TPTP Problem File: SLH0524^1.p

%------------------------------------------------------------------------------
% 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    : FSM_Tests/0036_FSM/prob_05193_199803__19414656_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2009 ( 449 unt; 738 typ;   0 def)
%            Number of atoms       : 4261 (1722 equ;   0 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 16843 ( 702   ~;  32   |; 527   &;13437   @)
%                                         (   0 <=>;2145  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  161 ( 160 usr)
%            Number of type conns  : 1814 (1814   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  581 ( 578 usr;  35 con; 0-5 aty)
%            Number of variables   : 4935 ( 530   ^;4035   !; 370   ?;4935   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:27:50.500
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__b_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__c_Mt__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_Itf__c_Mtf__c_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__c_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__b_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__a_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__b_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__a_J_J,type,
    list_P7671510403078805059_b_c_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
    list_l5832389826979350138od_b_c: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    set_Pr3138385297192119153od_b_c: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__a_J_J,type,
    set_Pr2175785243794050845_b_c_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__a_J_J,type,
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thf(ty_n_t__FSM__Ofsm_Itf__b_Mtf__c_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__FSM__Ofsm_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__b_Mtf__c_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_Itf__b_M_062_Itf__b_M_Eo_J_J_Mt__List__Olist_Itf__b_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    produc5460929350080682669_c_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mt__Nat__Onat_J,type,
    produc163404553760264935_c_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    produc2133359284458116493od_c_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    produc3550516665487611281od_b_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__b_J,type,
    produc8127812668234276542_b_c_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__a_J,type,
    produc8127812663931047741_b_c_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
    produc3963297410138542439list_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    set_li6436108459499378894od_b_c: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    set_se3344102371033692456od_b_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_Itf__b_M_Eo_J_Mt__List__Olist_Itf__b_J_J,type,
    produc8914622519208346940list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_Mtf__a_J,type,
    produc7386240147096006850_b_c_a: $tType ).

thf(ty_n_t__FSM__Ofsm_Itf__c_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_Mtf__b_J,type,
    fsm_c_6732129604652128891_b_c_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__List__Olist_Itf__b_J_J,type,
    produc1021604436286309741list_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__b_J_J,type,
    produc8757014435846287022list_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    list_P903359562653991662od_b_c: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    list_P832325522607645677od_b_b: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    list_P1396940483166286381od_a_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
    list_list_list_b: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    set_Product_prod_c_a: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    set_Product_prod_b_c: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    set_Product_prod_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_M_062_Itf__c_M_Eo_J_J_J,type,
    set_b_c_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    product_prod_c_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    product_prod_nat_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    set_option_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
    set_a_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    set_nat_a: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_Itf__c_J_J,type,
    option_set_c: $tType ).

thf(ty_n_t__FSM__Ofsm_Itf__b_Mtf__c_Mt__Nat__Onat_J,type,
    fsm_b_c_nat: $tType ).

thf(ty_n_t__FSM__Ofsm_It__Nat__Onat_Mtf__b_Mtf__c_J,type,
    fsm_nat_b_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__c_Mtf__a_J,type,
    product_prod_c_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    product_prod_b_c: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    product_prod_b_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    product_prod_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    product_prod_a_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__b_J_J,type,
    list_list_b: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
    set_list_b: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    set_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    set_a_a: $tType ).

thf(ty_n_t__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J,type,
    fsm_a_b_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
    set_a_o: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Option__Ooption_Itf__a_J,type,
    option_a: $tType ).

thf(ty_n_t__List__Olist_Itf__c_J,type,
    list_c: $tType ).

thf(ty_n_t__List__Olist_Itf__b_J,type,
    list_b: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__c,type,
    c: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (578)
thf(sy_c_AList_Omap__default_001tf__b_001tf__c,type,
    map_default_b_c: b > c > ( c > c ) > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_AList_Omap__default__rel_001tf__b_001tf__c,type,
    map_default_rel_b_c: produc2009451851958091931od_b_c > produc2009451851958091931od_b_c > $o ).

thf(sy_c_AList_Omap__entry_001tf__b_001tf__c,type,
    map_entry_b_c: b > ( c > c ) > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_AList_Omap__entry__rel_001tf__b_001tf__c,type,
    map_entry_rel_b_c: produc6452300523622238572od_b_c > produc6452300523622238572od_b_c > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    bNF_Gr4730963734287941769od_b_c: set_li336664667694595924od_b_c > list_l8907847357763382004od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    bNF_Gr8135241567188112771od_b_c: set_li6436108459499378894od_b_c > list_P903359562653991662od_b_c > set_Product_prod_b_c ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
    bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__b,type,
    bNF_Greatest_Succ_b: set_list_b > list_b > set_b ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_M_Eo_J,type,
    comple6218144523844345314_b_c_o: set_li313786449249895621_b_c_o > list_P903359562653991662od_b_c > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
    comple8317665133742190828_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
    complete_Sup_Sup_a_o: set_a_o > a > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__b_M_062_Itf__c_M_Eo_J_J,type,
    comple1411054825784354266_b_c_o: set_b_c_o > b > c > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    comple1250193257546725051od_b_c: set_se3924713247505902254od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    comple7928098250179550773od_b_c: set_se3344102371033692456od_b_c > set_Product_prod_b_c ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
    comple2307003609928055243_set_a: set_set_a > set_a ).

thf(sy_c_FSM_OLS_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    lS_lis2930931384350476499_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_FSM_OLS_001tf__a_001tf__b_001tf__c,type,
    lS_a_b_c: fsm_a_b_c > a > set_li6436108459499378894od_b_c ).

thf(sy_c_FSM_OLS_092_060_094sub_062i_092_060_094sub_062n_001tf__c_001t__Product____Type__Oprod_Itf__b_Mtf__c_J_001tf__b,type,
    lS_i_n6458876116888321724_b_c_b: fsm_c_6732129604652128891_b_c_b > c > set_li6436108459499378894od_b_c > set_li7348942078743790372_b_c_b ).

thf(sy_c_FSM_OLS_092_060_094sub_062i_092_060_094sub_062n__rel_001tf__c_001t__Product____Type__Oprod_Itf__b_Mtf__c_J_001tf__b,type,
    lS_i_n5679008109505604727_b_c_b: produc7449291840106446561od_b_c > produc7449291840106446561od_b_c > $o ).

thf(sy_c_FSM_OLS__acyclic_001tf__a_001tf__b_001tf__c,type,
    lS_acyclic_a_b_c: fsm_a_b_c > a > set_li6436108459499378894od_b_c ).

thf(sy_c_FSM_OLS__rel_001tf__a_001tf__b_001tf__c,type,
    lS_rel_a_b_c: produc7386240147096006850_b_c_a > produc7386240147096006850_b_c_a > $o ).

thf(sy_c_FSM_Oacyclic_001tf__a_001tf__b_001tf__c,type,
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% Relevant facts (1265)
thf(fact_0__092_060open_062map_Afst_Aio_A_061_Ax_A_D_Axs_092_060close_062,axiom,
    ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ io2 )
    = ( cons_b @ x @ xsa ) ) ).

% \<open>map fst io = x # xs\<close>
thf(fact_1_calculation_I2_J,axiom,
    member6330420149250801815od_b_c @ io @ ( lS_a_b_c @ m @ q2 ) ).

% calculation(2)
thf(fact_2_assms_I1_J,axiom,
    observable_a_b_c @ m ).

% assms(1)
thf(fact_3_assms_I2_J,axiom,
    member_a @ q @ ( states_a_b_c @ m ) ).

% assms(2)
thf(fact_4__092_060open_062set_A_Ilanguage__for__input_AM_Aq_H_Axs_J_A_061_A_123io_A_092_060in_062_ALS_AM_Aq_H_O_Amap_Afst_Aio_A_061_Axs_125_092_060close_062,axiom,
    ( ( set_li4480668622519654659od_b_c @ ( langua5143504920213240017_a_b_c @ m @ q2 @ xsa ) )
    = ( collec2280997390073109977od_b_c
      @ ^ [Io: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ m @ q2 ) )
          & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Io )
            = xsa ) ) ) ) ).

% \<open>set (language_for_input M q' xs) = {io \<in> LS M q'. map fst io = xs}\<close>
thf(fact_5__092_060open_062io_A_092_060in_062_ALS_AM_Aq_092_060close_062,axiom,
    member6330420149250801815od_b_c @ io2 @ ( lS_a_b_c @ m @ qa ) ).

% \<open>io \<in> LS M q\<close>
thf(fact_6_Cons_Oprems,axiom,
    member_a @ qa @ ( states_a_b_c @ m ) ).

% Cons.prems
thf(fact_7_map__eq__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) )
      = ( ! [X: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_8_Cons_OIH,axiom,
    ! [Q: a] :
      ( ( member_a @ Q @ ( states_a_b_c @ m ) )
     => ( ( set_li4480668622519654659od_b_c @ ( langua5143504920213240017_a_b_c @ m @ Q @ xsa ) )
        = ( collec2280997390073109977od_b_c
          @ ^ [Io: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ m @ Q ) )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Io )
                = xsa ) ) ) ) ) ).

% Cons.IH
thf(fact_9_calculation_I1_J,axiom,
    ( ( h_obs_a_b_c @ m @ qa @ x @ y )
    = ( some_a @ q2 ) ) ).

