list_nat,Xb: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( ( map_ta4023091503004262811c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa2
= ( cons_nat @ A4 @ As ) )
=> ( Y
!= ( map_ta4023091503004262811c_fm_i @ X @ As @ ( cons_Epistemic_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_422_map__tailrec__rev_Oelims,axiom,
! [X: nat > list_Epistemic_fm_i,Xa2: list_nat,Xb: list_l4360594050116504750c_fm_i,Y: list_l4360594050116504750c_fm_i] :
( ( ( map_ta589061831645167905c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa2
= ( cons_nat @ A4 @ As ) )
=> ( Y
!= ( map_ta589061831645167905c_fm_i @ X @ As @ ( cons_l8703137435948174558c_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_423_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa2: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa2
= ( cons_nat @ A4 @ As ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_424_map__tailrec__rev_Oelims,axiom,
! [X: epistemic_fm_i > epistemic_fm_i,Xa2: list_Epistemic_fm_i,Xb: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( ( map_ta2236153606606974709c_fm_i @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_Epistemic_fm_i )
=> ( Y != Xb ) )
=> ~ ! [A4: epistemic_fm_i,As: list_Epistemic_fm_i] :
( ( Xa2
= ( cons_Epistemic_fm_i @ A4 @ As ) )
=> ( Y
!= ( map_ta2236153606606974709c_fm_i @ X @ As @ ( cons_Epistemic_fm_i @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_425_in__set__insert,axiom,
! [X: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_426_in__set__insert,axiom,
! [X: set_se3485332733965609186c_fm_i,Xs: list_s8635415444435394792c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( set_se5076022966804059383c_fm_i @ Xs ) )
=> ( ( insert6197727299958409214c_fm_i @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_427_in__set__insert,axiom,
! [X: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_428_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_429_in__set__insert,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( insert903435116114438724c_fm_i @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_430_in__set__insert,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_431_subseqs__refl,axiom,
! [Xs: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ Xs @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) ) ).
% subseqs_refl
thf(fact_432_Cons__in__subseqsD,axiom,
! [Y: list_Epistemic_fm_i,Ys: list_l4360594050116504750c_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( member5742582314450595671c_fm_i @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) @ ( set_li8148249020643214275c_fm_i @ ( subseq3738671058005954685c_fm_i @ Xs ) ) )
=> ( member5742582314450595671c_fm_i @ Ys @ ( set_li8148249020643214275c_fm_i @ ( subseq3738671058005954685c_fm_i @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_433_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_434_Cons__in__subseqsD,axiom,
! [Y: epistemic_fm_i,Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ ( cons_Epistemic_fm_i @ Y @ Ys ) @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) )
=> ( member6475149752759350353c_fm_i @ Ys @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_435_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_i > list_Epistemic_fm_i,A: epistemic_fm_i,As2: list_Epistemic_fm_i,Bs: list_l4360594050116504750c_fm_i] :
( ( map_ta7280051635357636987c_fm_i @ F @ ( cons_Epistemic_fm_i @ A @ As2 ) @ Bs )
= ( map_ta7280051635357636987c_fm_i @ F @ As2 @ ( cons_l8703137435948174558c_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_436_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_i > nat,A: epistemic_fm_i,As2: list_Epistemic_fm_i,Bs: list_nat] :
( ( map_ta2419475119591317693_i_nat @ F @ ( cons_Epistemic_fm_i @ A @ As2 ) @ Bs )
= ( map_ta2419475119591317693_i_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_437_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_i > epistemic_fm_i,A: list_Epistemic_fm_i,As2: list_l4360594050116504750c_fm_i,Bs: list_Epistemic_fm_i] :
( ( map_ta4902254756410701423c_fm_i @ F @ ( cons_l8703137435948174558c_fm_i @ A @ As2 ) @ Bs )
= ( map_ta4902254756410701423c_fm_i @ F @ As2 @ ( cons_Epistemic_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_438_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_i > list_Epistemic_fm_i,A: list_Epistemic_fm_i,As2: list_l4360594050116504750c_fm_i,Bs: list_l4360594050116504750c_fm_i] :
( ( map_ta6471692776667290613c_fm_i @ F @ ( cons_l8703137435948174558c_fm_i @ A @ As2 ) @ Bs )
= ( map_ta6471692776667290613c_fm_i @ F @ As2 @ ( cons_l8703137435948174558c_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_439_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_i > nat,A: list_Epistemic_fm_i,As2: list_l4360594050116504750c_fm_i,Bs: list_nat] :
( ( map_ta8021280163625540547_i_nat @ F @ ( cons_l8703137435948174558c_fm_i @ A @ As2 ) @ Bs )
= ( map_ta8021280163625540547_i_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_440_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > epistemic_fm_i,A: nat,As2: list_nat,Bs: list_Epistemic_fm_i] :
( ( map_ta4023091503004262811c_fm_i @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta4023091503004262811c_fm_i @ F @ As2 @ ( cons_Epistemic_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_441_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > list_Epistemic_fm_i,A: nat,As2: list_nat,Bs: list_l4360594050116504750c_fm_i] :
( ( map_ta589061831645167905c_fm_i @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta589061831645167905c_fm_i @ F @ As2 @ ( cons_l8703137435948174558c_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_442_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A: nat,As2: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_443_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: epistemic_fm_i,As2: list_Epistemic_fm_i,Bs: list_Epistemic_fm_i] :
( ( map_ta2236153606606974709c_fm_i @ F @ ( cons_Epistemic_fm_i @ A @ As2 ) @ Bs )
= ( map_ta2236153606606974709c_fm_i @ F @ As2 @ ( cons_Epistemic_fm_i @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_444_map__tailrec__rev_Osimps_I1_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Bs: list_Epistemic_fm_i] :
( ( map_ta2236153606606974709c_fm_i @ F @ nil_Epistemic_fm_i @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_445_list__ex1__iff,axiom,
( list_e3983829164705222859c_fm_i
= ( ^ [P3: set_Epistemic_fm_i > $o,Xs3: list_s8081015415394010888c_fm_i] :
? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: set_Epistemic_fm_i] :
( ( ( member1104366573291651755c_fm_i @ Y3 @ ( set_se200842218512397079c_fm_i @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_446_list__ex1__iff,axiom,
( list_e1886195200064692907c_fm_i
= ( ^ [P3: set_se3485332733965609186c_fm_i > $o,Xs3: list_s8635415444435394792c_fm_i] :
? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ ( set_se5076022966804059383c_fm_i @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: set_se3485332733965609186c_fm_i] :
( ( ( member1461078328125707403c_fm_i @ Y3 @ ( set_se5076022966804059383c_fm_i @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_447_list__ex1__iff,axiom,
( list_ex1_set_nat
= ( ^ [P3: set_nat > $o,Xs3: list_set_nat] :
? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: set_nat] :
( ( ( member_set_nat2 @ Y3 @ ( set_set_nat2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_448_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P3: nat > $o,Xs3: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_449_list__ex1__iff,axiom,
( list_e1940110243566727793c_fm_i
= ( ^ [P3: list_Epistemic_fm_i > $o,Xs3: list_l4360594050116504750c_fm_i] :
? [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: list_Epistemic_fm_i] :
( ( ( member6475149752759350353c_fm_i @ Y3 @ ( set_li9010496130497932989c_fm_i @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_450_list__ex1__iff,axiom,
( list_e2031426328022727403c_fm_i
= ( ^ [P3: epistemic_fm_i > $o,Xs3: list_Epistemic_fm_i] :
? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs3 ) )
& ( P3 @ X2 )
& ! [Y3: epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_451_subset__code_I2_J,axiom,
! [A2: set_se7339729205154126530c_fm_i,Ys: list_s8635415444435394792c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A2 @ ( coset_5953259889648610005c_fm_i @ Ys ) )
= ( ! [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ ( set_se5076022966804059383c_fm_i @ Ys ) )
=> ~ ( member1461078328125707403c_fm_i @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_452_subset__code_I2_J,axiom,
! [A2: set_li8269783359719514504c_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A2 @ ( coset_1955608913136020891c_fm_i @ Ys ) )
= ( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Ys ) )
=> ~ ( member6475149752759350353c_fm_i @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_453_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_454_subset__code_I2_J,axiom,
! [A2: set_se3485332733965609186c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( coset_1342939534705115381c_fm_i @ Ys ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_455_subset__code_I2_J,axiom,
! [A2: set_set_nat,Ys: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( coset_set_nat @ Ys ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ~ ( member_set_nat2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_456_subset__code_I2_J,axiom,
! [A2: set_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( coset_Epistemic_fm_i @ Ys ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_457_List_Oinsert__def,axiom,
( insert2436375702790685214c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Xs3: list_s8081015415394010888c_fm_i] : ( if_lis2088983353317256142c_fm_i @ ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs3 ) ) @ Xs3 @ ( cons_s4962720389763977656c_fm_i @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_458_List_Oinsert__def,axiom,
( insert6197727299958409214c_fm_i
= ( ^ [X2: set_se3485332733965609186c_fm_i,Xs3: list_s8635415444435394792c_fm_i] : ( if_lis4466469686133009838c_fm_i @ ( member1461078328125707403c_fm_i @ X2 @ ( set_se5076022966804059383c_fm_i @ Xs3 ) ) @ Xs3 @ ( cons_s5806790718070763416c_fm_i @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_459_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X2: set_nat,Xs3: list_set_nat] : ( if_list_set_nat @ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_set_nat @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_460_List_Oinsert__def,axiom,
( insert903435116114438724c_fm_i
= ( ^ [X2: list_Epistemic_fm_i,Xs3: list_l4360594050116504750c_fm_i] : ( if_lis7843061787625270900c_fm_i @ ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xs3 ) ) @ Xs3 @ ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_461_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_462_List_Oinsert__def,axiom,
( insert177310195918387262c_fm_i
= ( ^ [X2: epistemic_fm_i,Xs3: list_Epistemic_fm_i] : ( if_lis3446954105117697518c_fm_i @ ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_i @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_463_can__select__set__list__ex1,axiom,
! [P2: nat > $o,A2: list_nat] :
( ( can_select_nat @ P2 @ ( set_nat2 @ A2 ) )
= ( list_ex1_nat @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_464_can__select__set__list__ex1,axiom,
! [P2: list_Epistemic_fm_i > $o,A2: list_l4360594050116504750c_fm_i] :
( ( can_se8059536771417300901c_fm_i @ P2 @ ( set_li9010496130497932989c_fm_i @ A2 ) )
= ( list_e1940110243566727793c_fm_i @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_465_can__select__set__list__ex1,axiom,
! [P2: epistemic_fm_i > $o,A2: list_Epistemic_fm_i] :
( ( can_se5173380744702956063c_fm_i @ P2 @ ( set_Epistemic_fm_i2 @ A2 ) )
= ( list_e2031426328022727403c_fm_i @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_466_remove__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_467_remove__code_I2_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( remove5552923401211475591c_fm_i @ X @ ( coset_Epistemic_fm_i @ Xs ) )
= ( coset_Epistemic_fm_i @ ( insert177310195918387262c_fm_i @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_468_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
~ ( lexord3686779039239843700c_fm_i @ Less @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ nil_li3019469239499002158c_fm_i ) ).
% ord.lexordp_eq_simps(3)
thf(fact_469_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_470_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
~ ( lexord491902653470561646c_fm_i @ Less @ ( cons_Epistemic_fm_i @ X @ Xs ) @ nil_Epistemic_fm_i ) ).
% ord.lexordp_eq_simps(3)
thf(fact_471_bind__simps_I1_J,axiom,
! [F: epistemic_fm_i > list_l4360594050116504750c_fm_i] :
( ( bind_E8896667925115755021c_fm_i @ nil_Epistemic_fm_i @ F )
= nil_li3019469239499002158c_fm_i ) ).
% bind_simps(1)
thf(fact_472_bind__simps_I1_J,axiom,
! [F: epistemic_fm_i > list_nat] :
( ( bind_E7905432516484261803_i_nat @ nil_Epistemic_fm_i @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_473_bind__simps_I1_J,axiom,
! [F: list_Epistemic_fm_i > list_Epistemic_fm_i] :
( ( bind_l6518871046168819457c_fm_i @ nil_li3019469239499002158c_fm_i @ F )
= nil_Epistemic_fm_i ) ).
% bind_simps(1)
thf(fact_474_bind__simps_I1_J,axiom,
! [F: list_Epistemic_fm_i > list_l4360594050116504750c_fm_i] :
( ( bind_l7810477099806436231c_fm_i @ nil_li3019469239499002158c_fm_i @ F )
= nil_li3019469239499002158c_fm_i ) ).
% bind_simps(1)
thf(fact_475_bind__simps_I1_J,axiom,
! [F: list_Epistemic_fm_i > list_nat] :
( ( bind_l3617442327463547825_i_nat @ nil_li3019469239499002158c_fm_i @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_476_bind__simps_I1_J,axiom,
! [F: nat > list_Epistemic_fm_i] :
( ( bind_n285676863042431113c_fm_i @ nil_nat @ F )
= nil_Epistemic_fm_i ) ).
% bind_simps(1)
thf(fact_477_bind__simps_I1_J,axiom,
! [F: nat > list_l4360594050116504750c_fm_i] :
( ( bind_n5408596032337950991c_fm_i @ nil_nat @ F )
= nil_li3019469239499002158c_fm_i ) ).
% bind_simps(1)
thf(fact_478_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_479_bind__simps_I1_J,axiom,
! [F: epistemic_fm_i > list_Epistemic_fm_i] :
( ( bind_E4122511559465885831c_fm_i @ nil_Epistemic_fm_i @ F )
= nil_Epistemic_fm_i ) ).