% calculation(1)
thf(fact_10_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X22 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_11_list_Oset__intros_I2_J,axiom,
    ! [Y: list_P903359562653991662od_b_c,X22: list_l8907847357763382004od_b_c,X21: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ X22 ) )
     => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_12_list_Oset__intros_I2_J,axiom,
    ! [Y: b,X22: list_b,X21: b] :
      ( ( member_b @ Y @ ( set_b2 @ X22 ) )
     => ( member_b @ Y @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_13_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_b_c,X22: list_P903359562653991662od_b_c,X21: product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ X22 ) )
     => ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_14_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_15_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P903359562653991662od_b_c,X22: list_l8907847357763382004od_b_c] : ( member6330420149250801815od_b_c @ X21 @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_16_list_Oset__intros_I1_J,axiom,
    ! [X21: b,X22: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_17_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] : ( member7862447941013992593od_b_c @ X21 @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_18_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z2: list_a] :
            ( A
           != ( cons_a @ E @ Z2 ) )
       => ~ ! [Z1: a,Z2: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z2 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_19_list_Oset__cases,axiom,
    ! [E: list_P903359562653991662od_b_c,A: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ E @ ( set_li4480668622519654659od_b_c @ A ) )
     => ( ! [Z2: list_l8907847357763382004od_b_c] :
            ( A
           != ( cons_l342771267123639716od_b_c @ E @ Z2 ) )
       => ~ ! [Z1: list_P903359562653991662od_b_c,Z2: list_l8907847357763382004od_b_c] :
              ( ( A
                = ( cons_l342771267123639716od_b_c @ Z1 @ Z2 ) )
             => ~ ( member6330420149250801815od_b_c @ E @ ( set_li4480668622519654659od_b_c @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_20_list_Oset__cases,axiom,
    ! [E: b,A: list_b] :
      ( ( member_b @ E @ ( set_b2 @ A ) )
     => ( ! [Z2: list_b] :
            ( A
           != ( cons_b @ E @ Z2 ) )
       => ~ ! [Z1: b,Z2: list_b] :
              ( ( A
                = ( cons_b @ Z1 @ Z2 ) )
             => ~ ( member_b @ E @ ( set_b2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_21_list_Oset__cases,axiom,
    ! [E: product_prod_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ E @ ( set_Product_prod_b_c2 @ A ) )
     => ( ! [Z2: list_P903359562653991662od_b_c] :
            ( A
           != ( cons_P4529483553340347422od_b_c @ E @ Z2 ) )
       => ~ ! [Z1: product_prod_b_c,Z2: list_P903359562653991662od_b_c] :
              ( ( A
                = ( cons_P4529483553340347422od_b_c @ Z1 @ Z2 ) )
             => ~ ( member7862447941013992593od_b_c @ E @ ( set_Product_prod_b_c2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_22_set__ConsD,axiom,
    ! [Y: a,X2: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_23_set__ConsD,axiom,
    ! [Y: list_P903359562653991662od_b_c,X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_24_set__ConsD,axiom,
    ! [Y: b,X2: b,Xs: list_b] :
      ( ( member_b @ Y @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_b @ Y @ ( set_b2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_25_set__ConsD,axiom,
    ! [Y: product_prod_b_c,X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_26__092_060open_062set_A_Ilanguage__for__input_AM_Aq_A_Ix_A_D_Axs_J_J_A_092_060subseteq_062_A_123io_A_092_060in_062_ALS_AM_Aq_O_Amap_Afst_Aio_A_061_Ax_A_D_Axs_125_092_060close_062,axiom,
    ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( langua5143504920213240017_a_b_c @ m @ qa @ ( cons_b @ x @ xsa ) ) )
    @ ( collec2280997390073109977od_b_c
      @ ^ [Io: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ m @ qa ) )
          & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Io )
            = ( cons_b @ x @ xsa ) ) ) ) ) ).

% \<open>set (language_for_input M q (x # xs)) \<subseteq> {io \<in> LS M q. map fst io = x # xs}\<close>
thf(fact_27_list_Oinject,axiom,
    ! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
      ( ( ( cons_b @ X21 @ X22 )
        = ( cons_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_28_list_Oinject,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c,Y21: product_prod_b_c,Y22: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X21 @ X22 )
        = ( cons_P4529483553340347422od_b_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_29__092_060open_062io_A_092_060in_062_A_123io_A_092_060in_062_ALS_AM_Aq_O_Amap_Afst_Aio_A_061_Ax_A_D_Axs_125_092_060close_062,axiom,
    ( member6330420149250801815od_b_c @ io2
    @ ( collec2280997390073109977od_b_c
      @ ^ [Io: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ m @ qa ) )
          & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Io )
            = ( cons_b @ x @ xsa ) ) ) ) ) ).

% \<open>io \<in> {io \<in> LS M q. map fst io = x # xs}\<close>
thf(fact_30__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_H_O_A_092_060lbrakk_062FSM_Oh__obs_AM_Aq_Ax_Ay_A_061_ASome_Aq_H_059_Aio_H_A_092_060in_062_ALS_AM_Aq_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q2: a] :
        ( ( ( h_obs_a_b_c @ m @ qa @ x @ y )
          = ( some_a @ Q2 ) )
       => ~ ( member6330420149250801815od_b_c @ io @ ( lS_a_b_c @ m @ Q2 ) ) ) ).

% \<open>\<And>thesis. (\<And>q'. \<lbrakk>FSM.h_obs M q x y = Some q'; io' \<in> LS M q'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_31__092_060open_062io_A_061_A_Ix_M_Ay_J_A_D_Aio_H_092_060close_062,axiom,
    ( io2
    = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ x @ y ) @ io ) ) ).

% \<open>io = (x, y) # io'\<close>
thf(fact_32_list_Osimps_I9_J,axiom,
    ! [F: b > b,X21: b,X22: list_b] :
      ( ( map_b_b @ F @ ( cons_b @ X21 @ X22 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_b_b @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_33_list_Osimps_I9_J,axiom,
    ! [F: b > product_prod_b_c,X21: b,X22: list_b] :
      ( ( map_b_4433156289202544749od_b_c @ F @ ( cons_b @ X21 @ X22 ) )
      = ( cons_P4529483553340347422od_b_c @ ( F @ X21 ) @ ( map_b_4433156289202544749od_b_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_34_list_Osimps_I9_J,axiom,
    ! [F: product_prod_b_c > b,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_Pr8222292247188198875_b_c_b @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_35_list_Osimps_I9_J,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( map_Pr6384634858812015940od_b_c @ F @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( cons_P4529483553340347422od_b_c @ ( F @ X21 ) @ ( map_Pr6384634858812015940od_b_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_36_h__obs__state,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Q3: a] :
      ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
        = ( some_a @ Q3 ) )
     => ( member_a @ Q3 @ ( states_a_b_c @ M ) ) ) ).

% h_obs_state
thf(fact_37_Cons__eq__map__D,axiom,
    ! [X2: b,Xs: list_b,F: b > b,Ys: list_b] :
      ( ( ( cons_b @ X2 @ Xs )
        = ( map_b_b @ F @ Ys ) )
     => ? [Z: b,Zs: list_b] :
          ( ( Ys
            = ( cons_b @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_b_b @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_38_Cons__eq__map__D,axiom,
    ! [X2: b,Xs: list_b,F: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_b @ X2 @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
     => ? [Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Ys
            = ( cons_P4529483553340347422od_b_c @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr8222292247188198875_b_c_b @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_39_Cons__eq__map__D,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: b > product_prod_b_c,Ys: list_b] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_b_4433156289202544749od_b_c @ F @ Ys ) )
     => ? [Z: b,Zs: list_b] :
          ( ( Ys
            = ( cons_b @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_b_4433156289202544749od_b_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_40_Cons__eq__map__D,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr6384634858812015940od_b_c @ F @ Ys ) )
     => ? [Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Ys
            = ( cons_P4529483553340347422od_b_c @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr6384634858812015940od_b_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_41_map__eq__Cons__D,axiom,
    ! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( cons_b @ Y @ Ys ) )
     => ? [Z: b,Zs: list_b] :
          ( ( Xs
            = ( cons_b @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_b_b @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_42_map__eq__Cons__D,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Y: b,Ys: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( cons_b @ Y @ Ys ) )
     => ? [Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( cons_P4529483553340347422od_b_c @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_Pr8222292247188198875_b_c_b @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_43_map__eq__Cons__D,axiom,
    ! [F: b > product_prod_b_c,Xs: list_b,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_b_4433156289202544749od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y @ Ys ) )
     => ? [Z: b,Zs: list_b] :
          ( ( Xs
            = ( cons_b @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_b_4433156289202544749od_b_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_44_map__eq__Cons__D,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y @ Ys ) )
     => ? [Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( cons_P4529483553340347422od_b_c @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_Pr6384634858812015940od_b_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_45_not__Cons__self2,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( cons_b @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_46_not__Cons__self2,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_47_Cons__eq__map__conv,axiom,
    ! [X2: b,Xs: list_b,F: b > b,Ys: list_b] :
      ( ( ( cons_b @ X2 @ Xs )
        = ( map_b_b @ F @ Ys ) )
      = ( ? [Z3: b,Zs2: list_b] :
            ( ( Ys
              = ( cons_b @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_b_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_48_Cons__eq__map__conv,axiom,
    ! [X2: b,Xs: list_b,F: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_b @ X2 @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
      = ( ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_Pr8222292247188198875_b_c_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_49_Cons__eq__map__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: b > product_prod_b_c,Ys: list_b] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_b_4433156289202544749od_b_c @ F @ Ys ) )
      = ( ? [Z3: b,Zs2: list_b] :
            ( ( Ys
              = ( cons_b @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_b_4433156289202544749od_b_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_50_Cons__eq__map__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr6384634858812015940od_b_c @ F @ Ys ) )
      = ( ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_Pr6384634858812015940od_b_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_51_map__eq__Cons__conv,axiom,
    ! [F: b > b,Xs: list_b,Y: b,Ys: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = ( cons_b @ Y @ Ys ) )
      = ( ? [Z3: b,Zs2: list_b] :
            ( ( Xs
              = ( cons_b @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_b_b @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_52_map__eq__Cons__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Y: b,Ys: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( cons_b @ Y @ Ys ) )
      = ( ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_Pr8222292247188198875_b_c_b @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_53_map__eq__Cons__conv,axiom,
    ! [F: b > product_prod_b_c,Xs: list_b,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_b_4433156289202544749od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y @ Ys ) )
      = ( ? [Z3: b,Zs2: list_b] :
            ( ( Xs
              = ( cons_b @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_b_4433156289202544749od_b_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_54_map__eq__Cons__conv,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y @ Ys ) )
      = ( ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_Pr6384634858812015940od_b_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_55_set__subset__Cons,axiom,
    ! [Xs: list_b,X2: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_56_set__subset__Cons,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X2: product_prod_b_c] : ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_57_set__subset__Cons,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,X2: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_58_subset__code_I1_J,axiom,
    ! [Xs: list_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( member_a @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_59_subset__code_I1_J,axiom,
    ! [Xs: list_P903359562653991662od_b_c,B: set_Product_prod_b_c] :
      ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ B )
      = ( ! [X: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
           => ( member7862447941013992593od_b_c @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_60_subset__code_I1_J,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ B )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ( member6330420149250801815od_b_c @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_61_ex__map__conv,axiom,
    ! [Ys: list_b,F: product_prod_b_c > b] :
      ( ( ? [Xs2: list_P903359562653991662od_b_c] :
            ( Ys
            = ( map_Pr8222292247188198875_b_c_b @ F @ Xs2 ) ) )
      = ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Ys ) )
           => ? [Y2: product_prod_b_c] :
                ( X
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_62_map__cong,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ( Xs = Ys )
     => ( ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
          = ( map_Pr8222292247188198875_b_c_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_63_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_64_map__idI,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_li8081432502061035332od_b_c @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_65_map__idI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c] :
      ( ! [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_66_map__ext,axiom,
    ! [Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ! [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_67_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_68_list_Omap__ident__strong,axiom,
    ! [T: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ! [Z: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Z @ ( set_li4480668622519654659od_b_c @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_li8081432502061035332od_b_c @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_69_list_Omap__ident__strong,axiom,
    ! [T: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c] :
      ( ! [Z: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z @ ( set_Product_prod_b_c2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr6384634858812015940od_b_c @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_70_list_Oinj__map__strong,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xa: list_P903359562653991662od_b_c,F: product_prod_b_c > b,Fa: product_prod_b_c > b] :
      ( ! [Z: product_prod_b_c,Za: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z @ ( set_Product_prod_b_c2 @ X2 ) )
         => ( ( member7862447941013992593od_b_c @ Za @ ( set_Product_prod_b_c2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr8222292247188198875_b_c_b @ F @ X2 )
          = ( map_Pr8222292247188198875_b_c_b @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_71_list_Omap__cong0,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ! [Z: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z @ ( set_Product_prod_b_c2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr8222292247188198875_b_c_b @ F @ X2 )
        = ( map_Pr8222292247188198875_b_c_b @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_72_list_Omap__cong,axiom,
    ! [X2: list_P903359562653991662od_b_c,Ya: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ( X2 = Ya )
     => ( ! [Z: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ Z @ ( set_Product_prod_b_c2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ F @ X2 )
          = ( map_Pr8222292247188198875_b_c_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_73__092_060open_062_Ix_M_Ay_J_A_D_Aio_H_A_092_060in_062_ALS_AM_Aq_092_060close_062,axiom,
    member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ x @ y ) @ io ) @ ( lS_a_b_c @ m @ qa ) ).

% \<open>(x, y) # io' \<in> LS M q\<close>
thf(fact_74__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_Aio_H_O_Aio_A_061_A_Ix_M_Ay_J_A_D_Aio_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Y3: c,Io2: list_P903359562653991662od_b_c] :
        ( io2
       != ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ x @ Y3 ) @ Io2 ) ) ).

% \<open>\<And>thesis. (\<And>y io'. io = (x, y) # io' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_75_option_Oinject,axiom,
    ! [X23: a,Y23: a] :
      ( ( ( some_a @ X23 )
        = ( some_a @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% option.inject
thf(fact_76_subsetI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ( member_a @ X3 @ B ) )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% subsetI
thf(fact_77_subsetI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A2 )
         => ( member6330420149250801815od_b_c @ X3 @ B ) )
     => ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ).

% subsetI
thf(fact_78_subset__antisym,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_79_order__refl,axiom,
    ! [X2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X2 @ X2 ) ).

% order_refl
thf(fact_80_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_81_dual__order_Orefl,axiom,
    ! [A: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A @ A ) ).

% dual_order.refl
thf(fact_82_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_83_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
    ! [A: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ A @ ( collec2280997390073109977od_b_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_89_Collect__cong,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( P @ X3 )
          = ( Q4 @ X3 ) )
     => ( ( collec2280997390073109977od_b_c @ P )
        = ( collec2280997390073109977od_b_c @ Q4 ) ) ) ).

% Collect_cong
thf(fact_90_Collect__cong,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q4 @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q4 ) ) ) ).

% Collect_cong
thf(fact_91_list__map__source__elem,axiom,
    ! [X2: a,F: list_P903359562653991662od_b_c > a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_92_list__map__source__elem,axiom,
    ! [X2: a,F: product_prod_b_c > a,Xs: list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( map_Pr8222292247188198874_b_c_a @ F @ Xs ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_93_list__map__source__elem,axiom,
    ! [X2: b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( member_b @ X2 @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_94_list__map__source__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_95_list__map__source__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_96_list__map__source__elem,axiom,
    ! [X2: product_prod_b_c,F: list_P903359562653991662od_b_c > product_prod_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( map_li8582383987200892350od_b_c @ F @ Xs ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_97_list__map__source__elem,axiom,
    ! [X2: product_prod_b_c,F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( X2
            = ( F @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_98_list__map__set__prop,axiom,
    ! [X2: b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,P: b > $o] :
      ( ( member_b @ X2 @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) )
     => ( ! [Y3: product_prod_b_c] : ( P @ ( F @ Y3 ) )
       => ( P @ X2 ) ) ) ).

% list_map_set_prop
thf(fact_99_set__map__elem,axiom,
    ! [Y: a,F: a > a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_a_a @ F @ Xs ) ) )
     => ~ ! [X3: a] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_100_set__map__elem,axiom,
    ! [Y: a,F: list_P903359562653991662od_b_c > a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_101_set__map__elem,axiom,
    ! [Y: a,F: product_prod_b_c > a,Xs: list_P903359562653991662od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_Pr8222292247188198874_b_c_a @ F @ Xs ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_102_set__map__elem,axiom,
    ! [Y: b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( member_b @ Y @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_103_set__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,Xs: list_a] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: a] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_104_set__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_105_set__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_106_set__map__elem,axiom,
    ! [Y: product_prod_b_c,F: a > product_prod_b_c,Xs: list_a] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( map_a_5072596589534360878od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: a] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_107_set__map__elem,axiom,
    ! [Y: product_prod_b_c,F: list_P903359562653991662od_b_c > product_prod_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( map_li8582383987200892350od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_108_set__map__elem,axiom,
    ! [Y: product_prod_b_c,F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( Y
              = ( F @ X3 ) )
           => ~ ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_109_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_110_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_111_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > product_prod_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member7862447941013992593od_b_c @ ( F @ X2 ) @ ( set_Product_prod_b_c2 @ ( map_a_5072596589534360878od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_112_map__set,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_113_map__set,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_114_map__set,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > product_prod_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( member7862447941013992593od_b_c @ ( F @ X2 ) @ ( set_Product_prod_b_c2 @ ( map_li8582383987200892350od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_115_map__set,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > a] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_Pr8222292247188198874_b_c_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_116_map__set,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( member_b @ ( F @ X2 ) @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_117_map__set,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( set_li4480668622519654659od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_118_map__set,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( member7862447941013992593od_b_c @ ( F @ X2 ) @ ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_119_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima8146652644187019584_c_b_c @ M @ Q @ Io3 ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_120_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima1559550560783484624_a_b_c @ M @ Q @ Io3 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_121_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X: a] : ( member_a @ X @ A3 )
          @ ^ [X: a] : ( member_a @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_122_less__eq__set__def,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( ord_le1261880702084804079_b_c_o
          @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A3 )
          @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_123_map__pair__fst,axiom,
    ! [F: b > c,Xs: list_b] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
        @ ( map_b_4433156289202544749od_b_c
          @ ^ [X: b] : ( product_Pair_b_c @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_124_order__antisym__conv,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X2 )
     => ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_125_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_126_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_127_ord__le__eq__subst,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
    ! [A: nat,B3: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
    ! [A: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_131_ord__eq__le__subst,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_li6436108459499378894od_b_c > nat,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B3: nat,C: nat] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_135_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_136_order__eq__refl,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( X2 = Y )
     => ( ord_le282488521294790766od_b_c @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_137_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_138_order__subst2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B3 ) @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_139_order__subst2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_140_order__subst2,axiom,
    ! [A: nat,B3: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_141_order__subst2,axiom,
    ! [A: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_142_order__subst1,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( F @ B3 ) )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_143_order__subst1,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B3: nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le282488521294790766od_b_c @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_144_order__subst1,axiom,
    ! [A: nat,F: set_li6436108459499378894od_b_c > nat,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_145_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_146_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] : ( Y4 = Z4 ) )
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
          & ( ord_le282488521294790766od_b_c @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_147_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_148_antisym,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_149_antisym,axiom,
    ! [A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_150_dual__order_Otrans,axiom,
    ! [B3: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ A )
     => ( ( ord_le282488521294790766od_b_c @ C @ B3 )
       => ( ord_le282488521294790766od_b_c @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_151_dual__order_Otrans,axiom,
    ! [B3: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_152_dual__order_Oantisym,axiom,
    ! [B3: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ A )
     => ( ( ord_le282488521294790766od_b_c @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_153_dual__order_Oantisym,axiom,
    ! [B3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B3 @ A )
     => ( ( ord_less_eq_nat @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_154_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] : ( Y4 = Z4 ) )
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
          & ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_155_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_156_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B3: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B3 ) ) ) ).