% bind_simps(1)
thf(fact_480_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_481_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_482_GreatestI2__order,axiom,
! [P2: nat > $o,X: nat,Q2: nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_483_GreatestI2__order,axiom,
! [P2: set_nat > $o,X: set_nat,Q2: set_nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ! [X3: set_nat] :
( ( P2 @ X3 )
=> ( ! [Y5: set_nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_5724808138429204845et_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_484_GreatestI2__order,axiom,
! [P2: ( $o > set_Epistemic_fm_i ) > $o,X: $o > set_Epistemic_fm_i,Q2: ( $o > set_Epistemic_fm_i ) > $o] :
( ( P2 @ X )
=> ( ! [Y2: $o > set_Epistemic_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le8883210156954716975c_fm_i @ Y2 @ X ) )
=> ( ! [X3: $o > set_Epistemic_fm_i] :
( ( P2 @ X3 )
=> ( ! [Y5: $o > set_Epistemic_fm_i] :
( ( P2 @ Y5 )
=> ( ord_le8883210156954716975c_fm_i @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_1616619477128590774c_fm_i @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_485_GreatestI2__order,axiom,
! [P2: set_se3485332733965609186c_fm_i > $o,X: set_se3485332733965609186c_fm_i,Q2: set_se3485332733965609186c_fm_i > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_se3485332733965609186c_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ Y2 @ X ) )
=> ( ! [X3: set_se3485332733965609186c_fm_i] :
( ( P2 @ X3 )
=> ( ! [Y5: set_se3485332733965609186c_fm_i] :
( ( P2 @ Y5 )
=> ( ord_le5389487502678872194c_fm_i @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_6562264480300907323c_fm_i @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_486_GreatestI2__order,axiom,
! [P2: set_set_nat > $o,X: set_set_nat,Q2: set_set_nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_set_nat] :
( ( P2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X ) )
=> ( ! [X3: set_set_nat] :
( ( P2 @ X3 )
=> ( ! [Y5: set_set_nat] :
( ( P2 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_1279421399067128355et_nat @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_487_GreatestI2__order,axiom,
! [P2: set_Epistemic_fm_i > $o,X: set_Epistemic_fm_i,Q2: set_Epistemic_fm_i > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_Epistemic_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ Y2 @ X ) )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
=> ( ! [Y5: set_Epistemic_fm_i] :
( ( P2 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ Y5 @ X3 ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( order_5154021046103009627c_fm_i @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_488_Greatest__equality,axiom,
! [P2: nat > $o,X: nat] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_489_Greatest__equality,axiom,
! [P2: set_nat > $o,X: set_nat] :
( ( P2 @ X )
=> ( ! [Y2: set_nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_490_Greatest__equality,axiom,
! [P2: ( $o > set_Epistemic_fm_i ) > $o,X: $o > set_Epistemic_fm_i] :
( ( P2 @ X )
=> ( ! [Y2: $o > set_Epistemic_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le8883210156954716975c_fm_i @ Y2 @ X ) )
=> ( ( order_1616619477128590774c_fm_i @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_491_Greatest__equality,axiom,
! [P2: set_se3485332733965609186c_fm_i > $o,X: set_se3485332733965609186c_fm_i] :
( ( P2 @ X )
=> ( ! [Y2: set_se3485332733965609186c_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ Y2 @ X ) )
=> ( ( order_6562264480300907323c_fm_i @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_492_Greatest__equality,axiom,
! [P2: set_set_nat > $o,X: set_set_nat] :
( ( P2 @ X )
=> ( ! [Y2: set_set_nat] :
( ( P2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X ) )
=> ( ( order_1279421399067128355et_nat @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_493_Greatest__equality,axiom,
! [P2: set_Epistemic_fm_i > $o,X: set_Epistemic_fm_i] :
( ( P2 @ X )
=> ( ! [Y2: set_Epistemic_fm_i] :
( ( P2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ Y2 @ X ) )
=> ( ( order_5154021046103009627c_fm_i @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_494_listrelp_Ocases,axiom,
! [R: epistemic_fm_i > list_Epistemic_fm_i > $o,A1: list_Epistemic_fm_i,A22: list_l4360594050116504750c_fm_i] :
( ( listre7628413289575147717c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_i )
=> ( A22 != nil_li3019469239499002158c_fm_i ) )
=> ~ ! [X3: epistemic_fm_i,Y2: list_Epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( A1
= ( cons_Epistemic_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_l4360594050116504750c_fm_i] :
( ( A22
= ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre7628413289575147717c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_495_listrelp_Ocases,axiom,
! [R: epistemic_fm_i > nat > $o,A1: list_Epistemic_fm_i,A22: list_nat] :
( ( listre7650074605423786739_i_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_i )
=> ( A22 != nil_nat ) )
=> ~ ! [X3: epistemic_fm_i,Y2: nat,Xs2: list_Epistemic_fm_i] :
( ( A1
= ( cons_Epistemic_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre7650074605423786739_i_nat @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_496_listrelp_Ocases,axiom,
! [R: list_Epistemic_fm_i > epistemic_fm_i > $o,A1: list_l4360594050116504750c_fm_i,A22: list_Epistemic_fm_i] :
( ( listre5250616410628212153c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_li3019469239499002158c_fm_i )
=> ( A22 != nil_Epistemic_fm_i ) )
=> ~ ! [X3: list_Epistemic_fm_i,Y2: epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] :
( ( A1
= ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_i] :
( ( A22
= ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre5250616410628212153c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_497_listrelp_Ocases,axiom,
! [R: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,A1: list_l4360594050116504750c_fm_i,A22: list_l4360594050116504750c_fm_i] :
( ( listre5721989502611914303c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_li3019469239499002158c_fm_i )
=> ( A22 != nil_li3019469239499002158c_fm_i ) )
=> ~ ! [X3: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] :
( ( A1
= ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_l4360594050116504750c_fm_i] :
( ( A22
= ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre5721989502611914303c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_498_listrelp_Ocases,axiom,
! [R: list_Epistemic_fm_i > nat > $o,A1: list_l4360594050116504750c_fm_i,A22: list_nat] :
( ( listre3862057929835193081_i_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_li3019469239499002158c_fm_i )
=> ( A22 != nil_nat ) )
=> ~ ! [X3: list_Epistemic_fm_i,Y2: nat,Xs2: list_l4360594050116504750c_fm_i] :
( ( A1
= ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre3862057929835193081_i_nat @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_499_listrelp_Ocases,axiom,
! [R: nat > epistemic_fm_i > $o,A1: list_nat,A22: list_Epistemic_fm_i] :
( ( listre30318951981956049c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_Epistemic_fm_i ) )
=> ~ ! [X3: nat,Y2: epistemic_fm_i,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_i] :
( ( A22
= ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre30318951981956049c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_500_listrelp_Ocases,axiom,
! [R: nat > list_Epistemic_fm_i > $o,A1: list_nat,A22: list_l4360594050116504750c_fm_i] :
( ( listre5653211634709596247c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_li3019469239499002158c_fm_i ) )
=> ~ ! [X3: nat,Y2: list_Epistemic_fm_i,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_l4360594050116504750c_fm_i] :
( ( A22
= ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre5653211634709596247c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_501_listrelp_Ocases,axiom,
! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( listrelp_nat_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listrelp_nat_nat @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_502_listrelp_Ocases,axiom,
! [R: epistemic_fm_i > epistemic_fm_i > $o,A1: list_Epistemic_fm_i,A22: list_Epistemic_fm_i] :
( ( listre3501123205157137215c_fm_i @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_i )
=> ( A22 != nil_Epistemic_fm_i ) )
=> ~ ! [X3: epistemic_fm_i,Y2: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( A1
= ( cons_Epistemic_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_i] :
( ( A22
= ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre3501123205157137215c_fm_i @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_503_member__remove,axiom,
! [X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i,A2: set_li8269783359719514504c_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( remove1193922379842187277c_fm_i @ Y @ A2 ) )
= ( ( member6475149752759350353c_fm_i @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_504_member__remove,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( remove1392734035364174951c_fm_i @ Y @ A2 ) )
= ( ( member1104366573291651755c_fm_i @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_505_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_506_member__remove,axiom,
! [X: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,A2: set_se7339729205154126530c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( remove2615947271592508999c_fm_i @ Y @ A2 ) )
= ( ( member1461078328125707403c_fm_i @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_507_member__remove,axiom,
! [X: set_nat,Y: set_nat,A2: set_set_nat] :
( ( member_set_nat2 @ X @ ( remove_set_nat @ Y @ A2 ) )
= ( ( member_set_nat2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_508_member__remove,axiom,
! [X: epistemic_fm_i,Y: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( remove5552923401211475591c_fm_i @ Y @ A2 ) )
= ( ( member6642669606046002379c_fm_i @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_509_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Y: list_Epistemic_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( lexord3686779039239843700c_fm_i @ Less @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord3686779039239843700c_fm_i @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_510_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_nat @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_511_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( lexord491902653470561646c_fm_i @ Less @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord491902653470561646c_fm_i @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_512_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,Ys: list_l4360594050116504750c_fm_i] : ( lexord3686779039239843700c_fm_i @ Less @ nil_li3019469239499002158c_fm_i @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_513_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_514_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,Ys: list_Epistemic_fm_i] : ( lexord491902653470561646c_fm_i @ Less @ nil_Epistemic_fm_i @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_515_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,Xs: list_l4360594050116504750c_fm_i] :
( ( lexord3686779039239843700c_fm_i @ Less @ Xs @ nil_li3019469239499002158c_fm_i )
= ( Xs = nil_li3019469239499002158c_fm_i ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_516_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_517_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( lexord491902653470561646c_fm_i @ Less @ Xs @ nil_Epistemic_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_518_ord_Olexordp__eq_Ocong,axiom,
lexord491902653470561646c_fm_i = lexord491902653470561646c_fm_i ).
% ord.lexordp_eq.cong
thf(fact_519_ord_Olexordp__eq__refl,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( lexord491902653470561646c_fm_i @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_520_can__select__def,axiom,
( can_se8059536771417300901c_fm_i
= ( ^ [P3: list_Epistemic_fm_i > $o,A5: set_li8269783359719514504c_fm_i] :
? [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: list_Epistemic_fm_i] :
( ( ( member6475149752759350353c_fm_i @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_521_can__select__def,axiom,
( can_se8904854918057400319c_fm_i
= ( ^ [P3: set_Epistemic_fm_i > $o,A5: set_se3485332733965609186c_fm_i] :
? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: set_Epistemic_fm_i] :
( ( ( member1104366573291651755c_fm_i @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_522_can__select__def,axiom,
( can_select_nat
= ( ^ [P3: nat > $o,A5: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_523_can__select__def,axiom,
( can_se1297080968488376799c_fm_i
= ( ^ [P3: set_se3485332733965609186c_fm_i > $o,A5: set_se7339729205154126530c_fm_i] :
? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: set_se3485332733965609186c_fm_i] :
( ( ( member1461078328125707403c_fm_i @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_524_can__select__def,axiom,
( can_select_set_nat
= ( ^ [P3: set_nat > $o,A5: set_set_nat] :
? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: set_nat] :
( ( ( member_set_nat2 @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_525_can__select__def,axiom,
( can_se5173380744702956063c_fm_i
= ( ^ [P3: epistemic_fm_i > $o,A5: set_Epistemic_fm_i] :
? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A5 )
& ( P3 @ X2 )
& ! [Y3: epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_526_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( A2 = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_527_ord_Olexordp__eq_OCons,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( Less @ X @ Y )
=> ( lexord3686779039239843700c_fm_i @ Less @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_528_ord_Olexordp__eq_OCons,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( Less @ X @ Y )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_529_ord_Olexordp__eq_OCons,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i,Y: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( Less @ X @ Y )
=> ( lexord491902653470561646c_fm_i @ Less @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_530_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexord3686779039239843700c_fm_i @ Less @ Xs @ Ys )
=> ( lexord3686779039239843700c_fm_i @ Less @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_531_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_nat @ Less @ Xs @ Ys )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_532_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i,Y: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexord491902653470561646c_fm_i @ Less @ Xs @ Ys )
=> ( lexord491902653470561646c_fm_i @ Less @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_533_ord_Olexordp__eq_ONil,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,Ys: list_l4360594050116504750c_fm_i] : ( lexord3686779039239843700c_fm_i @ Less @ nil_li3019469239499002158c_fm_i @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_534_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_535_ord_Olexordp__eq_ONil,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,Ys: list_Epistemic_fm_i] : ( lexord491902653470561646c_fm_i @ Less @ nil_Epistemic_fm_i @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_536_listrelp_OCons,axiom,
! [R: epistemic_fm_i > list_Epistemic_fm_i > $o,X: epistemic_fm_i,Y: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( R @ X @ Y )
=> ( ( listre7628413289575147717c_fm_i @ R @ Xs @ Ys )
=> ( listre7628413289575147717c_fm_i @ R @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_537_listrelp_OCons,axiom,
! [R: epistemic_fm_i > nat > $o,X: epistemic_fm_i,Y: nat,Xs: list_Epistemic_fm_i,Ys: list_nat] :
( ( R @ X @ Y )
=> ( ( listre7650074605423786739_i_nat @ R @ Xs @ Ys )
=> ( listre7650074605423786739_i_nat @ R @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_538_listrelp_OCons,axiom,
! [R: list_Epistemic_fm_i > epistemic_fm_i > $o,X: list_Epistemic_fm_i,Y: epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Ys: list_Epistemic_fm_i] :
( ( R @ X @ Y )
=> ( ( listre5250616410628212153c_fm_i @ R @ Xs @ Ys )
=> ( listre5250616410628212153c_fm_i @ R @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_539_listrelp_OCons,axiom,
! [R: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i,Ys: list_l4360594050116504750c_fm_i] :
( ( R @ X @ Y )
=> ( ( listre5721989502611914303c_fm_i @ R @ Xs @ Ys )
=> ( listre5721989502611914303c_fm_i @ R @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_540_listrelp_OCons,axiom,
! [R: list_Epistemic_fm_i > nat > $o,X: list_Epistemic_fm_i,Y: nat,Xs: list_l4360594050116504750c_fm_i,Ys: list_nat] :
( ( R @ X @ Y )
=> ( ( listre3862057929835193081_i_nat @ R @ Xs @ Ys )
=> ( listre3862057929835193081_i_nat @ R @ ( cons_l8703137435948174558c_fm_i @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_541_listrelp_OCons,axiom,
! [R: nat > epistemic_fm_i > $o,X: nat,Y: epistemic_fm_i,Xs: list_nat,Ys: list_Epistemic_fm_i] :
( ( R @ X @ Y )
=> ( ( listre30318951981956049c_fm_i @ R @ Xs @ Ys )
=> ( listre30318951981956049c_fm_i @ R @ ( cons_nat @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_542_listrelp_OCons,axiom,
! [R: nat > list_Epistemic_fm_i > $o,X: nat,Y: list_Epistemic_fm_i,Xs: list_nat,Ys: list_l4360594050116504750c_fm_i] :
( ( R @ X @ Y )
=> ( ( listre5653211634709596247c_fm_i @ R @ Xs @ Ys )
=> ( listre5653211634709596247c_fm_i @ R @ ( cons_nat @ X @ Xs ) @ ( cons_l8703137435948174558c_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_543_listrelp_OCons,axiom,
! [R: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( R @ X @ Y )
=> ( ( listrelp_nat_nat @ R @ Xs @ Ys )
=> ( listrelp_nat_nat @ R @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_544_listrelp_OCons,axiom,
! [R: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i,Y: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( R @ X @ Y )
=> ( ( listre3501123205157137215c_fm_i @ R @ Xs @ Ys )
=> ( listre3501123205157137215c_fm_i @ R @ ( cons_Epistemic_fm_i @ X @ Xs ) @ ( cons_Epistemic_fm_i @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_545_listrelp_ONil,axiom,
! [R: epistemic_fm_i > list_Epistemic_fm_i > $o] : ( listre7628413289575147717c_fm_i @ R @ nil_Epistemic_fm_i @ nil_li3019469239499002158c_fm_i ) ).
% listrelp.Nil
thf(fact_546_listrelp_ONil,axiom,
! [R: epistemic_fm_i > nat > $o] : ( listre7650074605423786739_i_nat @ R @ nil_Epistemic_fm_i @ nil_nat ) ).
% listrelp.Nil
thf(fact_547_listrelp_ONil,axiom,
! [R: list_Epistemic_fm_i > epistemic_fm_i > $o] : ( listre5250616410628212153c_fm_i @ R @ nil_li3019469239499002158c_fm_i @ nil_Epistemic_fm_i ) ).
% listrelp.Nil
thf(fact_548_listrelp_ONil,axiom,
! [R: list_Epistemic_fm_i > list_Epistemic_fm_i > $o] : ( listre5721989502611914303c_fm_i @ R @ nil_li3019469239499002158c_fm_i @ nil_li3019469239499002158c_fm_i ) ).
% listrelp.Nil
thf(fact_549_listrelp_ONil,axiom,
! [R: list_Epistemic_fm_i > nat > $o] : ( listre3862057929835193081_i_nat @ R @ nil_li3019469239499002158c_fm_i @ nil_nat ) ).
% listrelp.Nil
thf(fact_550_listrelp_ONil,axiom,
! [R: nat > epistemic_fm_i > $o] : ( listre30318951981956049c_fm_i @ R @ nil_nat @ nil_Epistemic_fm_i ) ).
% listrelp.Nil
thf(fact_551_listrelp_ONil,axiom,
! [R: nat > list_Epistemic_fm_i > $o] : ( listre5653211634709596247c_fm_i @ R @ nil_nat @ nil_li3019469239499002158c_fm_i ) ).
% listrelp.Nil
thf(fact_552_listrelp_ONil,axiom,
! [R: nat > nat > $o] : ( listrelp_nat_nat @ R @ nil_nat @ nil_nat ) ).
% listrelp.Nil
thf(fact_553_listrelp_ONil,axiom,
! [R: epistemic_fm_i > epistemic_fm_i > $o] : ( listre3501123205157137215c_fm_i @ R @ nil_Epistemic_fm_i @ nil_Epistemic_fm_i ) ).
% listrelp.Nil
thf(fact_554_list__bind__cong,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,F: epistemic_fm_i > list_Epistemic_fm_i,G2: epistemic_fm_i > list_Epistemic_fm_i] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( bind_E4122511559465885831c_fm_i @ Xs @ F )
= ( bind_E4122511559465885831c_fm_i @ Ys @ G2 ) ) ) ) ).
% list_bind_cong
thf(fact_555_ord_Olexordp__eq_Osimps,axiom,
( lexord3686779039239843700c_fm_i
= ( ^ [Less2: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,A12: list_l4360594050116504750c_fm_i,A23: list_l4360594050116504750c_fm_i] :
( ? [Ys3: list_l4360594050116504750c_fm_i] :
( ( A12 = nil_li3019469239499002158c_fm_i )
& ( A23 = Ys3 ) )
| ? [X2: list_Epistemic_fm_i,Y3: list_Epistemic_fm_i,Xs3: list_l4360594050116504750c_fm_i,Ys3: list_l4360594050116504750c_fm_i] :
( ( A12
= ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) )
& ( Less2 @ X2 @ Y3 ) )
| ? [X2: list_Epistemic_fm_i,Y3: list_Epistemic_fm_i,Xs3: list_l4360594050116504750c_fm_i,Ys3: list_l4360594050116504750c_fm_i] :
( ( A12
= ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X2 @ Y3 )
& ~ ( Less2 @ Y3 @ X2 )
& ( lexord3686779039239843700c_fm_i @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_556_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Ys3: list_nat] :
( ( A12 = nil_nat )
& ( A23 = Ys3 ) )
| ? [X2: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X2 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( Less2 @ X2 @ Y3 ) )
| ? [X2: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X2 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X2 @ Y3 )
& ~ ( Less2 @ Y3 @ X2 )
& ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_557_ord_Olexordp__eq_Osimps,axiom,
( lexord491902653470561646c_fm_i
= ( ^ [Less2: epistemic_fm_i > epistemic_fm_i > $o,A12: list_Epistemic_fm_i,A23: list_Epistemic_fm_i] :
( ? [Ys3: list_Epistemic_fm_i] :
( ( A12 = nil_Epistemic_fm_i )
& ( A23 = Ys3 ) )
| ? [X2: epistemic_fm_i,Y3: epistemic_fm_i,Xs3: list_Epistemic_fm_i,Ys3: list_Epistemic_fm_i] :
( ( A12
= ( cons_Epistemic_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) )
& ( Less2 @ X2 @ Y3 ) )
| ? [X2: epistemic_fm_i,Y3: epistemic_fm_i,Xs3: list_Epistemic_fm_i,Ys3: list_Epistemic_fm_i] :
( ( A12
= ( cons_Epistemic_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X2 @ Y3 )
& ~ ( Less2 @ Y3 @ X2 )
& ( lexord491902653470561646c_fm_i @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_558_ord_Olexordp__eq_Ocases,axiom,
! [Less: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,A1: list_l4360594050116504750c_fm_i,A22: list_l4360594050116504750c_fm_i] :
( ( lexord3686779039239843700c_fm_i @ Less @ A1 @ A22 )
=> ( ( A1 != nil_li3019469239499002158c_fm_i )
=> ( ! [X3: list_Epistemic_fm_i] :
( ? [Xs2: list_l4360594050116504750c_fm_i] :
( A1
= ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) )
=> ! [Y2: list_Epistemic_fm_i] :
( ? [Ys2: list_l4360594050116504750c_fm_i] :
( A22
= ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i,Xs2: list_l4360594050116504750c_fm_i] :
( ( A1
= ( cons_l8703137435948174558c_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_l4360594050116504750c_fm_i] :
( ( A22
= ( cons_l8703137435948174558c_fm_i @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexord3686779039239843700c_fm_i @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_559_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( lexordp_eq_nat @ Less @ A1 @ A22 )
=> ( ( A1 != nil_nat )
=> ( ! [X3: nat] :
( ? [Xs2: list_nat] :
( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_560_ord_Olexordp__eq_Ocases,axiom,
! [Less: epistemic_fm_i > epistemic_fm_i > $o,A1: list_Epistemic_fm_i,A22: list_Epistemic_fm_i] :
( ( lexord491902653470561646c_fm_i @ Less @ A1 @ A22 )
=> ( ( A1 != nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i] :
( ? [Xs2: list_Epistemic_fm_i] :
( A1
= ( cons_Epistemic_fm_i @ X3 @ Xs2 ) )
=> ! [Y2: epistemic_fm_i] :
( ? [Ys2: list_Epistemic_fm_i] :
( A22
= ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: epistemic_fm_i,Y2: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( A1
= ( cons_Epistemic_fm_i @ X3 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_i] :
( ( A22
= ( cons_Epistemic_fm_i @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexord491902653470561646c_fm_i @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_561_listrelp_Osimps,axiom,
( listre7628413289575147717c_fm_i
= ( ^ [R2: epistemic_fm_i > list_Epistemic_fm_i > $o,A12: list_Epistemic_fm_i,A23: list_l4360594050116504750c_fm_i] :
( ( ( A12 = nil_Epistemic_fm_i )
& ( A23 = nil_li3019469239499002158c_fm_i ) )
| ? [X2: epistemic_fm_i,Y3: list_Epistemic_fm_i,Xs3: list_Epistemic_fm_i,Ys3: list_l4360594050116504750c_fm_i] :
( ( A12
= ( cons_Epistemic_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre7628413289575147717c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_562_listrelp_Osimps,axiom,
( listre7650074605423786739_i_nat
= ( ^ [R2: epistemic_fm_i > nat > $o,A12: list_Epistemic_fm_i,A23: list_nat] :
( ( ( A12 = nil_Epistemic_fm_i )
& ( A23 = nil_nat ) )
| ? [X2: epistemic_fm_i,Y3: nat,Xs3: list_Epistemic_fm_i,Ys3: list_nat] :
( ( A12
= ( cons_Epistemic_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre7650074605423786739_i_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_563_listrelp_Osimps,axiom,
( listre5250616410628212153c_fm_i
= ( ^ [R2: list_Epistemic_fm_i > epistemic_fm_i > $o,A12: list_l4360594050116504750c_fm_i,A23: list_Epistemic_fm_i] :
( ( ( A12 = nil_li3019469239499002158c_fm_i )
& ( A23 = nil_Epistemic_fm_i ) )
| ? [X2: list_Epistemic_fm_i,Y3: epistemic_fm_i,Xs3: list_l4360594050116504750c_fm_i,Ys3: list_Epistemic_fm_i] :
( ( A12
= ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre5250616410628212153c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_564_listrelp_Osimps,axiom,
( listre5721989502611914303c_fm_i
= ( ^ [R2: list_Epistemic_fm_i > list_Epistemic_fm_i > $o,A12: list_l4360594050116504750c_fm_i,A23: list_l4360594050116504750c_fm_i] :
( ( ( A12 = nil_li3019469239499002158c_fm_i )
& ( A23 = nil_li3019469239499002158c_fm_i ) )
| ? [X2: list_Epistemic_fm_i,Y3: list_Epistemic_fm_i,Xs3: list_l4360594050116504750c_fm_i,Ys3: list_l4360594050116504750c_fm_i] :
( ( A12
= ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre5721989502611914303c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_565_listrelp_Osimps,axiom,
( listre3862057929835193081_i_nat
= ( ^ [R2: list_Epistemic_fm_i > nat > $o,A12: list_l4360594050116504750c_fm_i,A23: list_nat] :
( ( ( A12 = nil_li3019469239499002158c_fm_i )
& ( A23 = nil_nat ) )
| ? [X2: list_Epistemic_fm_i,Y3: nat,Xs3: list_l4360594050116504750c_fm_i,Ys3: list_nat] :
( ( A12
= ( cons_l8703137435948174558c_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre3862057929835193081_i_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_566_listrelp_Osimps,axiom,
( listre30318951981956049c_fm_i
= ( ^ [R2: nat > epistemic_fm_i > $o,A12: list_nat,A23: list_Epistemic_fm_i] :
( ( ( A12 = nil_nat )
& ( A23 = nil_Epistemic_fm_i ) )
| ? [X2: nat,Y3: epistemic_fm_i,Xs3: list_nat,Ys3: list_Epistemic_fm_i] :
( ( A12
= ( cons_nat @ X2 @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre30318951981956049c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_567_listrelp_Osimps,axiom,
( listre5653211634709596247c_fm_i
= ( ^ [R2: nat > list_Epistemic_fm_i > $o,A12: list_nat,A23: list_l4360594050116504750c_fm_i] :
( ( ( A12 = nil_nat )
& ( A23 = nil_li3019469239499002158c_fm_i ) )
| ? [X2: nat,Y3: list_Epistemic_fm_i,Xs3: list_nat,Ys3: list_l4360594050116504750c_fm_i] :
( ( A12
= ( cons_nat @ X2 @ Xs3 ) )
& ( A23
= ( cons_l8703137435948174558c_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre5653211634709596247c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_568_listrelp_Osimps,axiom,
( listrelp_nat_nat
= ( ^ [R2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( ( A12 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X2: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X2 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listrelp_nat_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_569_listrelp_Osimps,axiom,
( listre3501123205157137215c_fm_i
= ( ^ [R2: epistemic_fm_i > epistemic_fm_i > $o,A12: list_Epistemic_fm_i,A23: list_Epistemic_fm_i] :
( ( ( A12 = nil_Epistemic_fm_i )
& ( A23 = nil_Epistemic_fm_i ) )
| ? [X2: epistemic_fm_i,Y3: epistemic_fm_i,Xs3: list_Epistemic_fm_i,Ys3: list_Epistemic_fm_i] :
( ( A12
= ( cons_Epistemic_fm_i @ X2 @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) )
& ( R2 @ X2 @ Y3 )
& ( listre3501123205157137215c_fm_i @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_570_subset__subseqs,axiom,
! [X4: set_li8269783359719514504c_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ X4 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( member794313724012529841c_fm_i @ X4 @ ( image_1870194968711581023c_fm_i @ set_li9010496130497932989c_fm_i @ ( set_li8148249020643214275c_fm_i @ ( subseq3738671058005954685c_fm_i @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_571_subset__subseqs,axiom,
! [X4: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X4 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_572_subset__subseqs,axiom,
! [X4: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1461078328125707403c_fm_i @ X4 @ ( image_862176557672056415c_fm_i @ set_se200842218512397079c_fm_i @ ( set_li1699392436468070557c_fm_i @ ( subseq7739903765785359575c_fm_i @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_573_subset__subseqs,axiom,
! [X4: set_set_nat,Xs: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_set_nat @ X4 @ ( image_8726355809080528601et_nat @ set_set_nat2 @ ( set_list_set_nat2 @ ( subseqs_set_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_574_subset__subseqs,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X4 @ ( image_4310783338035737183c_fm_i @ set_Epistemic_fm_i2 @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_575_transpose__empty,axiom,
! [Xs: list_l2400788496623510452c_fm_i] :
( ( ( transp7757552713476573368c_fm_i @ Xs )
= nil_li3456212153916897716c_fm_i )
= ( ! [X2: list_l4360594050116504750c_fm_i] :
( ( member5742582314450595671c_fm_i @ X2 @ ( set_li8148249020643214275c_fm_i @ Xs ) )
=> ( X2 = nil_li3019469239499002158c_fm_i ) ) ) ) ).
% transpose_empty
thf(fact_576_transpose__empty,axiom,
! [Xs: list_list_nat] :
( ( ( transpose_nat @ Xs )
= nil_list_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( X2 = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_577_transpose__empty,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( ( transp2882070894626480306c_fm_i @ Xs )
= nil_li3019469239499002158c_fm_i )
= ( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( X2 = nil_Epistemic_fm_i ) ) ) ) ).
% transpose_empty
thf(fact_578_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_579_SuccI,axiom,
! [Kl: list_s8081015415394010888c_fm_i,K: set_Epistemic_fm_i,Kl2: set_li2934481837907835752c_fm_i] :
( ( member605545779687026225c_fm_i @ ( append5826191483316899613c_fm_i @ Kl @ ( cons_s4962720389763977656c_fm_i @ K @ nil_se670848885773549064c_fm_i ) ) @ Kl2 )
=> ( member1104366573291651755c_fm_i @ K @ ( bNF_Gr2376118584863155869c_fm_i @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_580_SuccI,axiom,
! [Kl: list_s8635415444435394792c_fm_i,K: set_se3485332733965609186c_fm_i,Kl2: set_li1928893501965799752c_fm_i] :
( ( member1400835821191078929c_fm_i @ ( append8732965211748620541c_fm_i @ Kl @ ( cons_s5806790718070763416c_fm_i @ K @ nil_se1681839564182969320c_fm_i ) ) @ Kl2 )
=> ( member1461078328125707403c_fm_i @ K @ ( bNF_Gr8592748930649239677c_fm_i @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_581_SuccI,axiom,
! [Kl: list_set_nat,K: set_nat,Kl2: set_list_set_nat] :
( ( member_list_set_nat @ ( append_set_nat @ Kl @ ( cons_set_nat @ K @ nil_set_nat ) ) @ Kl2 )
=> ( member_set_nat2 @ K @ ( bNF_Gr3282828795834814635et_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_582_SuccI,axiom,
! [Kl: list_l4360594050116504750c_fm_i,K: list_Epistemic_fm_i,Kl2: set_li9045521773454279566c_fm_i] :
( ( member5742582314450595671c_fm_i @ ( append1952357471990327235c_fm_i @ Kl @ ( cons_l8703137435948174558c_fm_i @ K @ nil_li3019469239499002158c_fm_i ) ) @ Kl2 )
=> ( member6475149752759350353c_fm_i @ K @ ( bNF_Gr7749281761897660227c_fm_i @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_583_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_584_SuccI,axiom,
! [Kl: list_Epistemic_fm_i,K: epistemic_fm_i,Kl2: set_li8269783359719514504c_fm_i] :
( ( member6475149752759350353c_fm_i @ ( append9179727448351703357c_fm_i @ Kl @ ( cons_Epistemic_fm_i @ K @ nil_Epistemic_fm_i ) ) @ Kl2 )
=> ( member6642669606046002379c_fm_i @ K @ ( bNF_Gr1093487169986386109c_fm_i @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_585_SuccD,axiom,
! [K: set_Epistemic_fm_i,Kl2: set_li2934481837907835752c_fm_i,Kl: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ K @ ( bNF_Gr2376118584863155869c_fm_i @ Kl2 @ Kl ) )
=> ( member605545779687026225c_fm_i @ ( append5826191483316899613c_fm_i @ Kl @ ( cons_s4962720389763977656c_fm_i @ K @ nil_se670848885773549064c_fm_i ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_586_SuccD,axiom,
! [K: set_se3485332733965609186c_fm_i,Kl2: set_li1928893501965799752c_fm_i,Kl: list_s8635415444435394792c_fm_i] :
( ( member1461078328125707403c_fm_i @ K @ ( bNF_Gr8592748930649239677c_fm_i @ Kl2 @ Kl ) )
=> ( member1400835821191078929c_fm_i @ ( append8732965211748620541c_fm_i @ Kl @ ( cons_s5806790718070763416c_fm_i @ K @ nil_se1681839564182969320c_fm_i ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_587_SuccD,axiom,
! [K: set_nat,Kl2: set_list_set_nat,Kl: list_set_nat] :
( ( member_set_nat2 @ K @ ( bNF_Gr3282828795834814635et_nat @ Kl2 @ Kl ) )
=> ( member_list_set_nat @ ( append_set_nat @ Kl @ ( cons_set_nat @ K @ nil_set_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_588_SuccD,axiom,
! [K: list_Epistemic_fm_i,Kl2: set_li9045521773454279566c_fm_i,Kl: list_l4360594050116504750c_fm_i] :
( ( member6475149752759350353c_fm_i @ K @ ( bNF_Gr7749281761897660227c_fm_i @ Kl2 @ Kl ) )
=> ( member5742582314450595671c_fm_i @ ( append1952357471990327235c_fm_i @ Kl @ ( cons_l8703137435948174558c_fm_i @ K @ nil_li3019469239499002158c_fm_i ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_589_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_590_SuccD,axiom,
! [K: epistemic_fm_i,Kl2: set_li8269783359719514504c_fm_i,Kl: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ K @ ( bNF_Gr1093487169986386109c_fm_i @ Kl2 @ Kl ) )
=> ( member6475149752759350353c_fm_i @ ( append9179727448351703357c_fm_i @ Kl @ ( cons_Epistemic_fm_i @ K @ nil_Epistemic_fm_i ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_591_remove__code_I1_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_592_remove__code_I1_J,axiom,
! [X: list_Epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( remove1193922379842187277c_fm_i @ X @ ( set_li9010496130497932989c_fm_i @ Xs ) )
= ( set_li9010496130497932989c_fm_i @ ( remove5890047714521709364c_fm_i @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_593_remove__code_I1_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( remove5552923401211475591c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_594_lexordp__eq__simps_I3_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% lexordp_eq_simps(3)
thf(fact_595_insort__is__Cons,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > nat,A: epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ X3 ) ) )
=> ( ( linord2163417920900579895_i_nat @ F @ A @ Xs )
= ( cons_Epistemic_fm_i @ A @ Xs ) ) ) ).
% insort_is_Cons
thf(fact_596_insort__is__Cons,axiom,
! [Xs: list_l4360594050116504750c_fm_i,F: list_Epistemic_fm_i > nat,A: list_Epistemic_fm_i] :
( ! [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ X3 ) ) )
=> ( ( linord1135965747126270013_i_nat @ F @ A @ Xs )
= ( cons_l8703137435948174558c_fm_i @ A @ Xs ) ) ) ).
% insort_is_Cons
thf(fact_597_insort__is__Cons,axiom,
! [Xs: list_nat,F: nat > nat,A: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ X3 ) ) )
=> ( ( linord8961336180081300637at_nat @ F @ A @ Xs )
= ( cons_nat @ A @ Xs ) ) ) ).