% linorder_wlog
thf(fact_157_order__trans,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z5: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ Z5 )
       => ( ord_le282488521294790766od_b_c @ X2 @ Z5 ) ) ) ).

% order_trans
thf(fact_158_order__trans,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z5 )
       => ( ord_less_eq_nat @ X2 @ Z5 ) ) ) ).

% order_trans
thf(fact_159_order_Otrans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ord_le282488521294790766od_b_c @ A @ C ) ) ) ).

% order.trans
thf(fact_160_order_Otrans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_161_order__antisym,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_162_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_163_ord__le__eq__trans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_le282488521294790766od_b_c @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_164_ord__le__eq__trans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_165_ord__eq__le__trans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A = B3 )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ord_le282488521294790766od_b_c @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_166_ord__eq__le__trans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( A = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_167_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] : ( Y4 = Z4 ) )
    = ( ^ [X: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ X @ Y2 )
          & ( ord_le282488521294790766od_b_c @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_168_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_169_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z5 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z5 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z5 )
           => ~ ( ord_less_eq_nat @ Z5 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z5 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z5 )
               => ~ ( ord_less_eq_nat @ Z5 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z5 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_170_nle__le,axiom,
    ! [A: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A )
        & ( B3 != A ) ) ) ).

% nle_le
thf(fact_171_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q4 ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q4 @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_172_Collect__mono__iff,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q4 ) )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( P @ X )
           => ( Q4 @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_173_set__eq__subset,axiom,
    ( ( ^ [Y4: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] : ( Y4 = Z4 ) )
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A3 @ B2 )
          & ( ord_le282488521294790766od_b_c @ B2 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_174_subset__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ C2 )
       => ( ord_le282488521294790766od_b_c @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_175_Collect__mono,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q4 @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q4 ) ) ) ).

% Collect_mono
thf(fact_176_Collect__mono,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( P @ X3 )
         => ( Q4 @ X3 ) )
     => ( ord_le282488521294790766od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ).

% Collect_mono
thf(fact_177_subset__refl,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A2 @ A2 ) ).

% subset_refl
thf(fact_178_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A3 )
         => ( member_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_179_subset__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
        ! [T2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ T2 @ A3 )
         => ( member6330420149250801815od_b_c @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_180_equalityD2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A2 = B )
     => ( ord_le282488521294790766od_b_c @ B @ A2 ) ) ).

% equalityD2
thf(fact_181_equalityD1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A2 = B )
     => ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ).

% equalityD1
thf(fact_182_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B2: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A3 )
         => ( member_a @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_183_subset__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
        ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ A3 )
         => ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_184_equalityE,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A2 = B )
     => ~ ( ( ord_le282488521294790766od_b_c @ A2 @ B )
         => ~ ( ord_le282488521294790766od_b_c @ B @ A2 ) ) ) ).

% equalityE
thf(fact_185_subsetD,axiom,
    ! [A2: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a @ C @ A2 )
       => ( member_a @ C @ B ) ) ) ).

% subsetD
thf(fact_186_subsetD,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( member6330420149250801815od_b_c @ C @ A2 )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% subsetD
thf(fact_187_in__mono,axiom,
    ! [A2: set_a,B: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a @ X2 @ A2 )
       => ( member_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_188_in__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( member6330420149250801815od_b_c @ X2 @ A2 )
       => ( member6330420149250801815od_b_c @ X2 @ B ) ) ) ).

% in_mono
thf(fact_189_h__obs__language__iff,axiom,
    ! [M: fsm_a_b_c,X2: b,Y: c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Io3 ) @ ( lS_a_b_c @ M @ Q ) )
        = ( ? [Q5: a] :
              ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
                = ( some_a @ Q5 ) )
              & ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q5 ) ) ) ) ) ) ).

% h_obs_language_iff
thf(fact_190_Collect__subset,axiom,
    ! [A2: set_a,P: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_191_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_192_Collect__subset,axiom,
    ! [A2: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ord_le282488521294790766od_b_c
      @ ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_193_old_Oprod_Oinject,axiom,
    ! [A: b,B3: c,A6: b,B6: c] :
      ( ( ( product_Pair_b_c @ A @ B3 )
        = ( product_Pair_b_c @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_194_prod_Oinject,axiom,
    ! [X1: b,X23: c,Y1: b,Y23: c] :
      ( ( ( product_Pair_b_c @ X1 @ X23 )
        = ( product_Pair_b_c @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_195_after__h__obs__prepend,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Q3: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
          = ( some_a @ Q3 ) )
       => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q3 ) )
         => ( ( after_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Io3 ) )
            = ( after_a_b_c @ M @ Q3 @ Io3 ) ) ) ) ) ).

% after_h_obs_prepend
thf(fact_196_h__obs__from__LS,axiom,
    ! [M: fsm_a_b_c,X2: b,Y: c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [Q2: a] :
              ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
             != ( some_a @ Q2 ) ) ) ) ).

% h_obs_from_LS
thf(fact_197_eq__fst__iff,axiom,
    ! [A: b,P2: product_prod_b_c] :
      ( ( A
        = ( product_fst_b_c @ P2 ) )
      = ( ? [B4: c] :
            ( P2
            = ( product_Pair_b_c @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_198_fst__conv,axiom,
    ! [X1: b,X23: c] :
      ( ( product_fst_b_c @ ( product_Pair_b_c @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_199_fst__eqD,axiom,
    ! [X2: b,Y: c,A: b] :
      ( ( ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_200_fstI,axiom,
    ! [X2: product_prod_b_c,Y: b,Z5: c] :
      ( ( X2
        = ( product_Pair_b_c @ Y @ Z5 ) )
     => ( ( product_fst_b_c @ X2 )
        = Y ) ) ).

% fstI
thf(fact_201_lists__of__length__elems,axiom,
    ! [Xs: list_P903359562653991662od_b_c,T3: list_P903359562653991662od_b_c,N: nat] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( lists_2361477089028216131od_b_c @ T3 @ N ) ) )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ T3 ) ) ) ).

% lists_of_length_elems
thf(fact_202_lists__of__length__elems,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,T3: list_l8907847357763382004od_b_c,N: nat] :
      ( ( member7934871825613078557od_b_c @ Xs @ ( set_li2649297040355570313od_b_c @ ( lists_3751910231668219977od_b_c @ T3 @ N ) ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ T3 ) ) ) ).

% lists_of_length_elems
thf(fact_203_maximal__prefix__in__language__properties_I2_J,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima8146652644187019584_c_b_c @ M @ Q @ Io3 ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Io3 ) ) ) ) ) ).

% maximal_prefix_in_language_properties(2)
thf(fact_204_maximal__prefix__in__language__properties_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima1559550560783484624_a_b_c @ M @ Q @ Io3 ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Io3 ) ) ) ) ) ).

% maximal_prefix_in_language_properties(2)
thf(fact_205_list_Omap__disc__iff,axiom,
    ! [F: product_prod_b_c > b,A: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ A )
        = nil_b )
      = ( A = nil_Product_prod_b_c ) ) ).

% list.map_disc_iff
thf(fact_206_Nil__is__map__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( nil_b
        = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% Nil_is_map_conv
thf(fact_207_map__is__Nil__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% map_is_Nil_conv
thf(fact_208_find__remove_H_Ocases,axiom,
    ! [X2: produc4714985347897971870list_b] :
      ( ! [P3: b > $o,Uu: list_b] :
          ( X2
         != ( produc1341956810442481360list_b @ P3 @ ( produc1564554178308465111list_b @ nil_b @ Uu ) ) )
     => ~ ! [P3: b > $o,X3: b,Xs3: list_b,Prev: list_b] :
            ( X2
           != ( produc1341956810442481360list_b @ P3 @ ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_209_find__remove_H_Ocases,axiom,
    ! [X2: produc5559707038353031495od_b_c] :
      ( ! [P3: product_prod_b_c > $o,Uu: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc1005531610559690873od_b_c @ P3 @ ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu ) ) )
     => ~ ! [P3: product_prod_b_c > $o,X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Prev: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc1005531610559690873od_b_c @ P3 @ ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_210_language__for__input_Osimps_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( langua5143504920213240017_a_b_c @ M @ Q @ nil_b )
      = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) ) ).