% insort_is_Cons
thf(fact_598_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_599_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X5: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_600_le__rel__bool__arg__iff,axiom,
( ord_le6517057662708076536c_fm_i
= ( ^ [X5: $o > $o > set_Epistemic_fm_i,Y6: $o > $o > set_Epistemic_fm_i] :
( ( ord_le8883210156954716975c_fm_i @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8883210156954716975c_fm_i @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_601_le__rel__bool__arg__iff,axiom,
( ord_le2204021973410629903c_fm_i
= ( ^ [X5: $o > set_se3485332733965609186c_fm_i,Y6: $o > set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le5389487502678872194c_fm_i @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_602_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X5: $o > set_set_nat,Y6: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_603_le__rel__bool__arg__iff,axiom,
( ord_le8883210156954716975c_fm_i
= ( ^ [X5: $o > set_Epistemic_fm_i,Y6: $o > set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3843937902494030498c_fm_i @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_604_bot__apply,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X2: epistemic_fm_i] : bot_bot_o ) ) ).
% bot_apply
thf(fact_605_image__eqI,axiom,
! [B2: epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B2
= ( F @ X ) )
=> ( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_606_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_607_image__eqI,axiom,
! [B2: nat,F: epistemic_fm_i > nat,X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B2
= ( F @ X ) )
=> ( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_2291338433772226221_i_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_608_image__eqI,axiom,
! [B2: epistemic_fm_i,F: nat > epistemic_fm_i,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_3894954817185171339c_fm_i @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_609_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_set_nat2 @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_610_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X: set_nat,A2: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat2 @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_611_image__eqI,axiom,
! [B2: set_nat,F: epistemic_fm_i > set_nat,X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B2
= ( F @ X ) )
=> ( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( member_set_nat2 @ B2 @ ( image_5232765980333981283et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_612_image__eqI,axiom,
! [B2: nat,F: list_Epistemic_fm_i > nat,X: list_Epistemic_fm_i,A2: set_li8269783359719514504c_fm_i] :
( ( B2
= ( F @ X ) )
=> ( ( member6475149752759350353c_fm_i @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_625688586424191411_i_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_613_image__eqI,axiom,
! [B2: nat,F: set_Epistemic_fm_i > nat,X: set_Epistemic_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( B2
= ( F @ X ) )
=> ( ( member1104366573291651755c_fm_i @ X @ A2 )
=> ( member_nat2 @ B2 @ ( image_9130930184777177229_i_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_614_image__eqI,axiom,
! [B2: list_Epistemic_fm_i,F: nat > list_Epistemic_fm_i,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( member6475149752759350353c_fm_i @ B2 @ ( image_2416842291298594577c_fm_i @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_615_empty__Collect__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P2 ) )
= ( ! [X2: epistemic_fm_i] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_616_empty__Collect__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P2 ) )
= ( ! [X2: set_Epistemic_fm_i] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_617_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_618_Collect__empty__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P2 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X2: epistemic_fm_i] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_619_Collect__empty__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P2 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X2: set_Epistemic_fm_i] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_620_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_621_all__not__in__conv,axiom,
! [A2: set_li8269783359719514504c_fm_i] :
( ( ! [X2: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ X2 @ A2 ) )
= ( A2 = bot_bo6562031733678445556c_fm_i ) ) ).
% all_not_in_conv
thf(fact_622_all__not__in__conv,axiom,
! [A2: set_se7339729205154126530c_fm_i] :
( ( ! [X2: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ X2 @ A2 ) )
= ( A2 = bot_bo4781621276559889198c_fm_i ) ) ).
% all_not_in_conv
thf(fact_623_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat2 @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_624_all__not__in__conv,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( ! [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ A2 ) )
= ( A2 = bot_bo145720340923748686c_fm_i ) ) ).
% all_not_in_conv
thf(fact_625_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_626_all__not__in__conv,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ! [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ A2 ) )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% all_not_in_conv
thf(fact_627_empty__iff,axiom,
! [C: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ C @ bot_bo6562031733678445556c_fm_i ) ).
% empty_iff
thf(fact_628_empty__iff,axiom,
! [C: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ C @ bot_bo4781621276559889198c_fm_i ) ).
% empty_iff
thf(fact_629_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat2 @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_630_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_631_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_632_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_633_same__append__eq,axiom,
! [Xs: list_l4360594050116504750c_fm_i,Ys: list_l4360594050116504750c_fm_i,Zs: list_l4360594050116504750c_fm_i] :
( ( ( append1952357471990327235c_fm_i @ Xs @ Ys )
= ( append1952357471990327235c_fm_i @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_634_finite__imageI,axiom,
! [F3: set_Epistemic_fm_i,H: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( finite3304564979551393739c_fm_i @ ( image_120052958407486725c_fm_i @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_635_finite__imageI,axiom,
! [F3: set_Epistemic_fm_i,H: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( finite_finite_nat @ ( image_2291338433772226221_i_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_636_finite__imageI,axiom,
! [F3: set_nat,H: nat > epistemic_fm_i] :
( ( finite_finite_nat @ F3 )
=> ( finite3304564979551393739c_fm_i @ ( image_3894954817185171339c_fm_i @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_637_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_638_subset__empty,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_empty
thf(fact_639_empty__subsetI,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% empty_subsetI
thf(fact_640_append_Oright__neutral,axiom,
! [A: list_Epistemic_fm_i] :
( ( append9179727448351703357c_fm_i @ A @ nil_Epistemic_fm_i )
= A ) ).
% append.right_neutral
thf(fact_641_append__Nil2,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( append9179727448351703357c_fm_i @ Xs @ nil_Epistemic_fm_i )
= Xs ) ).
% append_Nil2
thf(fact_642_append__self__conv,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Xs @ Ys )
= Xs )
= ( Ys = nil_Epistemic_fm_i ) ) ).
% append_self_conv
thf(fact_643_self__append__conv,axiom,
! [Y: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( Y
= ( append9179727448351703357c_fm_i @ Y @ Ys ) )
= ( Ys = nil_Epistemic_fm_i ) ) ).
% self_append_conv
thf(fact_644_append__self__conv2,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Xs @ Ys )
= Ys )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% append_self_conv2
thf(fact_645_self__append__conv2,axiom,
! [Y: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( Y
= ( append9179727448351703357c_fm_i @ Xs @ Y ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% self_append_conv2
thf(fact_646_Nil__is__append__conv,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( nil_Epistemic_fm_i
= ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( ( Xs = nil_Epistemic_fm_i )
& ( Ys = nil_Epistemic_fm_i ) ) ) ).
% Nil_is_append_conv
thf(fact_647_append__is__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Xs @ Ys )
= nil_Epistemic_fm_i )
= ( ( Xs = nil_Epistemic_fm_i )
& ( Ys = nil_Epistemic_fm_i ) ) ) ).
% append_is_Nil_conv
thf(fact_648_removeAll__id,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove817376155714978094c_fm_i @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_649_set__empty,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs )
= bot_bo4194595901900360558c_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty
thf(fact_650_set__empty2,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ Xs ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty2
thf(fact_651_append1__eq__conv,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i,Ys: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) )
= ( append9179727448351703357c_fm_i @ Ys @ ( cons_Epistemic_fm_i @ Y @ nil_Epistemic_fm_i ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_652_rev__image__eqI,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,B2: epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_653_ex__in__conv,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ? [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A2 ) )
= ( A2 != bot_bo4194595901900360558c_fm_i ) ) ).
% ex_in_conv
thf(fact_654_equals0I,axiom,
! [A2: set_Epistemic_fm_i] :
( ! [Y2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y2 @ A2 )
=> ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_655_equals0D,axiom,
! [A2: set_Epistemic_fm_i,A: epistemic_fm_i] :
( ( A2 = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A @ A2 ) ) ).
% equals0D
thf(fact_656_imageI,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( member6642669606046002379c_fm_i @ ( F @ X ) @ ( image_120052958407486725c_fm_i @ F @ A2 ) ) ) ).
% imageI
thf(fact_657_emptyE,axiom,
! [A: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_658_bot_Oextremum,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% bot.extremum
thf(fact_659_bot_Oextremum__unique,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_660_bot_Oextremum__uniqueI,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_661_image__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) @ ( image_120052958407486725c_fm_i @ F @ B ) ) ) ).
% image_mono
thf(fact_662_image__subsetI,axiom,
! [A2: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( member6642669606046002379c_fm_i @ ( F @ X3 ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_663_subset__imageE,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ~ ! [C3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C3 @ A2 )
=> ( B
!= ( image_120052958407486725c_fm_i @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_664_subset__image__iff,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
= ( ? [AA: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ AA @ A2 )
& ( B
= ( image_120052958407486725c_fm_i @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_665_all__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ! [B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ A2 )
=> ( P2 @ ( image_120052958407486725c_fm_i @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_666_Cons__eq__appendI,axiom,
! [X: epistemic_fm_i,Xs1: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append9179727448351703357c_fm_i @ Xs1 @ Zs ) )
=> ( ( cons_Epistemic_fm_i @ X @ Xs )
= ( append9179727448351703357c_fm_i @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_667_append__Cons,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( append9179727448351703357c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) @ Ys )
= ( cons_Epistemic_fm_i @ X @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_668_append__Nil,axiom,
! [Ys: list_Epistemic_fm_i] :
( ( append9179727448351703357c_fm_i @ nil_Epistemic_fm_i @ Ys )
= Ys ) ).
% append_Nil
thf(fact_669_append_Oleft__neutral,axiom,
! [A: list_Epistemic_fm_i] :
( ( append9179727448351703357c_fm_i @ nil_Epistemic_fm_i @ A )
= A ) ).
% append.left_neutral
thf(fact_670_eq__Nil__appendI,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( Xs = Ys )
=> ( Xs
= ( append9179727448351703357c_fm_i @ nil_Epistemic_fm_i @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_671_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_672_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_673_infinite__imp__nonempty,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ( S != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_674_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_675_removeAll_Osimps_I2_J,axiom,
! [X: epistemic_fm_i,Y: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( X = Y )
=> ( ( remove817376155714978094c_fm_i @ X @ ( cons_Epistemic_fm_i @ Y @ Xs ) )
= ( remove817376155714978094c_fm_i @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( remove817376155714978094c_fm_i @ X @ ( cons_Epistemic_fm_i @ Y @ Xs ) )
= ( cons_Epistemic_fm_i @ Y @ ( remove817376155714978094c_fm_i @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_676_removeAll_Osimps_I1_J,axiom,
! [X: epistemic_fm_i] :
( ( remove817376155714978094c_fm_i @ X @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% removeAll.simps(1)
thf(fact_677_finite__surj,axiom,
! [A2: set_Epistemic_fm_i,B: set_nat,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2291338433772226221_i_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_678_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_679_finite__surj,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ( finite3304564979551393739c_fm_i @ B ) ) ) ).
% finite_surj
thf(fact_680_finite__surj,axiom,
! [A2: set_nat,B: set_Epistemic_fm_i,F: nat > epistemic_fm_i] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
=> ( finite3304564979551393739c_fm_i @ B ) ) ) ).
% finite_surj
thf(fact_681_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_682_finite__subset__image,axiom,
! [B: set_nat,F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2291338433772226221_i_nat @ F @ A2 ) )
=> ? [C3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C3 @ A2 )
& ( finite3304564979551393739c_fm_i @ C3 )
& ( B
= ( image_2291338433772226221_i_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_683_finite__subset__image,axiom,
! [B: set_Epistemic_fm_i,F: nat > epistemic_fm_i,A2: set_nat] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_3894954817185171339c_fm_i @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_684_finite__subset__image,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ? [C3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C3 @ A2 )
& ( finite3304564979551393739c_fm_i @ C3 )
& ( B
= ( image_120052958407486725c_fm_i @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_685_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
& ( P2 @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 )
& ( P2 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_686_ex__finite__subset__image,axiom,
! [F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i,P2: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_2291338433772226221_i_nat @ F @ A2 ) )
& ( P2 @ B5 ) ) )
= ( ? [B5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A2 )
& ( P2 @ ( image_2291338433772226221_i_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_687_ex__finite__subset__image,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat,P2: set_Epistemic_fm_i > $o] :
( ( ? [B5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
& ( P2 @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 )
& ( P2 @ ( image_3894954817185171339c_fm_i @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_688_ex__finite__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ? [B5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
& ( P2 @ B5 ) ) )
= ( ? [B5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A2 )
& ( P2 @ ( image_120052958407486725c_fm_i @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_689_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 ) )
=> ( P2 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_690_all__finite__subset__image,axiom,
! [F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i,P2: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_2291338433772226221_i_nat @ F @ A2 ) ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A2 ) )
=> ( P2 @ ( image_2291338433772226221_i_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_691_all__finite__subset__image,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat,P2: set_Epistemic_fm_i > $o] :
( ( ! [B5: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ ( image_3894954817185171339c_fm_i @ F @ A2 ) ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 ) )
=> ( P2 @ ( image_3894954817185171339c_fm_i @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_692_all__finite__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ! [B5: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ ( image_120052958407486725c_fm_i @ F @ A2 ) ) )
=> ( P2 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A2 ) )
=> ( P2 @ ( image_120052958407486725c_fm_i @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_693_rev__induct,axiom,
! [P2: list_Epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( P2 @ nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append9179727448351703357c_fm_i @ Xs2 @ ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_694_rev__exhaust,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( Xs != nil_Epistemic_fm_i )
=> ~ ! [Ys2: list_Epistemic_fm_i,Y2: epistemic_fm_i] :
( Xs
!= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ Y2 @ nil_Epistemic_fm_i ) ) ) ) ).
% rev_exhaust
thf(fact_695_Cons__eq__append__conv,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X @ Xs )
= ( append9179727448351703357c_fm_i @ Ys @ Zs ) )
= ( ( ( Ys = nil_Epistemic_fm_i )
& ( ( cons_Epistemic_fm_i @ X @ Xs )
= Zs ) )
| ? [Ys4: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X @ Ys4 )
= Ys )
& ( Xs
= ( append9179727448351703357c_fm_i @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_696_append__eq__Cons__conv,axiom,
! [Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i,X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Ys @ Zs )
= ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( ( ( Ys = nil_Epistemic_fm_i )
& ( Zs
= ( cons_Epistemic_fm_i @ X @ Xs ) ) )
| ? [Ys4: list_Epistemic_fm_i] :
( ( Ys
= ( cons_Epistemic_fm_i @ X @ Ys4 ) )
& ( ( append9179727448351703357c_fm_i @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_697_rev__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( Xs != nil_Epistemic_fm_i )
=> ( ! [X3: epistemic_fm_i] : ( P2 @ ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) )
=> ( ! [X3: epistemic_fm_i,Xs2: list_Epistemic_fm_i] :
( ( Xs2 != nil_Epistemic_fm_i )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append9179727448351703357c_fm_i @ Xs2 @ ( cons_Epistemic_fm_i @ X3 @ nil_Epistemic_fm_i ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_698_split__list__first__prop__iff,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_Epistemic_fm_i,X2: epistemic_fm_i] :
( ? [Zs2: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys3 @ ( cons_Epistemic_fm_i @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_699_split__list__last__prop__iff,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys3: list_Epistemic_fm_i,X2: epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys3 @ ( cons_Epistemic_fm_i @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_700_in__set__conv__decomp__first,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys3 @ ( cons_Epistemic_fm_i @ X @ Zs2 ) ) )
& ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_701_in__set__conv__decomp__last,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys3 @ ( cons_Epistemic_fm_i @ X @ Zs2 ) ) )
& ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_702_split__list__first__propE,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i] :
( ? [Zs3: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_703_split__list__last__propE,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_704_split__list__first__prop,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i] :
( ? [Zs3: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_705_split__list__last__prop,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
& ( P2 @ X3 )
& ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ ( set_Epistemic_fm_i2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_706_in__set__conv__decomp,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys3 @ ( cons_Epistemic_fm_i @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_707_append__Cons__eq__iff,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Xs4: list_Epistemic_fm_i,Ys5: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( ( ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ Ys ) )
= ( append9179727448351703357c_fm_i @ Xs4 @ ( cons_Epistemic_fm_i @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_708_split__list__propE,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ~ ! [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i] :
( ? [Zs3: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_709_split__list__first,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X @ Zs3 ) ) )
& ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_710_split__list__prop,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X6 ) )
=> ? [Ys2: list_Epistemic_fm_i,X3: epistemic_fm_i] :
( ? [Zs3: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X3 @ Zs3 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_711_split__list__last,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X @ Zs3 ) ) )
& ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_712_split__list,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( Xs
= ( append9179727448351703357c_fm_i @ Ys2 @ ( cons_Epistemic_fm_i @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_713_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_714_finite__has__maximal,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_715_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_716_finite__has__minimal,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_717_empty__set,axiom,
( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% empty_set
thf(fact_718_transpose_Osimps_I2_J,axiom,
! [Xss2: list_l4360594050116504750c_fm_i] :
( ( transp2882070894626480306c_fm_i @ ( cons_l8703137435948174558c_fm_i @ nil_Epistemic_fm_i @ Xss2 ) )
= ( transp2882070894626480306c_fm_i @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_719_subset__emptyI,axiom,
! [A2: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_720_subseqs__powset,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( image_4310783338035737183c_fm_i @ set_Epistemic_fm_i2 @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) )
= ( pow_Epistemic_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_721_concat__eq__append__conv,axiom,
! [Xss2: list_l4360594050116504750c_fm_i,Ys: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( concat2780896577252151011c_fm_i @ Xss2 )
= ( append9179727448351703357c_fm_i @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_li3019469239499002158c_fm_i )
=> ( ( Ys = nil_Epistemic_fm_i )
& ( Zs = nil_Epistemic_fm_i ) ) )
& ( ( Xss2 != nil_li3019469239499002158c_fm_i )
=> ? [Xss1: list_l4360594050116504750c_fm_i,Xs3: list_Epistemic_fm_i,Xs5: list_Epistemic_fm_i,Xss22: list_l4360594050116504750c_fm_i] :
( ( Xss2
= ( append1952357471990327235c_fm_i @ Xss1 @ ( cons_l8703137435948174558c_fm_i @ ( append9179727448351703357c_fm_i @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append9179727448351703357c_fm_i @ ( concat2780896577252151011c_fm_i @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append9179727448351703357c_fm_i @ Xs5 @ ( concat2780896577252151011c_fm_i @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_722_butlast__snoc,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( butlas2200625824590931828c_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_723_finite__Pow__iff,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ ( pow_Epistemic_fm_i @ A2 ) )
= ( finite3304564979551393739c_fm_i @ A2 ) ) ).
% finite_Pow_iff
thf(fact_724_finite__Pow__iff,axiom,
! [A2: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_Pow_iff
thf(fact_725_Pow__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( pow_Epistemic_fm_i @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% Pow_iff
thf(fact_726_PowI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( member1104366573291651755c_fm_i @ A2 @ ( pow_Epistemic_fm_i @ B ) ) ) ).
% PowI
thf(fact_727_concat__eq__Nil__conv,axiom,
! [Xss2: list_l4360594050116504750c_fm_i] :
( ( ( concat2780896577252151011c_fm_i @ Xss2 )
= nil_Epistemic_fm_i )
= ( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xss2 ) )
=> ( X2 = nil_Epistemic_fm_i ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_728_Nil__eq__concat__conv,axiom,
! [Xss2: list_l4360594050116504750c_fm_i] :
( ( nil_Epistemic_fm_i
= ( concat2780896577252151011c_fm_i @ Xss2 ) )
= ( ! [X2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X2 @ ( set_li9010496130497932989c_fm_i @ Xss2 ) )
=> ( X2 = nil_Epistemic_fm_i ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_729_Pow__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( pow_Epistemic_fm_i @ A2 ) @ ( pow_Epistemic_fm_i @ B ) ) ) ).
% Pow_mono
thf(fact_730_PowD,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( pow_Epistemic_fm_i @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% PowD
thf(fact_731_butlast_Osimps_I1_J,axiom,
( ( butlas2200625824590931828c_fm_i @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% butlast.simps(1)
thf(fact_732_in__set__butlastD,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ ( butlas2200625824590931828c_fm_i @ Xs ) ) )
=> ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_733_concat_Osimps_I1_J,axiom,
( ( concat2780896577252151011c_fm_i @ nil_li3019469239499002158c_fm_i )
= nil_Epistemic_fm_i ) ).
% concat.simps(1)
thf(fact_734_butlast_Osimps_I2_J,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( ( Xs = nil_Epistemic_fm_i )
=> ( ( butlas2200625824590931828c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= nil_Epistemic_fm_i ) )
& ( ( Xs != nil_Epistemic_fm_i )
=> ( ( butlas2200625824590931828c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( cons_Epistemic_fm_i @ X @ ( butlas2200625824590931828c_fm_i @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_735_butlast__append,axiom,
! [Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( Ys = nil_Epistemic_fm_i )
=> ( ( butlas2200625824590931828c_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( butlas2200625824590931828c_fm_i @ Xs ) ) )
& ( ( Ys != nil_Epistemic_fm_i )
=> ( ( butlas2200625824590931828c_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( append9179727448351703357c_fm_i @ Xs @ ( butlas2200625824590931828c_fm_i @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_736_in__set__butlast__appendI,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ ( butlas2200625824590931828c_fm_i @ Xs ) ) )
| ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ ( butlas2200625824590931828c_fm_i @ Ys ) ) ) )
=> ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ ( butlas2200625824590931828c_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_737_maps__simps_I2_J,axiom,
! [F: epistemic_fm_i > list_Epistemic_fm_i] :
( ( maps_E1743072755862344257c_fm_i @ F @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% maps_simps(2)
thf(fact_738_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_739_Min_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_740_Min__antimono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_741_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_742_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ X @ A6 ) ) ) ) ) ).
% Min.boundedE
thf(fact_743_eq__Min__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M2
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat2 @ M2 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ M2 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_744_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_745_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_746_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_747_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_748_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_749_Min__eq__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M2 )
= ( ( member_nat2 @ M2 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ M2 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_750_infinite__Pow,axiom,
! [A2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A2 )
=> ~ ( finite7933139204641697195c_fm_i @ ( pow_Epistemic_fm_i @ A2 ) ) ) ).
% infinite_Pow
thf(fact_751_infinite__Pow,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ~ ( finite1152437895449049373et_nat @ ( pow_nat @ A2 ) ) ) ).
% infinite_Pow
thf(fact_752_bot__empty__eq,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_753_append__butlast__last__id,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( Xs != nil_Epistemic_fm_i )
=> ( ( append9179727448351703357c_fm_i @ ( butlas2200625824590931828c_fm_i @ Xs ) @ ( cons_Epistemic_fm_i @ ( last_Epistemic_fm_i @ Xs ) @ nil_Epistemic_fm_i ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_754_last__appendL,axiom,
! [Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( Ys = nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( last_Epistemic_fm_i @ Xs ) ) ) ).
% last_appendL
thf(fact_755_last__appendR,axiom,
! [Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( Ys != nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( last_Epistemic_fm_i @ Ys ) ) ) ).
% last_appendR
thf(fact_756_last__snoc,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( last_Epistemic_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) )
= X ) ).
% last_snoc
thf(fact_757_last__ConsR,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( Xs != nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( last_Epistemic_fm_i @ Xs ) ) ) ).
% last_ConsR
thf(fact_758_last__ConsL,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( Xs = nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_759_last_Osimps,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( ( Xs = nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( last_Epistemic_fm_i @ Xs ) ) ) ) ).
% last.simps
thf(fact_760_last__in__set,axiom,
! [As2: list_Epistemic_fm_i] :
( ( As2 != nil_Epistemic_fm_i )
=> ( member6642669606046002379c_fm_i @ ( last_Epistemic_fm_i @ As2 ) @ ( set_Epistemic_fm_i2 @ As2 ) ) ) ).
% last_in_set
thf(fact_761_last__append,axiom,
! [Ys: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( Ys = nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( last_Epistemic_fm_i @ Xs ) ) )
& ( ( Ys != nil_Epistemic_fm_i )
=> ( ( last_Epistemic_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( last_Epistemic_fm_i @ Ys ) ) ) ) ).
% last_append
thf(fact_762_longest__common__suffix,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
? [Ss: list_Epistemic_fm_i,Xs6: list_Epistemic_fm_i,Ys6: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Xs6 @ Ss ) )
& ( Ys
= ( append9179727448351703357c_fm_i @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_Epistemic_fm_i )
| ( Ys6 = nil_Epistemic_fm_i )
| ( ( last_Epistemic_fm_i @ Xs6 )
!= ( last_Epistemic_fm_i @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_763_snoc__eq__iff__butlast,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) )
= Ys )
= ( ( Ys != nil_Epistemic_fm_i )
& ( ( butlas2200625824590931828c_fm_i @ Ys )
= Xs )
& ( ( last_Epistemic_fm_i @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_764_Pow__set_I1_J,axiom,
( ( pow_Epistemic_fm_i @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) )
= ( insert7698009978809854162c_fm_i @ bot_bo4194595901900360558c_fm_i @ bot_bo145720340923748686c_fm_i ) ) ).
% Pow_set(1)
thf(fact_765_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_766_Sup__fin_Osubset__imp,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A2 ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_767_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_768_Inf__fin_Osubset__imp,axiom,
! [A2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_769_rotate1_Osimps_I2_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( rotate3540958592676483701c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( append9179727448351703357c_fm_i @ Xs @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) ) ).
% rotate1.simps(2)
thf(fact_770_Max_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ).
% Max.subset_imp
thf(fact_771_insert__iff,axiom,
! [A: epistemic_fm_i,B2: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member6642669606046002379c_fm_i @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_772_insertCI,axiom,
! [A: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A @ B )
=> ( A = B2 ) )
=> ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertCI
thf(fact_773_singletonI,axiom,
! [A: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_774_finite__insert,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ A2 ) )
= ( finite3304564979551393739c_fm_i @ A2 ) ) ).
% finite_insert
thf(fact_775_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_776_insert__subset,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ A2 ) @ B )
= ( ( member6642669606046002379c_fm_i @ X @ B )
& ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_777_rotate1__is__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( ( rotate3540958592676483701c_fm_i @ Xs )
= nil_Epistemic_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% rotate1_is_Nil_conv
thf(fact_778_set__rotate1,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( rotate3540958592676483701c_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_rotate1
thf(fact_779_singleton__insert__inj__eq,axiom,
! [B2: epistemic_fm_i,A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_780_singleton__insert__inj__eq_H,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A @ A2 )
= ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A = B2 )
& ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_781_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_782_List_Oset__insert,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( insert177310195918387262c_fm_i @ X @ Xs ) )
= ( insert7817948997695205106c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_783_Max_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_784_Max__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B4: nat] :
( ( member_nat2 @ B4 @ A2 )
=> ( ord_less_eq_nat @ B4 @ A ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Max_insert2
thf(fact_785_mk__disjoint__insert,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ A2 )
=> ? [B6: set_Epistemic_fm_i] :
( ( A2
= ( insert7817948997695205106c_fm_i @ A @ B6 ) )
& ~ ( member6642669606046002379c_fm_i @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_786_insert__eq__iff,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ B2 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ A @ A2 )
= ( insert7817948997695205106c_fm_i @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_Epistemic_fm_i] :
( ( A2
= ( insert7817948997695205106c_fm_i @ B2 @ C4 ) )
& ~ ( member6642669606046002379c_fm_i @ B2 @ C4 )
& ( B
= ( insert7817948997695205106c_fm_i @ A @ C4 ) )
& ~ ( member6642669606046002379c_fm_i @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_787_insert__absorb,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( insert7817948997695205106c_fm_i @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_788_insert__ident,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X @ A2 )
= ( insert7817948997695205106c_fm_i @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_789_Set_Oset__insert,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ~ ! [B6: set_Epistemic_fm_i] :
( ( A2
= ( insert7817948997695205106c_fm_i @ X @ B6 ) )
=> ( member6642669606046002379c_fm_i @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_790_insertI2,axiom,
! [A: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ B )
=> ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertI2
thf(fact_791_insertI1,axiom,
! [A: epistemic_fm_i,B: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A @ B ) ) ).
% insertI1
thf(fact_792_insertE,axiom,
! [A: epistemic_fm_i,B2: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member6642669606046002379c_fm_i @ A @ A2 ) ) ) ).
% insertE
thf(fact_793_singleton__iff,axiom,
! [B2: epistemic_fm_i,A: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_794_singletonD,axiom,
! [B2: epistemic_fm_i,A: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_795_finite_OinsertI,axiom,
! [A2: set_Epistemic_fm_i,A: epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_796_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_797_insert__mono,axiom,
! [C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,A: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ C2 ) @ ( insert7817948997695205106c_fm_i @ A @ D ) ) ) ).
% insert_mono
thf(fact_798_subset__insert,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_799_subset__insertI,axiom,
! [B: set_Epistemic_fm_i,A: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A @ B ) ) ).
% subset_insertI
thf(fact_800_subset__insertI2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_801_insert__subsetI,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_802_rotate1_Osimps_I1_J,axiom,
( ( rotate3540958592676483701c_fm_i @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% rotate1.simps(1)
thf(fact_803_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_804_Inf__fin__le__Sup__fin,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A2 ) @ ( lattic3744900211830597177c_fm_i @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_805_finite_Ocases,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( A != bot_bo4194595901900360558c_fm_i )
=> ~ ! [A7: set_Epistemic_fm_i] :
( ? [A4: epistemic_fm_i] :
( A
= ( insert7817948997695205106c_fm_i @ A4 @ A7 ) )
=> ~ ( finite3304564979551393739c_fm_i @ A7 ) ) ) ) ).
% finite.cases
thf(fact_806_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat2 @ A4 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_807_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( ( A3 = bot_bo4194595901900360558c_fm_i )
| ? [A5: set_Epistemic_fm_i,B3: epistemic_fm_i] :
( ( A3
= ( insert7817948997695205106c_fm_i @ B3 @ A5 ) )
& ( finite3304564979551393739c_fm_i @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_808_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A5: set_nat,B3: nat] :
( ( A3
= ( insert_nat2 @ B3 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_809_finite__induct,axiom,
! [F3: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X3 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_810_finite__induct,axiom,
! [F3: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat2 @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat2 @ X3 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_811_finite__ne__induct,axiom,
! [F3: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i] : ( P2 @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X3: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( F4 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_812_finite__ne__induct,axiom,
! [F3: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat2 @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat2 @ X3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_813_infinite__finite__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X3 @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_814_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat2 @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat2 @ X3 @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_815_subset__singletonD,axiom,
! [A2: set_Epistemic_fm_i,X: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A2 = bot_bo4194595901900360558c_fm_i )
| ( A2
= ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_816_subset__singleton__iff,axiom,
! [X4: set_Epistemic_fm_i,A: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) )
= ( ( X4 = bot_bo4194595901900360558c_fm_i )
| ( X4
= ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_817_Max__ge,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max_ge
thf(fact_818_Max__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_819_Max__eq__if,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ B )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
& ( ord_less_eq_nat @ X3 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_820_Max_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_821_Max__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_822_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_823_Inf__fin_OcoboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,A: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( member1104366573291651755c_fm_i @ A @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_824_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_825_Sup__fin_OcoboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,A: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( member1104366573291651755c_fm_i @ A @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( lattic3744900211830597177c_fm_i @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_826_Fpow__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( finite7994277067554142856c_fm_i @ A2 ) @ ( finite7994277067554142856c_fm_i @ B ) ) ) ).
% Fpow_mono
thf(fact_827_finite__subset__induct,axiom,
! [F3: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat2 @ A4 @ A2 )
=> ( ~ ( member_nat2 @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat2 @ A4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_828_finite__subset__induct,axiom,
! [F3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A2 )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A4: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A4 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_829_finite__subset__induct_H,axiom,
! [F3: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat2 @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ~ ( member_nat2 @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat2 @ A4 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_830_finite__subset__induct_H,axiom,
! [F3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A2 )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A4: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F4 @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A4 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_831_Max__eq__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A2 )
= M2 )
= ( ( member_nat2 @ M2 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_832_Max__ge__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_833_eq__Max__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M2
= ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ( member_nat2 @ M2 @ A2 )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_834_Max_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_835_Max_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_836_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B4: nat] :
( ( member_nat2 @ B4 @ A2 )
=> ( ord_less_eq_nat @ A @ B4 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_837_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ X @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_838_Inf__fin_OboundedE,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ ( lattic8495942116038605215c_fm_i @ A2 ) )
=> ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_839_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_840_Inf__fin_OboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ! [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ A4 ) )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( lattic8495942116038605215c_fm_i @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_841_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_842_Sup__fin_OboundedE,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A2 ) @ X )
=> ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ A6 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_843_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_844_Sup__fin_OboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ! [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ A4 @ X ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_845_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_846_Inf__fin_Obounded__iff,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ ( lattic8495942116038605215c_fm_i @ A2 ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_847_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_848_Sup__fin_Obounded__iff,axiom,
! [A2: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ( A2 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A2 ) @ X )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_849_Max__mono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% Max_mono
thf(fact_850_sorted__list__of__set__nonempty,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A2 )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_851_set__removeAll,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% set_removeAll
thf(fact_852_Compl__subset__Compl__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ A2 ) @ ( uminus7605083404480446585c_fm_i @ B ) )
= ( ord_le3843937902494030498c_fm_i @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_853_Compl__anti__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ B ) @ ( uminus7605083404480446585c_fm_i @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_854_DiffI,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_855_Diff__iff,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A2 )
& ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_856_finite__Diff2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) )
= ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_Diff2
thf(fact_857_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_858_finite__Diff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_859_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_860_Diff__insert0,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ B ) )
= ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_861_insert__Diff1,axiom,
! [X: epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ A2 ) @ B )
= ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_862_subset__Compl__singleton,axiom,
! [A2: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( uminus7605083404480446585c_fm_i @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_863_Diff__eq__empty__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_864_finite__Diff__insert,axiom,
! [A2: set_Epistemic_fm_i,A: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ B ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_865_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_866_DiffE,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_867_DiffD1,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A2 ) ) ).