% language_for_input.simps(1)
thf(fact_211_find__index_Ocases,axiom,
    ! [X2: produc8914622519208346940list_b] :
      ( ! [F2: b > $o] :
          ( X2
         != ( produc6309431049009149422list_b @ F2 @ nil_b ) )
     => ~ ! [F2: b > $o,X3: b,Xs3: list_b] :
            ( X2
           != ( produc6309431049009149422list_b @ F2 @ ( cons_b @ X3 @ Xs3 ) ) ) ) ).

% find_index.cases
thf(fact_212_find__index_Ocases,axiom,
    ! [X2: produc7816210539016439438od_b_c] :
      ( ! [F2: product_prod_b_c > $o] :
          ( X2
         != ( produc7392021306980666816od_b_c @ F2 @ nil_Product_prod_b_c ) )
     => ~ ! [F2: product_prod_b_c > $o,X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc7392021306980666816od_b_c @ F2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ).

% find_index.cases
thf(fact_213_sorted__wrt_Ocases,axiom,
    ! [X2: produc5185152304234826110list_b] :
      ( ! [P3: b > b > $o] :
          ( X2
         != ( produc8193136575784045678list_b @ P3 @ nil_b ) )
     => ~ ! [P3: b > b > $o,X3: b,Ys2: list_b] :
            ( X2
           != ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_214_sorted__wrt_Ocases,axiom,
    ! [X2: produc2416979464461413415od_b_c] :
      ( ! [P3: product_prod_b_c > product_prod_b_c > $o] :
          ( X2
         != ( produc6092892548594537111od_b_c @ P3 @ nil_Product_prod_b_c ) )
     => ~ ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_215_successively_Ocases,axiom,
    ! [X2: produc5185152304234826110list_b] :
      ( ! [P3: b > b > $o] :
          ( X2
         != ( produc8193136575784045678list_b @ P3 @ nil_b ) )
     => ( ! [P3: b > b > $o,X3: b] :
            ( X2
           != ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X3 @ nil_b ) ) )
       => ~ ! [P3: b > b > $o,X3: b,Y3: b,Xs3: list_b] :
              ( X2
             != ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X3 @ ( cons_b @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_216_successively_Ocases,axiom,
    ! [X2: produc2416979464461413415od_b_c] :
      ( ! [P3: product_prod_b_c > product_prod_b_c > $o] :
          ( X2
         != ( produc6092892548594537111od_b_c @ P3 @ nil_Product_prod_b_c ) )
     => ( ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c] :
            ( X2
           != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) )
       => ~ ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Y3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_217_splice_Ocases,axiom,
    ! [X2: produc3963297410138542439list_b] :
      ( ! [Ys2: list_b] :
          ( X2
         != ( produc1564554178308465111list_b @ nil_b @ Ys2 ) )
     => ~ ! [X3: b,Xs3: list_b,Ys2: list_b] :
            ( X2
           != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_218_splice_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_219_shuffles_Ocases,axiom,
    ! [X2: produc3963297410138542439list_b] :
      ( ! [Ys2: list_b] :
          ( X2
         != ( produc1564554178308465111list_b @ nil_b @ Ys2 ) )
     => ( ! [Xs3: list_b] :
            ( X2
           != ( produc1564554178308465111list_b @ Xs3 @ nil_b ) )
       => ~ ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
              ( X2
             != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_220_shuffles_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ( ! [Xs3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ Xs3 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_221_is__prefix_Ocases,axiom,
    ! [X2: produc3963297410138542439list_b] :
      ( ! [Uu: list_b] :
          ( X2
         != ( produc1564554178308465111list_b @ nil_b @ Uu ) )
     => ( ! [X3: b,Xs3: list_b] :
            ( X2
           != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ nil_b ) )
       => ~ ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
              ( X2
             != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_222_is__prefix_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Uu: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu ) )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_223_transpose_Ocases,axiom,
    ! [X2: list_list_b] :
      ( ( X2 != nil_list_b )
     => ( ! [Xss: list_list_b] :
            ( X2
           != ( cons_list_b @ nil_b @ Xss ) )
       => ~ ! [X3: b,Xs3: list_b,Xss: list_list_b] :
              ( X2
             != ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_224_transpose_Ocases,axiom,
    ! [X2: list_l8907847357763382004od_b_c] :
      ( ( X2 != nil_li8071162985807626740od_b_c )
     => ( ! [Xss: list_l8907847357763382004od_b_c] :
            ( X2
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
              ( X2
             != ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_225_prefixes__set__Nil,axiom,
    ! [Xs: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) ) ).

% prefixes_set_Nil
thf(fact_226_list_Odistinct_I1_J,axiom,
    ! [X21: b,X22: list_b] :
      ( nil_b
     != ( cons_b @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_227_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( nil_Product_prod_b_c
     != ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_228_list_OdiscI,axiom,
    ! [List: list_b,X21: b,X22: list_b] :
      ( ( List
        = ( cons_b @ X21 @ X22 ) )
     => ( List != nil_b ) ) ).

% list.discI
thf(fact_229_list_OdiscI,axiom,
    ! [List: list_P903359562653991662od_b_c,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( List
        = ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
     => ( List != nil_Product_prod_b_c ) ) ).

% list.discI
thf(fact_230_list_Oexhaust,axiom,
    ! [Y: list_b] :
      ( ( Y != nil_b )
     => ~ ! [X212: b,X222: list_b] :
            ( Y
           != ( cons_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_231_list_Oexhaust,axiom,
    ! [Y: list_P903359562653991662od_b_c] :
      ( ( Y != nil_Product_prod_b_c )
     => ~ ! [X212: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
            ( Y
           != ( cons_P4529483553340347422od_b_c @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_232_min__list_Ocases,axiom,
    ! [X2: list_b] :
      ( ! [X3: b,Xs3: list_b] :
          ( X2
         != ( cons_b @ X3 @ Xs3 ) )
     => ( X2 = nil_b ) ) ).

% min_list.cases
thf(fact_233_min__list_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
          ( X2
         != ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
     => ( X2 = nil_Product_prod_b_c ) ) ).

% min_list.cases
thf(fact_234_remdups__adj_Ocases,axiom,
    ! [X2: list_b] :
      ( ( X2 != nil_b )
     => ( ! [X3: b] :
            ( X2
           != ( cons_b @ X3 @ nil_b ) )
       => ~ ! [X3: b,Y3: b,Xs3: list_b] :
              ( X2
             != ( cons_b @ X3 @ ( cons_b @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_235_remdups__adj_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Y3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( X2
             != ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_236_neq__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
      = ( ? [Y2: b,Ys3: list_b] :
            ( Xs
            = ( cons_b @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_237_neq__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
      = ( ? [Y2: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( Xs
            = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_238_list__induct2_H,axiom,
    ! [P: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
      ( ( P @ nil_b @ nil_b )
     => ( ! [X3: b,Xs3: list_b] : ( P @ ( cons_b @ X3 @ Xs3 ) @ nil_b )
       => ( ! [Y3: b,Ys2: list_b] : ( P @ nil_b @ ( cons_b @ Y3 @ Ys2 ) )
         => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_239_list__induct2_H,axiom,
    ! [P: list_b > list_P903359562653991662od_b_c > $o,Xs: list_b,Ys: list_P903359562653991662od_b_c] :
      ( ( P @ nil_b @ nil_Product_prod_b_c )
     => ( ! [X3: b,Xs3: list_b] : ( P @ ( cons_b @ X3 @ Xs3 ) @ nil_Product_prod_b_c )
       => ( ! [Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] : ( P @ nil_b @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) )
         => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_240_list__induct2_H,axiom,
    ! [P: list_P903359562653991662od_b_c > list_b > $o,Xs: list_P903359562653991662od_b_c,Ys: list_b] :
      ( ( P @ nil_Product_prod_b_c @ nil_b )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_b )
       => ( ! [Y3: b,Ys2: list_b] : ( P @ nil_Product_prod_b_c @ ( cons_b @ Y3 @ Ys2 ) )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_241_list__induct2_H,axiom,
    ! [P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_Product_prod_b_c )
       => ( ! [Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] : ( P @ nil_Product_prod_b_c @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_242_list__nonempty__induct,axiom,
    ! [Xs: list_b,P: list_b > $o] :
      ( ( Xs != nil_b )
     => ( ! [X3: b] : ( P @ ( cons_b @ X3 @ nil_b ) )
       => ( ! [X3: b,Xs3: list_b] :
              ( ( Xs3 != nil_b )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_b @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_243_list__nonempty__induct,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( Xs3 != nil_Product_prod_b_c )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_244_linear__order__from__list__position_H_Ocases,axiom,
    ! [X2: list_b] :
      ( ( X2 != nil_b )
     => ~ ! [X3: b,Xs3: list_b] :
            ( X2
           != ( cons_b @ X3 @ Xs3 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_245_linear__order__from__list__position_H_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_246_list_Osimps_I8_J,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ nil_Product_prod_b_c )
      = nil_b ) ).

% list.simps(8)
thf(fact_247_prefixes__prefixes,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Ys @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
     => ( ( member6330420149250801815od_b_c @ Zs3 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
       => ( ( member6330420149250801815od_b_c @ Ys @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Zs3 ) ) )
          | ( member6330420149250801815od_b_c @ Zs3 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) ) ) ) ) ).