% DiffD1
thf(fact_868_DiffD2,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_869_Diff__infinite__finite,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_870_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_871_double__diff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B @ ( minus_1543973939109951465c_fm_i @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_872_Diff__subset,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_873_Diff__mono,axiom,
! [A2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_874_insert__Diff__if,axiom,
! [X: epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ A2 ) @ B )
= ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ A2 ) @ B )
= ( insert7817948997695205106c_fm_i @ X @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_875_subset__Compl__self__eq,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( uminus7605083404480446585c_fm_i @ A2 ) )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_Compl_self_eq
thf(fact_876_Diff__insert__absorb,axiom,
! [X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ A2 ) @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_877_insert__Diff,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( insert7817948997695205106c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_878_subset__Diff__insert,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ X @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_879_compl__coset,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( uminus7605083404480446585c_fm_i @ ( coset_Epistemic_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2 @ Xs ) ) ).
% compl_coset
thf(fact_880_coset__def,axiom,
( coset_Epistemic_fm_i
= ( ^ [Xs3: list_Epistemic_fm_i] : ( uminus7605083404480446585c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_881_is__singletonI_H,axiom,
! [A2: set_Epistemic_fm_i] :
( ( A2 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,Y2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( ( member6642669606046002379c_fm_i @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A2 ) ) ) ).
% is_singletonI'
thf(fact_882_infinite__remove,axiom,
! [S: set_Epistemic_fm_i,A: epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ ( insert7817948997695205106c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% infinite_remove
thf(fact_883_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_884_infinite__coinduct,axiom,
! [X4: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( X4 @ A2 )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( X4 @ A7 )
=> ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
& ( ( X4 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_885_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A7: set_nat] :
( ( X4 @ A7 )
=> ? [X6: nat] :
( ( member_nat2 @ X6 @ A7 )
& ( ( X4 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_886_finite__empty__induct,axiom,
! [A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A4: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_887_finite__empty__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat2 @ A4 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_888_Diff__single__insert,axiom,
! [A2: set_Epistemic_fm_i,X: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_889_subset__insert__iff,axiom,
! [A2: set_Epistemic_fm_i,X: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ B ) )
= ( ( ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) @ B ) )
& ( ~ ( member6642669606046002379c_fm_i @ X @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_890_in__image__insert__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,X: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ! [C3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ C3 @ B )
=> ~ ( member6642669606046002379c_fm_i @ X @ C3 ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( image_2698589095351095557c_fm_i @ ( insert7817948997695205106c_fm_i @ X ) @ B ) )
= ( ( member6642669606046002379c_fm_i @ X @ A2 )
& ( member1104366573291651755c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_891_finite__remove__induct,axiom,
! [B: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat2 @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_892_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_893_remove__induct,axiom,
! [P2: set_nat > $o,B: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat2 @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_894_remove__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,B: set_Epistemic_fm_i] :
( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_895_compl__le__compl__iff,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ X ) @ ( uminus7605083404480446585c_fm_i @ Y ) )
= ( ord_le3843937902494030498c_fm_i @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_896_diff__shunt__var,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_897_ComplI,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( member6642669606046002379c_fm_i @ C @ ( uminus7605083404480446585c_fm_i @ A2 ) ) ) ).
% ComplI
thf(fact_898_Compl__iff,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( uminus7605083404480446585c_fm_i @ A2 ) )
= ( ~ ( member6642669606046002379c_fm_i @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_899_ComplD,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( uminus7605083404480446585c_fm_i @ A2 ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ A2 ) ) ).
% ComplD
thf(fact_900_compl__le__swap2,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ Y ) @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_901_compl__le__swap1,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ ( uminus7605083404480446585c_fm_i @ X ) )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( uminus7605083404480446585c_fm_i @ Y ) ) ) ).
% compl_le_swap1
thf(fact_902_compl__mono,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ Y ) @ ( uminus7605083404480446585c_fm_i @ X ) ) ) ).
% compl_mono
thf(fact_903_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_904_infinite__Func,axiom,
! [A2: set_Epistemic_fm_i,B1: epistemic_fm_i,B22: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ B1 @ ( insert7817948997695205106c_fm_i @ B22 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ( B1 != B22 )
=> ~ ( finite4369126853184676732c_fm_i @ ( bNF_We489590214134045204c_fm_i @ A2 @ B ) ) ) ) ) ).
% infinite_Func
thf(fact_905_infinite__Func,axiom,
! [A2: set_nat,B1: epistemic_fm_i,B22: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ B1 @ ( insert7817948997695205106c_fm_i @ B22 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ( B1 != B22 )
=> ~ ( finite2015836717251302962c_fm_i @ ( bNF_We724601159059981372c_fm_i @ A2 @ B ) ) ) ) ) ).
% infinite_Func
thf(fact_906_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_907_Inf__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_908_finite__Int,axiom,
! [F3: set_Epistemic_fm_i,G3: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ F3 )
| ( finite3304564979551393739c_fm_i @ G3 ) )
=> ( finite3304564979551393739c_fm_i @ ( inf_in3450601097109690352c_fm_i @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_909_finite__Int,axiom,
! [F3: set_nat,G3: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G3 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_910_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
= ( ( ord_le3843937902494030498c_fm_i @ C2 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_911_Int__insert__left__if0,axiom,
! [A: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_912_Int__insert__left__if1,axiom,
! [A: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_913_Int__insert__right__if0,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_914_Int__insert__right__if1,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ B ) )
= ( insert7817948997695205106c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_915_insert__disjoint_I1_J,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ A2 ) @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A @ B )
& ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_916_insert__disjoint_I2_J,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ A2 ) @ B ) )
= ( ~ ( member6642669606046002379c_fm_i @ A @ B )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_917_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_i,A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A @ A2 ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A @ B )
& ( ( inf_in3450601097109690352c_fm_i @ B @ A2 )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_918_disjoint__insert_I2_J,axiom,
! [A2: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B2 @ A2 )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_919_in__set__remove1,axiom,
! [A: epistemic_fm_i,B2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( A != B2 )
=> ( ( member6642669606046002379c_fm_i @ A @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ B2 @ Xs ) ) )
= ( member6642669606046002379c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_920_inf__Sup__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_921_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_922_Int__insert__right,axiom,
! [A: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ B ) )
= ( insert7817948997695205106c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_923_Int__insert__left,axiom,
! [A: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_924_Int__mono,axiom,
! [A2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_925_Int__lower1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_926_Int__lower2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_927_Int__absorb1,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_928_Int__absorb2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_929_Int__greatest,axiom,
! [C2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_930_Int__Collect__mono,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P2 ) ) @ ( inf_in3450601097109690352c_fm_i @ B @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_931_disjoint__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A2 )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_932_Int__emptyI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ~ ( member6642669606046002379c_fm_i @ X3 @ B ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_933_remove1_Osimps_I2_J,axiom,
! [X: epistemic_fm_i,Y: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( X = Y )
=> ( ( remove8001893566138066622c_fm_i @ X @ ( cons_Epistemic_fm_i @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove8001893566138066622c_fm_i @ X @ ( cons_Epistemic_fm_i @ Y @ Xs ) )
= ( cons_Epistemic_fm_i @ Y @ ( remove8001893566138066622c_fm_i @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_934_remove1_Osimps_I1_J,axiom,
! [X: epistemic_fm_i] :
( ( remove8001893566138066622c_fm_i @ X @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% remove1.simps(1)
thf(fact_935_notin__set__remove1,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_936_remove1__idem,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_937_Inf__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_938_set__remove1__subset,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_939_remove1__append,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( append9179727448351703357c_fm_i @ ( remove8001893566138066622c_fm_i @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( append9179727448351703357c_fm_i @ Xs @ ( remove8001893566138066622c_fm_i @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_940_inf__shunt,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ X @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X @ ( uminus7605083404480446585c_fm_i @ Y ) ) ) ).
% inf_shunt
thf(fact_941_disjoint__eq__subset__Compl,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A2 @ ( uminus7605083404480446585c_fm_i @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_942_remove1__split,axiom,
! [A: epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( ( remove8001893566138066622c_fm_i @ A @ Xs )
= Ys )
= ( ? [Ls: list_Epistemic_fm_i,Rs: list_Epistemic_fm_i] :
( ( Xs
= ( append9179727448351703357c_fm_i @ Ls @ ( cons_Epistemic_fm_i @ A @ Rs ) ) )
& ~ ( member6642669606046002379c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Ls ) )
& ( Ys
= ( append9179727448351703357c_fm_i @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_943_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H @ ( inf_inf_nat @ X3 @ Y2 ) )
= ( inf_inf_nat @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_944_Inf__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_945_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat2 @ ( inf_inf_nat @ X3 @ Y2 ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_946_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_947_IntI,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_948_Int__iff,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A2 )
& ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_949_le__inf__iff,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ Y @ Z3 ) )
= ( ( ord_le3843937902494030498c_fm_i @ X @ Y )
& ( ord_le3843937902494030498c_fm_i @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_950_inf_Obounded__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_951_IntE,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_952_IntD1,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A2 ) ) ).
% IntD1
thf(fact_953_IntD2,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_954_inf_OcoboundedI2,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_955_inf_OcoboundedI1,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_956_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_957_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_958_inf_Ocobounded2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_959_inf_Ocobounded1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_960_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( A3
= ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_961_inf__greatest,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_962_inf_OboundedI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_963_inf_OboundedE,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_964_inf__absorb2,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( inf_in3450601097109690352c_fm_i @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_965_inf__absorb1,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( inf_in3450601097109690352c_fm_i @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_966_inf_Oabsorb2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_967_inf_Oabsorb1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_968_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_969_inf__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_970_inf_OorderI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% inf.orderI
thf(fact_971_inf_OorderE,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( A
= ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_972_le__infI2,axiom,
! [B2: set_Epistemic_fm_i,X: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_973_le__infI1,axiom,
! [A: set_Epistemic_fm_i,X: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_974_inf__mono,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_975_le__infI,axiom,
! [X: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_976_le__infE,axiom,
! [X: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ A @ B2 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X @ A )
=> ~ ( ord_le3843937902494030498c_fm_i @ X @ B2 ) ) ) ).
% le_infE
thf(fact_977_inf__le2,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_978_inf__le1,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_979_inf__sup__ord_I1_J,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_980_inf__sup__ord_I2_J,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_981_set__remove1__eq,axiom,
! [Xs: list_Epistemic_fm_i,X: epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% set_remove1_eq
thf(fact_982_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_983_Sup__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_984_finite__Un,axiom,
! [F3: set_Epistemic_fm_i,G3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F3 @ G3 ) )
= ( ( finite3304564979551393739c_fm_i @ F3 )
& ( finite3304564979551393739c_fm_i @ G3 ) ) ) ).
% finite_Un
thf(fact_985_finite__Un,axiom,
! [F3: set_nat,G3: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G3 ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_986_Un__subset__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_987_le__sup__iff,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) @ Z3 )
= ( ( ord_le3843937902494030498c_fm_i @ X @ Z3 )
& ( ord_le3843937902494030498c_fm_i @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_988_sup_Obounded__iff,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A )
= ( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
& ( ord_le3843937902494030498c_fm_i @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_989_set__append,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ).
% set_append
thf(fact_990_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_991_distinct__append,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( ( distin5379319190628067867c_fm_i @ Xs )
& ( distin5379319190628067867c_fm_i @ Ys )
& ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% distinct_append
thf(fact_992_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_993_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_994_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
=> ~ ! [A8: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A8 @ A2 )
=> ! [B7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B7 @ B )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_995_Un__absorb2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_996_Un__absorb1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_997_Un__upper2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% Un_upper2
thf(fact_998_Un__upper1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% Un_upper1
thf(fact_999_Un__least,axiom,
! [A2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1000_Un__mono,axiom,
! [A2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1001_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1002_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1003_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1004_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1005_sup_Ocobounded2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_1006_sup_Ocobounded1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_1007_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( A3
= ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1008_sup_OboundedI,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1009_sup_OboundedE,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1010_sup__absorb2,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( sup_su1936195050962291414c_fm_i @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1011_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( sup_su1936195050962291414c_fm_i @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1012_sup_Oabsorb2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_1013_sup_Oabsorb1,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_1014_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z4 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y2 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1015_sup_OorderI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ).
% sup.orderI
thf(fact_1016_sup_OorderE,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( A
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_1017_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1018_sup__least,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_1019_sup__mono,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1020_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_1021_le__supI2,axiom,
! [X: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_1022_le__supI1,axiom,
! [X: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_1023_sup__ge2,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) ) ).
% sup_ge2
thf(fact_1024_sup__ge1,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) ) ).
% sup_ge1
thf(fact_1025_le__supI,axiom,
! [A: set_Epistemic_fm_i,X: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ X )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_1026_le__supE,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ X )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ X )
=> ~ ( ord_le3843937902494030498c_fm_i @ B2 @ X ) ) ) ).
% le_supE
thf(fact_1027_inf__sup__ord_I3_J,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1028_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1029_Sup__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1030_distinct_Osimps_I1_J,axiom,
distin5379319190628067867c_fm_i @ nil_Epistemic_fm_i ).
% distinct.simps(1)
thf(fact_1031_distinct__length__2__or__more,axiom,
! [A: epistemic_fm_i,B2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ A @ ( cons_Epistemic_fm_i @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ A @ Xs ) )
& ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1032_infinite__Un,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) )
= ( ~ ( finite3304564979551393739c_fm_i @ S )
| ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_Un
thf(fact_1033_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1034_Un__infinite,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1035_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1036_finite__UnI,axiom,
! [F3: set_Epistemic_fm_i,G3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( finite3304564979551393739c_fm_i @ G3 )
=> ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_1037_finite__UnI,axiom,
! [F3: set_nat,G3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G3 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_1038_finite__distinct__list,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ? [Xs2: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs2 )
= A2 )
& ( distin5379319190628067867c_fm_i @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1039_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A2 )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1040_distinct__singleton,axiom,
! [X: epistemic_fm_i] : ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) ).
% distinct_singleton
thf(fact_1041_distinct_Osimps_I2_J,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ X @ Xs ) )
= ( ~ ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( distin5379319190628067867c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1042_distrib__sup__le,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ Y @ Z3 ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1043_distrib__inf__le,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X @ Z3 ) ) @ ( inf_in3450601097109690352c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1044_Un__Int__assoc__eq,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1045_Diff__subset__conv,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A2 @ B ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1046_Diff__partition,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_1047_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1048_distinct__concat,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin4780646717405202849c_fm_i @ Xs )
=> ( ! [Ys2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys2 ) )
=> ( ! [Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( Ys2 != Zs3 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys2 ) @ ( set_Epistemic_fm_i2 @ Zs3 ) )
= bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1049_maximal__def,axiom,
! [A2: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ S )
= ( ! [P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ).
% maximal_def
thf(fact_1050_distinct__concat__iff,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) )
= ( ( distin4780646717405202849c_fm_i @ ( remove5890047714521709364c_fm_i @ nil_Epistemic_fm_i @ Xs ) )
& ! [Ys3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys3 ) )
& ! [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( member6475149752759350353c_fm_i @ Zs2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( Ys3 != Zs2 ) )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys3 ) @ ( set_Epistemic_fm_i2 @ Zs2 ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% distinct_concat_iff
thf(fact_1051_not__distinct__decomp,axiom,
! [Ws: list_Epistemic_fm_i] :
( ~ ( distin5379319190628067867c_fm_i @ Ws )
=> ? [Xs2: list_Epistemic_fm_i,Ys2: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i,Y2: epistemic_fm_i] :
( Ws
= ( append9179727448351703357c_fm_i @ Xs2 @ ( append9179727448351703357c_fm_i @ ( cons_Epistemic_fm_i @ Y2 @ nil_Epistemic_fm_i ) @ ( append9179727448351703357c_fm_i @ Ys2 @ ( append9179727448351703357c_fm_i @ ( cons_Epistemic_fm_i @ Y2 @ nil_Epistemic_fm_i ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1052_not__distinct__conv__prefix,axiom,
! [As2: list_Epistemic_fm_i] :
( ( ~ ( distin5379319190628067867c_fm_i @ As2 ) )
= ( ? [Xs3: list_Epistemic_fm_i,Y3: epistemic_fm_i,Ys3: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Xs3 ) )
& ( distin5379319190628067867c_fm_i @ Xs3 )
& ( As2
= ( append9179727448351703357c_fm_i @ Xs3 @ ( cons_Epistemic_fm_i @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1053_shunt1,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ Y ) @ Z3 )
= ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ ( uminus7605083404480446585c_fm_i @ Y ) @ Z3 ) ) ) ).
% shunt1
thf(fact_1054_shunt2,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X @ ( uminus7605083404480446585c_fm_i @ Y ) ) @ Z3 )
= ( ord_le3843937902494030498c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ Y @ Z3 ) ) ) ).
% shunt2
thf(fact_1055_sup__neg__inf,axiom,
! [P: set_Epistemic_fm_i,Q: set_Epistemic_fm_i,R: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ P @ ( sup_su1936195050962291414c_fm_i @ Q @ R ) )
= ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ P @ ( uminus7605083404480446585c_fm_i @ Q ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1056_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H @ ( sup_sup_nat @ X3 @ Y2 ) )
= ( sup_sup_nat @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1057_Sup__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1058_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat2 @ ( sup_sup_nat @ X3 @ Y2 ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1059_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1060_Inf__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1061_set__union,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( union_Epistemic_fm_i @ Xs @ Ys ) )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ).
% set_union
thf(fact_1062_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( ! [P4: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_1063_UnCI,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ A2 ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1064_Un__iff,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A2 )
| ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_1065_UnE,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_1066_UnI1,axiom,
! [C: epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A2 )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1067_UnI2,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1068_MCS__axioms,axiom,
! [A2: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) ) ).
% MCS_axioms
thf(fact_1069_extendS__def,axiom,
! [A2: epistemic_fm_i > $o,N2: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N2 @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ S @ N2 @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N2 @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N2 @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A2 ) @ S @ N2 @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_1070_chains__extend,axiom,
! [C: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i,Z3: set_Epistemic_fm_i] :
( ( member1461078328125707403c_fm_i @ C @ ( chains830531755891748338c_fm_i @ S ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ S )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ C )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Z3 ) )
=> ( member1461078328125707403c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ Z3 @ bot_bo145720340923748686c_fm_i ) @ C ) @ ( chains830531755891748338c_fm_i @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1071_consistent__disjuncts,axiom,
! [A2: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A2 @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P @ Q ) @ V )
=> ( ( episte2285483198712856234tent_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P @ bot_bo4194595901900360558c_fm_i ) @ V ) )
| ( episte2285483198712856234tent_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_1072_Zorn__Lemma2,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ ( chains830531755891748338c_fm_i @ A2 ) )
=> ? [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
& ! [Xb2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xb2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ Xb2 @ Xa ) ) ) )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1073_chainsD,axiom,
! [C: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i,X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( member1461078328125707403c_fm_i @ C @ ( chains830531755891748338c_fm_i @ S ) )
=> ( ( member1104366573291651755c_fm_i @ X @ C )
=> ( ( member1104366573291651755c_fm_i @ Y @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Y )
| ( ord_le3843937902494030498c_fm_i @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_1074_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_1075_inj__on__Cons1,axiom,
! [X: epistemic_fm_i,A2: set_li8269783359719514504c_fm_i] : ( inj_on8045994122548803953c_fm_i @ ( cons_Epistemic_fm_i @ X ) @ A2 ) ).
% inj_on_Cons1
thf(fact_1076_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_1077_finite__image__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( ( finite3304564979551393739c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
= ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1078_finite__image__iff,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat] :
( ( inj_on6046519332259356447c_fm_i @ F @ A2 )
=> ( ( finite3304564979551393739c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1079_finite__image__iff,axiom,
! [F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i] :
( ( inj_on4442902948846411329_i_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_2291338433772226221_i_nat @ F @ A2 ) )
= ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1080_finite__image__iff,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_1081_finite__imageD,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_imageD
thf(fact_1082_finite__imageD,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat] :
( ( finite3304564979551393739c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
=> ( ( inj_on6046519332259356447c_fm_i @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_1083_finite__imageD,axiom,
! [F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i] :
( ( finite_finite_nat @ ( image_2291338433772226221_i_nat @ F @ A2 ) )
=> ( ( inj_on4442902948846411329_i_nat @ F @ A2 )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ).
% finite_imageD
thf(fact_1084_finite__imageD,axiom,
! [F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_1085_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr1261947904930325089at_nat,Consistent: set_nat > $o,S: set_nat] :
( ( maxima4501515429357695765rd_nat @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ S )
& ( finite_finite_nat @ S2 )
& ~ ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1086_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
& ( finite3304564979551393739c_fm_i @ S2 )
& ~ ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1087_endo__inj__surj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ A2 )
=> ( ( inj_on_nat_nat @ F @ A2 )
=> ( ( image_nat_nat @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1088_endo__inj__surj,axiom,
! [A2: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) @ A2 )
=> ( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( ( image_120052958407486725c_fm_i @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1089_inj__on__finite,axiom,
! [F: epistemic_fm_i > nat,A2: set_Epistemic_fm_i,B: set_nat] :
( ( inj_on4442902948846411329_i_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_2291338433772226221_i_nat @ F @ A2 ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1090_inj__on__finite,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1091_inj__on__finite,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1092_inj__on__finite,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat,B: set_Epistemic_fm_i] :
( ( inj_on6046519332259356447c_fm_i @ F @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1093_finite__surj__inj,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F @ A2 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1094_finite__surj__inj,axiom,
! [A2: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ( inj_on6844387624173906033c_fm_i @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1095_inj__on__image__mem__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i,A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ B )
=> ( ( member6642669606046002379c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( member6642669606046002379c_fm_i @ ( F @ A ) @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
= ( member6642669606046002379c_fm_i @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1096_inj__img__insertE,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,X: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X @ B )
= ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ~ ! [X7: epistemic_fm_i,A8: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X7 @ A8 )
=> ( ( A2
= ( insert7817948997695205106c_fm_i @ X7 @ A8 ) )
=> ( ( X
= ( F @ X7 ) )
=> ( B
!= ( image_120052958407486725c_fm_i @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1097_subset__image__inj,axiom,
! [S: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ ( image_120052958407486725c_fm_i @ F @ T2 ) )
= ( ? [U: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ U @ T2 )
& ( inj_on6844387624173906033c_fm_i @ F @ U )
& ( S
= ( image_120052958407486725c_fm_i @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_1098_one__set__greater,axiom,
! [A2: set_Epistemic_fm_i,A9: set_Epistemic_fm_i] :
( ? [F5: epistemic_fm_i > epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F5 @ A2 ) @ A9 )
& ( inj_on6844387624173906033c_fm_i @ F5 @ A2 ) )
| ? [G4: epistemic_fm_i > epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ G4 @ A9 ) @ A2 )
& ( inj_on6844387624173906033c_fm_i @ G4 @ A9 ) ) ) ).
% one_set_greater
thf(fact_1099_Min_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_1100_Min_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_1101_Min__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min_insert
thf(fact_1102_min__def,axiom,
( ord_mi5813855470740368105c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( if_set2943457721768014536c_fm_i @ ( ord_le3843937902494030498c_fm_i @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_1103_min__absorb1,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( ord_mi5813855470740368105c_fm_i @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1104_min__absorb2,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( ord_mi5813855470740368105c_fm_i @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1105_Min_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ).
% Min.in_idem
thf(fact_1106_hom__Min__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H @ ( ord_min_nat @ X3 @ Y2 ) )
= ( ord_min_nat @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic8721135487736765967in_nat @ N ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1107_Min_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset
thf(fact_1108_Min_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1109_Min_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat2 @ ( ord_min_nat @ X3 @ Y2 ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ) ).
% Min.closed
thf(fact_1110_Min_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( lattic8721135487736765967in_nat @ B ) ) ) ) ) ) ) ).
% Min.union
thf(fact_1111_Max_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_1112_Max_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_1113_Max__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ).
% Max_insert
thf(fact_1114_Max_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.in_idem
thf(fact_1115_max__absorb2,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ( ord_ma5652343050864542807c_fm_i @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1116_max__absorb1,axiom,
! [Y: set_Epistemic_fm_i,X: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X )
=> ( ( ord_ma5652343050864542807c_fm_i @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_1117_max__def,axiom,
( ord_ma5652343050864542807c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( if_set2943457721768014536c_fm_i @ ( ord_le3843937902494030498c_fm_i @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_1118_hom__Max__commute,axiom,
! [H: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H @ ( ord_max_nat @ X3 @ Y2 ) )
= ( ord_max_nat @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H @ ( lattic8265883725875713057ax_nat @ N ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_1119_Max_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B ) @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ).
% Max.subset
thf(fact_1120_Max_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat2 @ ( ord_max_nat @ X3 @ Y2 ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ) ).
% Max.closed
thf(fact_1121_Max_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_1122_Max_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ) ) ).
% Max.union
thf(fact_1123_inj__image__Compl__subset,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat] :
( ( inj_on6046519332259356447c_fm_i @ F @ top_top_set_nat )
=> ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus7605083404480446585c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1124_the__inv__into__into,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,X: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ A2 )
=> ( ( member6642669606046002379c_fm_i @ X @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( member6642669606046002379c_fm_i @ ( the_in689375056060326579c_fm_i @ A2 @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1125_UNIV__I,axiom,
! [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ top_to8364301187473967186c_fm_i ) ).
% UNIV_I
thf(fact_1126_UNIV__I,axiom,
! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1127_finite__Plus__UNIV__iff,axiom,
( ( finite7028649318107984188c_fm_i @ top_to795506832365754251c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1128_finite__Plus__UNIV__iff,axiom,
( ( finite3794849440691794892_i_nat @ top_to6930978477685885395_i_nat )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1129_finite__Plus__UNIV__iff,axiom,
( ( finite1315171008826062962c_fm_i @ top_to8843800515868777337c_fm_i )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1130_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1131_Int__UNIV,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1132_max__top,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% max_top
thf(fact_1133_max__top2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% max_top2
thf(fact_1134_min__top,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ top_top_set_nat @ X )
= X ) ).
% min_top
thf(fact_1135_min__top2,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ X @ top_top_set_nat )
= X ) ).
% min_top2
thf(fact_1136_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_1137_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_1138_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_1139_Diff__UNIV,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1140_finite__compl,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( finite3304564979551393739c_fm_i @ ( uminus7605083404480446585c_fm_i @ A2 ) )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_compl
thf(fact_1141_finite__compl,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_1142_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1143_inj__image__mem__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ top_to8364301187473967186c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ ( F @ A ) @ ( image_120052958407486725c_fm_i @ F @ A2 ) )
= ( member6642669606046002379c_fm_i @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1144_inj__image__mem__iff,axiom,
! [F: nat > epistemic_fm_i,A: nat,A2: set_nat] :
( ( inj_on6046519332259356447c_fm_i @ F @ top_top_set_nat )
=> ( ( member6642669606046002379c_fm_i @ ( F @ A ) @ ( image_3894954817185171339c_fm_i @ F @ A2 ) )
= ( member_nat2 @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_1145_range__ex1__eq,axiom,
! [F: nat > epistemic_fm_i,B2: epistemic_fm_i] :
( ( inj_on6046519332259356447c_fm_i @ F @ top_top_set_nat )
=> ( ( member6642669606046002379c_fm_i @ B2 @ ( image_3894954817185171339c_fm_i @ F @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B2
= ( F @ X2 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1146_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X2: nat] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1147_surjI,axiom,
! [G2: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G2 @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1148_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1149_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1150_ex__new__if__finite,axiom,
! [A2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ A2 )
=> ? [A4: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1151_ex__new__if__finite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A2 )
=> ? [A4: nat] :
~ ( member_nat2 @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1152_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1153_finite__prod,axiom,
( ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_1154_finite__prod,axiom,
( ( finite7666249648261937720_i_nat @ top_to7957592587874534535_i_nat )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1155_finite__prod,axiom,
( ( finite5186571216396205790c_fm_i @ top_to647042589202650669c_fm_i )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_1156_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1157_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_1158_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite7666249648261937720_i_nat @ top_to7957592587874534535_i_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1159_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite5186571216396205790c_fm_i @ top_to647042589202650669c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_1160_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1161_Finite__Set_Ofinite__set,axiom,
( ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% Finite_Set.finite_set
thf(fact_1162_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1163_UNIV__witness,axiom,
? [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ top_to8364301187473967186c_fm_i ) ).
% UNIV_witness
thf(fact_1164_UNIV__witness,axiom,
? [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1165_UNIV__eq__I,axiom,
! [A2: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( top_to8364301187473967186c_fm_i = A2 ) ) ).
% UNIV_eq_I
thf(fact_1166_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X3: nat] : ( member_nat2 @ X3 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_1167_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1168_subset__UNIV,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ top_to8364301187473967186c_fm_i ) ).
% subset_UNIV
thf(fact_1169_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_1170_top__greatest,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ top_to8364301187473967186c_fm_i ) ).
% top_greatest
thf(fact_1171_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1172_top_Oextremum__unique,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A )
= ( A = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_unique
thf(fact_1173_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1174_top_Oextremum__uniqueI,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A )
=> ( A = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_uniqueI
thf(fact_1175_Int__UNIV__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1176_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1177_infinite__UNIV__fm,axiom,
~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ).
% infinite_UNIV_fm
thf(fact_1178_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1179_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat2 @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_1180_Un__UNIV__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_1181_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_1182_range__eqI,axiom,
! [B2: epistemic_fm_i,F: nat > epistemic_fm_i,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member6642669606046002379c_fm_i @ B2 @ ( image_3894954817185171339c_fm_i @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_1183_rangeI,axiom,
! [F: nat > epistemic_fm_i,X: nat] : ( member6642669606046002379c_fm_i @ ( F @ X ) @ ( image_3894954817185171339c_fm_i @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_1184_range__subsetD,axiom,
! [F: nat > epistemic_fm_i,B: set_Epistemic_fm_i,I: nat] :
( ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ top_top_set_nat ) @ B )
=> ( member6642669606046002379c_fm_i @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_1185_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_1186_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_1187_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1188_infinite__iff__countable__subset,axiom,
! [S: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ S ) )
= ( ? [F2: nat > epistemic_fm_i] :
( ( inj_on6046519332259356447c_fm_i @ F2 @ top_top_set_nat )
& ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1189_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_1190_infinite__countable__subset,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ? [F5: nat > epistemic_fm_i] :
( ( inj_on6046519332259356447c_fm_i @ F5 @ top_top_set_nat )
& ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F5 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_1191_Compl__partition2,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_1192_Compl__partition,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_1193_UNIV__coset,axiom,
( top_to8364301187473967186c_fm_i
= ( coset_Epistemic_fm_i @ nil_Epistemic_fm_i ) ) ).
% UNIV_coset
thf(fact_1194_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_1195_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% MCS.infinite_UNIV
thf(fact_1196_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_nat > $o] :
( ( maxima5350474959086842340CS_nat @ Consistent )
=> ~ ( finite_finite_nat @ top_top_set_nat ) ) ).
% MCS.infinite_UNIV
thf(fact_1197_sup__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1198_sup__shunt,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X @ Y )
= top_to8364301187473967186c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1199_boolean__algebra_Ocomplement__unique,axiom,
! [A: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1200_finite__UNIV__inj__surj,axiom,
! [F: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( inj_on6844387624173906033c_fm_i @ F @ top_to8364301187473967186c_fm_i )
=> ( ( image_120052958407486725c_fm_i @ F @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1201_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1202_finite__UNIV__surj__inj,axiom,
! [F: epistemic_fm_i > epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( ( image_120052958407486725c_fm_i @ F @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i )
=> ( inj_on6844387624173906033c_fm_i @ F @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1203_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1204_inj__image__subset__iff,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat,B: set_nat] :
( ( inj_on6046519332259356447c_fm_i @ F @ top_top_set_nat )
=> ( ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) @ ( image_3894954817185171339c_fm_i @ F @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1205_inj__image__subset__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( inj_on6844387624173906033c_fm_i @ F @ top_to8364301187473967186c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A2 ) @ ( image_120052958407486725c_fm_i @ F @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A2 @ B ) ) ) ).