% prefixes_prefixes
thf(fact_248_prefixes__Cons_I1_J,axiom,
    ! [X2: b,Xs: list_b,Y: b,Ys: list_b] :
      ( ( member_list_b @ ( cons_b @ X2 @ Xs ) @ ( set_list_b2 @ ( prefixes_b @ ( cons_b @ Y @ Ys ) ) ) )
     => ( X2 = Y ) ) ).

% prefixes_Cons(1)
thf(fact_249_prefixes__Cons_I1_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( cons_P4529483553340347422od_b_c @ Y @ Ys ) ) ) )
     => ( X2 = Y ) ) ).

% prefixes_Cons(1)
thf(fact_250_prefixes__Cons_I2_J,axiom,
    ! [X2: b,Xs: list_b,Y: b,Ys: list_b] :
      ( ( member_list_b @ ( cons_b @ X2 @ Xs ) @ ( set_list_b2 @ ( prefixes_b @ ( cons_b @ Y @ Ys ) ) ) )
     => ( member_list_b @ Xs @ ( set_list_b2 @ ( prefixes_b @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_251_prefixes__Cons_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( cons_P4529483553340347422od_b_c @ Y @ Ys ) ) ) )
     => ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_252_prefixes__prefix__subset,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) ) ) ).

% prefixes_prefix_subset
thf(fact_253_after__h__obs,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Q3: a] :
      ( ( observable_a_b_c @ M )
     => ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
          = ( some_a @ Q3 ) )
       => ( ( after_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_h_obs
thf(fact_254_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_b_c] :
      ~ ! [A5: b,B5: c] :
          ( Y
         != ( product_Pair_b_c @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_255_surj__pair,axiom,
    ! [P2: product_prod_b_c] :
    ? [X3: b,Y3: c] :
      ( P2
      = ( product_Pair_b_c @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_256_prod__cases,axiom,
    ! [P: product_prod_b_c > $o,P2: product_prod_b_c] :
      ( ! [A5: b,B5: c] : ( P @ ( product_Pair_b_c @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_257_Pair__inject,axiom,
    ! [A: b,B3: c,A6: b,B6: c] :
      ( ( ( product_Pair_b_c @ A @ B3 )
        = ( product_Pair_b_c @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_258_set__map__subset,axiom,
    ! [X2: a,Xs: list_a,T: a,F: a > a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_a_a @ F @ ( cons_a @ X2 @ nil_a ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_259_set__map__subset,axiom,
    ! [X2: b,Xs: list_b,T: a,F: b > a] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_b_a @ F @ ( cons_b @ X2 @ nil_b ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_b_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_260_set__map__subset,axiom,
    ! [X2: a,Xs: list_a,T: product_prod_b_c,F: a > product_prod_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_a_5072596589534360878od_b_c @ F @ ( cons_a @ X2 @ nil_a ) ) ) )
       => ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_a_5072596589534360878od_b_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_261_set__map__subset,axiom,
    ! [X2: b,Xs: list_b,T: product_prod_b_c,F: b > product_prod_b_c] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_b_4433156289202544749od_b_c @ F @ ( cons_b @ X2 @ nil_b ) ) ) )
       => ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_b_4433156289202544749od_b_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_262_set__map__subset,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,T: a,F: product_prod_b_c > a] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_Pr8222292247188198874_b_c_a @ F @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_Pr8222292247188198874_b_c_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_263_set__map__subset,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,T: b,F: product_prod_b_c > b] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ( member_b @ T @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) ) )
       => ( member_b @ T @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_264_set__map__subset,axiom,
    ! [X2: a,Xs: list_a,T: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member6330420149250801815od_b_c @ T @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ ( cons_a @ X2 @ nil_a ) ) ) )
       => ( member6330420149250801815od_b_c @ T @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_265_set__map__subset,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,T: a,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ ( cons_l342771267123639716od_b_c @ X2 @ nil_li8071162985807626740od_b_c ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_266_set__map__subset,axiom,
    ! [X2: b,Xs: list_b,T: list_P903359562653991662od_b_c,F: b > list_P903359562653991662od_b_c] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ( member6330420149250801815od_b_c @ T @ ( set_li4480668622519654659od_b_c @ ( map_b_608464085996867187od_b_c @ F @ ( cons_b @ X2 @ nil_b ) ) ) )
       => ( member6330420149250801815od_b_c @ T @ ( set_li4480668622519654659od_b_c @ ( map_b_608464085996867187od_b_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_267_set__map__subset,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,T: product_prod_b_c,F: product_prod_b_c > product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) ) )
       => ( member7862447941013992593od_b_c @ T @ ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_268_after__is__state,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Io3: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) )
       => ( member6330420149250801815od_b_c @ ( after_4052058690717316294_c_b_c @ M @ Q @ Io3 ) @ ( states7681702920031268536_c_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_269_after__is__state,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
       => ( member_a @ ( after_a_b_c @ M @ Q @ Io3 ) @ ( states_a_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_270_prefixes_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( prefixes_b @ ( cons_b @ X2 @ Xs ) )
      = ( cons_list_b @ nil_b @ ( map_list_b_list_b @ ( cons_b @ X2 ) @ ( prefixes_b @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_271_prefixes_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( prefix1131979855692807669od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 ) @ ( prefix1131979855692807669od_b_c @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_272_delete__aux_Ocases,axiom,
    ! [X2: produc6475121930738208726od_b_c] :
      ( ! [K: b] :
          ( X2
         != ( produc561751498933632326od_b_c @ K @ nil_Product_prod_b_c ) )
     => ~ ! [K: b,K2: b,V: c,Xs3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc561751498933632326od_b_c @ K @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_273_pred__subset__eq,axiom,
    ! [R: set_a,S: set_a] :
      ( ( ord_less_eq_a_o
        @ ^ [X: a] : ( member_a @ X @ R )
        @ ^ [X: a] : ( member_a @ X @ S ) )
      = ( ord_less_eq_set_a @ R @ S ) ) ).

% pred_subset_eq
thf(fact_274_pred__subset__eq,axiom,
    ! [R: set_li6436108459499378894od_b_c,S: set_li6436108459499378894od_b_c] :
      ( ( ord_le1261880702084804079_b_c_o
        @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ R )
        @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ S ) )
      = ( ord_le282488521294790766od_b_c @ R @ S ) ) ).

% pred_subset_eq
thf(fact_275_subrelI,axiom,
    ! [R2: set_Product_prod_b_c,S2: set_Product_prod_b_c] :
      ( ! [X3: b,Y3: c] :
          ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ R2 )
         => ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le253122037897321832od_b_c @ R2 @ S2 ) ) ).

% subrelI
thf(fact_276_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( ( ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ R ) )
        = ( ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_277_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( ord_less_eq_b_c_o
        @ ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ R )
        @ ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ S ) )
      = ( ord_le253122037897321832od_b_c @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_278_longest__common__prefix_Ocases,axiom,
    ! [X2: produc3963297410138542439list_b] :
      ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
          ( X2
         != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_b] :
            ( X2
           != ( produc1564554178308465111list_b @ nil_b @ Uv ) )
       => ~ ! [Uu: list_b] :
              ( X2
             != ( produc1564554178308465111list_b @ Uu @ nil_b ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_279_longest__common__prefix_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uv ) )
       => ~ ! [Uu: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc3076307724246486423od_b_c @ Uu @ nil_Product_prod_b_c ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_280_clearjunk_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ~ ! [P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_281_map__default_Ocases,axiom,
    ! [X2: produc2009451851958091931od_b_c] :
      ( ! [K: b,V: c,F2: c > c] :
          ( X2
         != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V @ ( produc1101678828523717360od_b_c @ F2 @ nil_Product_prod_b_c ) ) ) )
     => ~ ! [K: b,V: c,F2: c > c,P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V @ ( produc1101678828523717360od_b_c @ F2 @ ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_282_map__entry_Ocases,axiom,
    ! [X2: produc6452300523622238572od_b_c] :
      ( ! [K: b,F2: c > c] :
          ( X2
         != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F2 @ nil_Product_prod_b_c ) ) )
     => ~ ! [K: b,F2: c > c,P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F2 @ ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_283_maximal__prefix__in__language_Osimps_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Io3: list_P903359562653991662od_b_c] :
      ( ( maxima1559550560783484624_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Io3 ) )
      = ( case_o7419520253229056539_b_c_a @ nil_Product_prod_b_c
        @ ^ [Q5: a] : ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ ( maxima1559550560783484624_a_b_c @ M @ Q5 @ Io3 ) )
        @ ( h_obs_a_b_c @ M @ Q @ X2 @ Y ) ) ) ).

% maximal_prefix_in_language.simps(2)
thf(fact_284_maximal__prefix__in__language_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( maxima1559550560783484624_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Product_prod_b_c )
         => ( Y != nil_Product_prod_b_c ) )
       => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
             => ( Y
               != ( case_o7419520253229056539_b_c_a @ nil_Product_prod_b_c
                  @ ^ [Q5: a] : ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ ( maxima1559550560783484624_a_b_c @ X2 @ Q5 @ Io4 ) )
                  @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ) ) ).