% inj_image_subset_iff
thf(fact_1206_surj__Compl__image__subset,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1207_surj__Compl__image__subset,axiom,
! [F: nat > epistemic_fm_i,A2: set_nat] :
( ( ( image_3894954817185171339c_fm_i @ F @ top_top_set_nat )
= top_to8364301187473967186c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( uminus7605083404480446585c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ A2 ) ) @ ( image_3894954817185171339c_fm_i @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1208_MCS_Ointro,axiom,
! [Consistent: set_nat > $o] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ! [S4: set_nat,S2: set_nat] :
( ( Consistent @ S4 )
=> ( ( ord_less_eq_set_nat @ S2 @ S4 )
=> ( Consistent @ S2 ) ) )
=> ( ! [S4: set_nat] :
( ~ ( Consistent @ S4 )
=> ? [S5: set_nat] :
( ( ord_less_eq_set_nat @ S5 @ S4 )
& ( finite_finite_nat @ S5 )
& ~ ( Consistent @ S5 ) ) )
=> ( maxima5350474959086842340CS_nat @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_1209_MCS_Ointro,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ! [S4: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( Consistent @ S4 )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S4 )
=> ( Consistent @ S2 ) ) )
=> ( ! [S4: set_Epistemic_fm_i] :
( ~ ( Consistent @ S4 )
=> ? [S5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S5 @ S4 )
& ( finite3304564979551393739c_fm_i @ S5 )
& ~ ( Consistent @ S5 ) ) )
=> ( maxima1924290493700099598c_fm_i @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_1210_MCS__def,axiom,
( maxima5350474959086842340CS_nat
= ( ^ [Consistent2: set_nat > $o] :
( ~ ( finite_finite_nat @ top_top_set_nat )
& ! [S6: set_nat,S7: set_nat] :
( ( Consistent2 @ S6 )
=> ( ( ord_less_eq_set_nat @ S7 @ S6 )
=> ( Consistent2 @ S7 ) ) )
& ! [S6: set_nat] :
( ~ ( Consistent2 @ S6 )
=> ? [S7: set_nat] :
( ( ord_less_eq_set_nat @ S7 @ S6 )
& ( finite_finite_nat @ S7 )
& ~ ( Consistent2 @ S7 ) ) ) ) ) ) ).
% MCS_def
thf(fact_1211_MCS__def,axiom,
( maxima1924290493700099598c_fm_i
= ( ^ [Consistent2: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ! [S6: set_Epistemic_fm_i,S7: set_Epistemic_fm_i] :
( ( Consistent2 @ S6 )
=> ( ( ord_le3843937902494030498c_fm_i @ S7 @ S6 )
=> ( Consistent2 @ S7 ) ) )
& ! [S6: set_Epistemic_fm_i] :
( ~ ( Consistent2 @ S6 )
=> ? [S7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S7 @ S6 )
& ( finite3304564979551393739c_fm_i @ S7 )
& ~ ( Consistent2 @ S7 ) ) ) ) ) ) ).
% MCS_def
thf(fact_1212_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1213_finite__option__UNIV,axiom,
( ( finite4430841290055254033c_fm_i @ top_to7034302889105699736c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% finite_option_UNIV
thf(fact_1214_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1215_cofinite__bot,axiom,
( ( cofini2965452609477090530c_fm_i = bot_bo3270057444204665492c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% cofinite_bot
thf(fact_1216_cofinite__bot,axiom,
( ( cofinite_nat = bot_bot_filter_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% cofinite_bot
thf(fact_1217_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1218_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1219_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1220_mono__Un,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) @ ( F @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% mono_Un
thf(fact_1221_mono__sup,axiom,
! [F: nat > set_Epistemic_fm_i,A2: nat,B: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) @ ( F @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% mono_sup
thf(fact_1222_mono__sup,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) @ ( F @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% mono_sup
thf(fact_1223_mono__inf,axiom,
! [F: nat > set_Epistemic_fm_i,A2: nat,B: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ ( inf_inf_nat @ A2 @ B ) ) @ ( inf_in3450601097109690352c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_1224_mono__inf,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) @ ( inf_in3450601097109690352c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_1225_mono__Int,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ ( inf_in3450601097109690352c_fm_i @ A2 @ B ) ) @ ( inf_in3450601097109690352c_fm_i @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% mono_Int
thf(fact_1226_mono__imp__mono__on,axiom,
! [F: nat > set_Epistemic_fm_i,A2: set_nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( monoto3888311813380466057c_fm_i @ A2 @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% mono_imp_mono_on
thf(fact_1227_mono__imp__mono__on,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( monoto9149264989429730663c_fm_i @ A2 @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% mono_imp_mono_on
thf(fact_1228_monoI,axiom,
! [F: nat > set_Epistemic_fm_i] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% monoI
thf(fact_1229_monoI,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% monoI
thf(fact_1230_monoE,axiom,
! [F: nat > set_Epistemic_fm_i,X: nat,Y: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1231_monoE,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1232_monoD,axiom,
! [F: nat > set_Epistemic_fm_i,X: nat,Y: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1233_monoD,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ top_to3675301018284359218c_fm_i @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Y )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1234_max__of__mono,axiom,
! [F: nat > set_Epistemic_fm_i,M2: nat,N2: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_ma5652343050864542807c_fm_i @ ( F @ M2 ) @ ( F @ N2 ) )
= ( F @ ( ord_max_nat @ M2 @ N2 ) ) ) ) ).
% max_of_mono
thf(fact_1235_min__of__mono,axiom,
! [F: nat > set_Epistemic_fm_i,M2: nat,N2: nat] :
( ( monoto3888311813380466057c_fm_i @ top_top_set_nat @ ord_less_eq_nat @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_mi5813855470740368105c_fm_i @ ( F @ M2 ) @ ( F @ N2 ) )
= ( F @ ( ord_min_nat @ M2 @ N2 ) ) ) ) ).
% min_of_mono
thf(fact_1236_mono__on__subset,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( monoto9149264989429730663c_fm_i @ A2 @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A2 )
=> ( monoto9149264989429730663c_fm_i @ B @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F ) ) ) ).
% mono_on_subset
thf(fact_1237_ord_Omono__on__subset,axiom,
! [A2: set_Epistemic_fm_i,Less_eq: epistemic_fm_i > epistemic_fm_i > $o,F: epistemic_fm_i > set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( monoto8024655975081467719c_fm_i @ A2 @ Less_eq @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( monoto8024655975081467719c_fm_i @ B @ Less_eq @ ord_le3843937902494030498c_fm_i @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_1238_mono__onI,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [R3: set_Epistemic_fm_i,S8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ R3 @ A2 )
=> ( ( member1104366573291651755c_fm_i @ S8 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ R3 @ S8 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ R3 ) @ ( F @ S8 ) ) ) ) )
=> ( monoto9149264989429730663c_fm_i @ A2 @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% mono_onI
thf(fact_1239_mono__onD,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,R: set_Epistemic_fm_i,S9: set_Epistemic_fm_i] :
( ( monoto9149264989429730663c_fm_i @ A2 @ ord_le3843937902494030498c_fm_i @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( member1104366573291651755c_fm_i @ R @ A2 )
=> ( ( member1104366573291651755c_fm_i @ S9 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ R @ S9 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ R ) @ ( F @ S9 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1240_ord_Omono__on__def,axiom,
! [A2: set_Epistemic_fm_i,Less_eq: epistemic_fm_i > epistemic_fm_i > $o,F: epistemic_fm_i > set_Epistemic_fm_i] :
( ( monoto8024655975081467719c_fm_i @ A2 @ Less_eq @ ord_le3843937902494030498c_fm_i @ F )
= ( ! [R2: epistemic_fm_i,S10: epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ R2 @ A2 )
& ( member6642669606046002379c_fm_i @ S10 @ A2 )
& ( Less_eq @ R2 @ S10 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ R2 ) @ ( F @ S10 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_1241_ord_Omono__onI,axiom,
! [A2: set_Epistemic_fm_i,Less_eq: epistemic_fm_i > epistemic_fm_i > $o,F: epistemic_fm_i > set_Epistemic_fm_i] :
( ! [R3: epistemic_fm_i,S8: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ R3 @ A2 )
=> ( ( member6642669606046002379c_fm_i @ S8 @ A2 )
=> ( ( Less_eq @ R3 @ S8 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ R3 ) @ ( F @ S8 ) ) ) ) )
=> ( monoto8024655975081467719c_fm_i @ A2 @ Less_eq @ ord_le3843937902494030498c_fm_i @ F ) ) ).
% ord.mono_onI
thf(fact_1242_ord_Omono__onD,axiom,
! [A2: set_Epistemic_fm_i,Less_eq: epistemic_fm_i > epistemic_fm_i > $o,F: epistemic_fm_i > set_Epistemic_fm_i,R: epistemic_fm_i,S9: epistemic_fm_i] :
( ( monoto8024655975081467719c_fm_i @ A2 @ Less_eq @ ord_le3843937902494030498c_fm_i @ F )
=> ( ( member6642669606046002379c_fm_i @ R @ A2 )
=> ( ( member6642669606046002379c_fm_i @ S9 @ A2 )
=> ( ( Less_eq @ R @ S9 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ R ) @ ( F @ S9 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_1243_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ A )
= nil_Epistemic_fm_i )
= ( A = nil_Epistemic_fm_i ) ) ).
% list.map_disc_iff
thf(fact_1244_Nil__is__map__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( nil_Epistemic_fm_i
= ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% Nil_is_map_conv
thf(fact_1245_map__is__Nil__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= nil_Epistemic_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% map_is_Nil_conv
thf(fact_1246_list_Oset__map,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,V2: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( map_Ep2755178516647988292c_fm_i @ F @ V2 ) )
= ( image_120052958407486725c_fm_i @ F @ ( set_Epistemic_fm_i2 @ V2 ) ) ) ).
% list.set_map
thf(fact_1247_list_Omap__ident__strong,axiom,
! [T3: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( set_Epistemic_fm_i2 @ T3 ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_1248_map__idI,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1249_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% list.simps(8)
thf(fact_1250_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( cons_Epistemic_fm_i @ Y @ Ys ) )
= ( ? [Z5: epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Xs
= ( cons_Epistemic_fm_i @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_Ep2755178516647988292c_fm_i @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1251_Cons__eq__map__conv,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ F @ Ys ) )
= ( ? [Z5: epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Ys
= ( cons_Epistemic_fm_i @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_Ep2755178516647988292c_fm_i @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1252_map__eq__Cons__D,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,Y: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( cons_Epistemic_fm_i @ Y @ Ys ) )
=> ? [Z4: epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Xs
= ( cons_Epistemic_fm_i @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Ep2755178516647988292c_fm_i @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1253_Cons__eq__map__D,axiom,
! [X: epistemic_fm_i,Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ F @ Ys ) )
=> ? [Z4: epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( Ys
= ( cons_Epistemic_fm_i @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_Ep2755178516647988292c_fm_i @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1254_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( cons_Epistemic_fm_i @ ( F @ X21 ) @ ( map_Ep2755178516647988292c_fm_i @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1255_image__set,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( image_120052958407486725c_fm_i @ F @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) ) ) ).
% image_set
thf(fact_1256_distinct__set__subseqs,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( distin8272202008718213115c_fm_i @ ( map_li712259506734541342c_fm_i @ set_Epistemic_fm_i2 @ ( subseq859285874047815415c_fm_i @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_1257_finite__UN,axiom,
! [A2: set_Epistemic_fm_i,B: epistemic_fm_i > set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( finite3304564979551393739c_fm_i @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ B @ A2 ) ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A2 )
=> ( finite3304564979551393739c_fm_i @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1258_finite__UN,axiom,
! [A2: set_Epistemic_fm_i,B: epistemic_fm_i > set_nat] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_5232765980333981283et_nat @ B @ A2 ) ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A2 )
=> ( finite_finite_nat @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1259_finite__UN,axiom,
! [A2: set_nat,B: nat > set_Epistemic_fm_i] :
( ( finite_finite_nat @ A2 )
=> ( ( finite3304564979551393739c_fm_i @ ( comple8437045806746640111c_fm_i @ ( image_3332579458907315563c_fm_i @ B @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( finite3304564979551393739c_fm_i @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1260_finite__UN,axiom,
! [A2: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( finite_finite_nat @ ( B @ X2 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1261_finite__Union,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A2 )
=> ( ! [M3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ M3 @ A2 )
=> ( finite3304564979551393739c_fm_i @ M3 ) )
=> ( finite3304564979551393739c_fm_i @ ( comple8437045806746640111c_fm_i @ A2 ) ) ) ) ).
% finite_Union
thf(fact_1262_finite__Union,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [M3: set_nat] :
( ( member_set_nat2 @ M3 @ A2 )
=> ( finite_finite_nat @ M3 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_1263_Zorn__Lemma,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ ( chains830531755891748338c_fm_i @ A2 ) )
=> ( member1104366573291651755c_fm_i @ ( comple8437045806746640111c_fm_i @ X3 ) @ A2 ) )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1264_sorted__wrt__mono__rel,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > epistemic_fm_i > $o,Q2: epistemic_fm_i > epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i,Y2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( member6642669606046002379c_fm_i @ Y2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( P2 @ X3 @ Y2 )
=> ( Q2 @ X3 @ Y2 ) ) ) )
=> ( ( sorted73444948313710223c_fm_i @ P2 @ Xs )
=> ( sorted73444948313710223c_fm_i @ Q2 @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1265_sorted__wrt_Osimps_I1_J,axiom,
! [P2: epistemic_fm_i > epistemic_fm_i > $o] : ( sorted73444948313710223c_fm_i @ P2 @ nil_Epistemic_fm_i ) ).
% sorted_wrt.simps(1)
thf(fact_1266_sorted__wrt__append,axiom,
! [P2: epistemic_fm_i > epistemic_fm_i > $o,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( sorted73444948313710223c_fm_i @ P2 @ ( append9179727448351703357c_fm_i @ Xs @ Ys ) )
= ( ( sorted73444948313710223c_fm_i @ P2 @ Xs )
& ( sorted73444948313710223c_fm_i @ P2 @ Ys )
& ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ! [Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( P2 @ X2 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1267_sorted__wrt1,axiom,
! [P2: epistemic_fm_i > epistemic_fm_i > $o,X: epistemic_fm_i] : ( sorted73444948313710223c_fm_i @ P2 @ ( cons_Epistemic_fm_i @ X @ nil_Epistemic_fm_i ) ) ).
% sorted_wrt1
thf(fact_1268_finite__UnionD,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( comple8437045806746640111c_fm_i @ A2 ) )
=> ( finite7933139204641697195c_fm_i @ A2 ) ) ).
% finite_UnionD
thf(fact_1269_finite__UnionD,axiom,
! [A2: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_1270_finite__sorted__distinct__unique,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [X3: list_nat] :
( ( ( set_nat2 @ X3 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X3 )
& ( distinct_nat @ X3 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1271_finite__subset__Union,axiom,
! [A2: set_nat,B8: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
=> ~ ! [F6: set_set_nat] :
( ( finite1152437895449049373et_nat @ F6 )
=> ( ( ord_le6893508408891458716et_nat @ F6 @ B8 )
=> ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1272_finite__subset__Union,axiom,
! [A2: set_Epistemic_fm_i,B8: set_se3485332733965609186c_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ ( comple8437045806746640111c_fm_i @ B8 ) )
=> ~ ! [F6: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F6 )
=> ( ( ord_le5389487502678872194c_fm_i @ F6 @ B8 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A2 @ ( comple8437045806746640111c_fm_i @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1273_Sup__SUP__eq,axiom,
( comple5681178248825093806fm_i_o
= ( ^ [S6: set_Epistemic_fm_i_o,X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ ( comple8437045806746640111c_fm_i @ ( image_5459497653761036832c_fm_i @ collec4904205187116291597c_fm_i @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1274_cSup__eq__maximum,axiom,
! [Z3: set_Epistemic_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ X4 )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Z3 ) )
=> ( ( comple8437045806746640111c_fm_i @ X4 )
= Z3 ) ) ) ).
% cSup_eq_maximum
% Helper facts (15)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X: list_Epistemic_fm_i,Y: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X: list_s8081015415394010888c_fm_i,Y: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X: list_s8081015415394010888c_fm_i,Y: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X: list_l4360594050116504750c_fm_i,Y: list_l4360594050116504750c_fm_i] :
( ( if_lis7843061787625270900c_fm_i @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X: list_l4360594050116504750c_fm_i,Y: list_l4360594050116504750c_fm_i] :
( ( if_lis7843061787625270900c_fm_i @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J_T,axiom,
! [X: list_s8635415444435394792c_fm_i,Y: list_s8635415444435394792c_fm_i] :
( ( if_lis4466469686133009838c_fm_i @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J_T,axiom,
! [X: list_s8635415444435394792c_fm_i,Y: list_s8635415444435394792c_fm_i] :
( ( if_lis4466469686133009838c_fm_i @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ~ ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ sa ) @ v )
| ~ ( member6642669606046002379c_fm_i @ a2 @ v )
| ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ a2 @ sa ) ) @ v ) ) ).
%------------------------------------------------------------------------------