% maximal_prefix_in_language.elims
thf(fact_285_after__transition,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Q3: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y @ Q3 ) ) ) @ ( transitions_a_b_c @ M ) )
       => ( ( after_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_transition
thf(fact_286_scheme,axiom,
    ! [Y: a,F: a > list_a,X2: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( F @ X2 ) ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_a_list_a @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_287_scheme,axiom,
    ! [Y: a,F: list_P903359562653991662od_b_c > list_a,X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( F @ X2 ) ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_li1600214416573329882list_a @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_288_scheme,axiom,
    ! [Y: a,F: product_prod_b_c > list_a,X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( F @ X2 ) ) )
     => ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
       => ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_Pr6622659521024808032list_a @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_289_scheme,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: a > list_l8907847357763382004od_b_c,X2: a,Xs: list_a] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X2 ) ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_a_3331650471236623290od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_290_scheme,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X2 ) ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_li6701939492004672586od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_291_scheme,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: product_prod_b_c > list_l8907847357763382004od_b_c,X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X2 ) ) )
     => ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
       => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_Pr2837759038969554384od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_292_scheme,axiom,
    ! [Y: product_prod_b_c,F: a > list_P903359562653991662od_b_c,X2: a,Xs: list_a] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X2 ) ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_293_scheme,axiom,
    ! [Y: product_prod_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X2 ) ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_294_scheme,axiom,
    ! [Y: product_prod_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X2 ) ) )
     => ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
       => ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) ) ) ) ).

% scheme
thf(fact_295_h__obs__language__single__transition__iff,axiom,
    ! [M: fsm_a_b_c,X2: b,Y: c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) @ ( lS_a_b_c @ M @ Q ) )
        = ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
         != none_a ) ) ) ).

% h_obs_language_single_transition_iff
thf(fact_296_language__for__input_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_b,Y: list_l8907847357763382004od_b_c] :
      ( ( ( langua5143504920213240017_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_b )
         => ( Y
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( Xb
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( concat8136016674533020335od_b_c
                  @ ( map_c_4452387693327778872od_b_c
                    @ ^ [Y2: c] :
                        ( case_o6498834556914106773_b_c_a @ nil_li8071162985807626740od_b_c
                        @ ^ [Q5: a] : ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) ) @ ( langua5143504920213240017_a_b_c @ X2 @ Q5 @ Xs3 ) )
                        @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y2 ) )
                    @ ( output4084235763626582864_a_b_c @ X2 ) ) ) ) ) ) ) ).

% language_for_input.elims
thf(fact_297_not__None__eq,axiom,
    ! [X2: option_a] :
      ( ( X2 != none_a )
      = ( ? [Y2: a] :
            ( X2
            = ( some_a @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_298_not__Some__eq,axiom,
    ! [X2: option_a] :
      ( ( ! [Y2: a] :
            ( X2
           != ( some_a @ Y2 ) ) )
      = ( X2 = none_a ) ) ).

% not_Some_eq
thf(fact_299_fsm__transition__source,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) ) ) ).

% fsm_transition_source
thf(fact_300_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l8907847357763382004od_b_c] :
      ( ( nil_Product_prod_b_c
        = ( concat3284542956341965481od_b_c @ Xss2 ) )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xss2 ) )
           => ( X = nil_Product_prod_b_c ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_301_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l8907847357763382004od_b_c] :
      ( ( ( concat3284542956341965481od_b_c @ Xss2 )
        = nil_Product_prod_b_c )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xss2 ) )
           => ( X = nil_Product_prod_b_c ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_302_concat__map__singleton,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( concat_b
        @ ( map_Pr6622659525328036833list_b
          @ ^ [X: product_prod_b_c] : ( cons_b @ ( F @ X ) @ nil_b )
          @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_303_option_Odistinct_I1_J,axiom,
    ! [X23: a] :
      ( none_a
     != ( some_a @ X23 ) ) ).

% option.distinct(1)
thf(fact_304_option_OdiscI,axiom,
    ! [Option: option_a,X23: a] :
      ( ( Option
        = ( some_a @ X23 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_305_option_Oexhaust,axiom,
    ! [Y: option_a] :
      ( ( Y != none_a )
     => ~ ! [X24: a] :
            ( Y
           != ( some_a @ X24 ) ) ) ).

% option.exhaust
thf(fact_306_split__option__ex,axiom,
    ( ( ^ [P5: option_a > $o] :
        ? [X4: option_a] : ( P5 @ X4 ) )
    = ( ^ [P6: option_a > $o] :
          ( ( P6 @ none_a )
          | ? [X: a] : ( P6 @ ( some_a @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_307_split__option__all,axiom,
    ( ( ^ [P5: option_a > $o] :
        ! [X4: option_a] : ( P5 @ X4 ) )
    = ( ^ [P6: option_a > $o] :
          ( ( P6 @ none_a )
          & ! [X: a] : ( P6 @ ( some_a @ X ) ) ) ) ) ).

% split_option_all
thf(fact_308_combine__options__cases,axiom,
    ! [X2: option_a,P: option_a > option_a > $o,Y: option_a] :
      ( ( ( X2 = none_a )
       => ( P @ X2 @ Y ) )
     => ( ( ( Y = none_a )
         => ( P @ X2 @ Y ) )
       => ( ! [A5: a,B5: a] :
              ( ( X2
                = ( some_a @ A5 ) )
             => ( ( Y
                  = ( some_a @ B5 ) )
               => ( P @ X2 @ Y ) ) )
         => ( P @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_309_set__concat__elem,axiom,
    ! [X2: a,Xss2: list_list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( concat_a @ Xss2 ) ) )
     => ~ ! [Xs3: list_a] :
            ( ( member_list_a @ Xs3 @ ( set_list_a2 @ Xss2 ) )
           => ~ ( member_a @ X2 @ ( set_a2 @ Xs3 ) ) ) ) ).

% set_concat_elem
thf(fact_310_set__concat__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xss2: list_l5832389826979350138od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ Xss2 ) ) )
     => ~ ! [Xs3: list_l8907847357763382004od_b_c] :
            ( ( member7934871825613078557od_b_c @ Xs3 @ ( set_li2649297040355570313od_b_c @ Xss2 ) )
           => ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs3 ) ) ) ) ).

% set_concat_elem
thf(fact_311_set__concat__elem,axiom,
    ! [X2: product_prod_b_c,Xss2: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ Xss2 ) ) )
     => ~ ! [Xs3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Xs3 @ ( set_li4480668622519654659od_b_c @ Xss2 ) )
           => ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs3 ) ) ) ) ).

% set_concat_elem
thf(fact_312_concat__map__elem,axiom,
    ! [Y: a,F: a > list_a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_a_list_a @ F @ Xs ) ) ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ~ ( member_a @ Y @ ( set_a2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_313_concat__map__elem,axiom,
    ! [Y: a,F: list_P903359562653991662od_b_c > list_a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_li1600214416573329882list_a @ F @ Xs ) ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ~ ( member_a @ Y @ ( set_a2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_314_concat__map__elem,axiom,
    ! [Y: a,F: product_prod_b_c > list_a,Xs: list_P903359562653991662od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( concat_a @ ( map_Pr6622659521024808032list_a @ F @ Xs ) ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
           => ~ ( member_a @ Y @ ( set_a2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_315_concat__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: a > list_l8907847357763382004od_b_c,Xs: list_a] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_a_3331650471236623290od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ~ ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_316_concat__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_li6701939492004672586od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ~ ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_317_concat__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: product_prod_b_c > list_l8907847357763382004od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_Pr2837759038969554384od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
           => ~ ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_318_concat__map__elem,axiom,
    ! [Y: product_prod_b_c,F: a > list_P903359562653991662od_b_c,Xs: list_a] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ~ ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_319_concat__map__elem,axiom,
    ! [Y: product_prod_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ~ ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_320_concat__map__elem,axiom,
    ! [Y: product_prod_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) )
     => ~ ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
           => ~ ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( F @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_321_set__concat__map__elem,axiom,
    ! [X2: a,F: list_P903359562653991662od_b_c > list_a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_li1600214416573329882list_a @ F @ Xs ) ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( member_a @ X2 @ ( set_a2 @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_322_set__concat__map__elem,axiom,
    ! [X2: a,F: product_prod_b_c > list_a,Xs: list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_Pr6622659521024808032list_a @ F @ Xs ) ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( member_a @ X2 @ ( set_a2 @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_323_set__concat__map__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_li6701939492004672586od_b_c @ F @ Xs ) ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_324_set__concat__map__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: product_prod_b_c > list_l8907847357763382004od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_Pr2837759038969554384od_b_c @ F @ Xs ) ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_325_set__concat__map__elem,axiom,
    ! [X2: product_prod_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) )
     => ? [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_326_set__concat__map__elem,axiom,
    ! [X2: product_prod_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) )
     => ? [X3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( F @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_327_map__concat,axiom,
    ! [F: product_prod_b_c > b,Xs: list_l8907847357763382004od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( concat3284542956341965481od_b_c @ Xs ) )
      = ( concat_b @ ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_328_h__obs__None,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c] :
      ( ( observable_a_b_c @ M )
     => ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
          = none_a )
        = ( ~ ? [Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y @ Q5 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) ) ).

% h_obs_None
thf(fact_329_set__concat__map__sublist,axiom,
    ! [X2: a,F: product_prod_b_c > list_a,Xs: list_P903359562653991662od_b_c,Xs4: list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_Pr6622659521024808032list_a @ F @ Xs ) ) ) )
     => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Xs4 ) )
       => ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_Pr6622659521024808032list_a @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_330_set__concat__map__sublist,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: product_prod_b_c > list_l8907847357763382004od_b_c,Xs: list_P903359562653991662od_b_c,Xs4: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_Pr2837759038969554384od_b_c @ F @ Xs ) ) ) )
     => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Xs4 ) )
       => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_Pr2837759038969554384od_b_c @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_331_set__concat__map__sublist,axiom,
    ! [X2: product_prod_b_c,F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Xs4: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) )
     => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Xs4 ) )
       => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_332_set__concat__map__sublist,axiom,
    ! [X2: a,F: list_P903359562653991662od_b_c > list_a,Xs: list_l8907847357763382004od_b_c,Xs4: list_l8907847357763382004od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_li1600214416573329882list_a @ F @ Xs ) ) ) )
     => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Xs4 ) )
       => ( member_a @ X2 @ ( set_a2 @ ( concat_a @ ( map_li1600214416573329882list_a @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_333_set__concat__map__sublist,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c,Xs4: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_li6701939492004672586od_b_c @ F @ Xs ) ) ) )
     => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Xs4 ) )
       => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ ( map_li6701939492004672586od_b_c @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_334_set__concat__map__sublist,axiom,
    ! [X2: product_prod_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Xs4: list_l8907847357763382004od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) )
     => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Xs4 ) )
       => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs4 ) ) ) ) ) ) ).

% set_concat_map_sublist
thf(fact_335_observable__alt__def,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q1: a,X: b,Y2: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y2 @ Q12 ) ) ) @ ( transitions_a_b_c @ M2 ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y2 @ Q13 ) ) ) @ ( transitions_a_b_c @ M2 ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_336_language__for__input_Osimps_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Xs: list_b] :
      ( ( langua5143504920213240017_a_b_c @ M @ Q @ ( cons_b @ X2 @ Xs ) )
      = ( concat8136016674533020335od_b_c
        @ ( map_c_4452387693327778872od_b_c
          @ ^ [Y2: c] :
              ( case_o6498834556914106773_b_c_a @ nil_li8071162985807626740od_b_c
              @ ^ [Q5: a] : ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) ) @ ( langua5143504920213240017_a_b_c @ M @ Q5 @ Xs ) )
              @ ( h_obs_a_b_c @ M @ Q @ X2 @ Y2 ) )
          @ ( output4084235763626582864_a_b_c @ M ) ) ) ) ).

% language_for_input.simps(2)
thf(fact_337_language__for__input_Opelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_b,Y: list_l8907847357763382004od_b_c] :
      ( ( ( langua5143504920213240017_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P4587740637805451515list_b @ langua2270551176343202290_a_b_c @ ( produc5717594475321894882list_b @ X2 @ ( produc6670463076781050526list_b @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_b )
           => ( ( Y
                = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) )
             => ~ ( accp_P4587740637805451515list_b @ langua2270551176343202290_a_b_c @ ( produc5717594475321894882list_b @ X2 @ ( produc6670463076781050526list_b @ Xa @ nil_b ) ) ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( Xb
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( concat8136016674533020335od_b_c
                      @ ( map_c_4452387693327778872od_b_c
                        @ ^ [Y2: c] :
                            ( case_o6498834556914106773_b_c_a @ nil_li8071162985807626740od_b_c
                            @ ^ [Q5: a] : ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) ) @ ( langua5143504920213240017_a_b_c @ X2 @ Q5 @ Xs3 ) )
                            @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y2 ) )
                        @ ( output4084235763626582864_a_b_c @ X2 ) ) ) )
                 => ~ ( accp_P4587740637805451515list_b @ langua2270551176343202290_a_b_c @ ( produc5717594475321894882list_b @ X2 @ ( produc6670463076781050526list_b @ Xa @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ) ) ).

% language_for_input.pelims
thf(fact_338_product__code,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( produc4408432528021480786od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Ys ) )
      = ( set_Pr292865031892111420od_b_c
        @ ( concat5338053325402172392od_b_c
          @ ( map_li3082895796472359171od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] : ( map_li1896284803375742973od_b_c @ ( produc3076307724246486423od_b_c @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_339_product__code,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( produc6025674572879852236od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_Product_prod_b_c2 @ Ys ) )
      = ( set_Pr6339120575877180470od_b_c
        @ ( concat8285913752383688162od_b_c
          @ ( map_li994475539454223869od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] : ( map_Pr3717219041064202493od_b_c @ ( produc106421975150051729od_b_c @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_340_product__code,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( produc7386353291125108952od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_li4480668622519654659od_b_c @ Ys ) )
      = ( set_Pr4145341638095603010od_b_c
        @ ( concat6092134814602110702od_b_c
          @ ( map_Pr16825066796629647od_b_c
            @ ^ [X: product_prod_b_c] : ( map_li68304194280966531od_b_c @ ( produc1467100693395308445od_b_c @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_341_product__code,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( produc5200162086477412690od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ys ) )
      = ( set_Pr7267323205303737404od_b_c
        @ ( concat9136675030996222952od_b_c
          @ ( map_Pr7701655480119177609od_b_c
            @ ^ [X: product_prod_b_c] : ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_342_product__code,axiom,
    ! [Xs: list_b,Ys: list_c] :
      ( ( product_product_b_c @ ( set_b2 @ Xs ) @ ( set_c2 @ Ys ) )
      = ( set_Product_prod_b_c2
        @ ( concat3284542956341965481od_b_c
          @ ( map_b_608464085996867187od_b_c
            @ ^ [X: b] : ( map_c_3793715988870728620od_b_c @ ( product_Pair_b_c @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_343_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_b,Xss2: list_list_b] :
      ( ( product_lists_b @ ( cons_list_b @ Xs @ Xss2 ) )
      = ( concat_list_b
        @ ( map_b_list_list_b
          @ ^ [X: b] : ( map_list_b_list_b @ ( cons_b @ X ) @ ( product_lists_b @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_344_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xss2: list_l8907847357763382004od_b_c] :
      ( ( produc3104363228908302167od_b_c @ ( cons_l342771267123639716od_b_c @ Xs @ Xss2 ) )
      = ( concat8136016674533020335od_b_c
        @ ( map_Pr2837759038969554384od_b_c
          @ ^ [X: product_prod_b_c] : ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X ) @ ( produc3104363228908302167od_b_c @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_345_transitions__from__set,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( transi6113640857693861514_a_b_c @ M @ Q )
        = ( collec8405073525749625505od_c_a
          @ ^ [T2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M ) )
              & ( ( produc4225943634317630866od_c_a @ T2 )
                = Q ) ) ) ) ) ).

% transitions_from_set
thf(fact_346_maximal__prefix__in__language_Opelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( maxima1559550560783484624_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P4000836841735168996od_b_c @ maxima8135688396739918515_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ( ( Y = nil_Product_prod_b_c )
             => ~ ( accp_P4000836841735168996od_b_c @ maxima8135688396739918515_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) ) )
         => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
               => ( ( Y
                    = ( case_o7419520253229056539_b_c_a @ nil_Product_prod_b_c
                      @ ^ [Q5: a] : ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ ( maxima1559550560783484624_a_b_c @ X2 @ Q5 @ Io4 ) )
                      @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) )
                 => ~ ( accp_P4000836841735168996od_b_c @ maxima8135688396739918515_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) ) ) ) ) ) ) ) ) ).

% maximal_prefix_in_language.pelims
thf(fact_347_case__optionE,axiom,
    ! [P: $o,Q4: a > $o,X2: option_a] :
      ( ( case_option_o_a @ P @ Q4 @ X2 )
     => ( ( ( X2 = none_a )
         => ~ P )
       => ~ ! [Y3: a] :
              ( ( X2
                = ( some_a @ Y3 ) )
             => ~ ( Q4 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_348_product__concat__map,axiom,
    ( product_b_c
    = ( ^ [Xs2: list_b,Ys3: list_c] :
          ( concat3284542956341965481od_b_c
          @ ( map_b_608464085996867187od_b_c
            @ ^ [X: b] : ( map_c_3793715988870728620od_b_c @ ( product_Pair_b_c @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_349_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( set_Product_prod_a_a2
        @ ( concat6071998529708183912od_a_a
          @ ( map_a_6618942900564114931od_a_a
            @ ^ [X: a] : ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec3336397797384452498od_a_a
        @ ^ [Xy: product_prod_a_a] :
            ( ( member_a @ ( product_fst_a_a @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member_a @ ( product_snd_a_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_350_concat__pair__set,axiom,
    ! [Ys: list_l8907847357763382004od_b_c,Xs: list_a] :
      ( ( set_Pr316496599637395756od_b_c
        @ ( concat8748367952757864152od_b_c
          @ ( map_a_2611511242289716067od_b_c
            @ ^ [X: a] : ( map_li7377147270969669229od_b_c @ ( produc6078649392988585351od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec8003727573348059138od_b_c
        @ ^ [Xy: produc2768647787938385943od_b_c] :
            ( ( member_a @ ( produc7198897215385899379od_b_c @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member6330420149250801815od_b_c @ ( produc1062780392962831797od_b_c @ Xy ) @ ( set_li4480668622519654659od_b_c @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_351_concat__pair__set,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_a] :
      ( ( set_Pr6515338384934575398od_b_c
        @ ( concat5395573113327685842od_b_c
          @ ( map_a_1999234684872493149od_b_c
            @ ^ [X: a] : ( map_Pr6703800383548559469od_b_c @ ( produc6867761804911831425od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec4552200074903125756od_b_c
        @ ^ [Xy: produc3550516665487611281od_b_c] :
            ( ( member_a @ ( produc3728844965657535341od_b_c @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member7862447941013992593od_b_c @ ( produc8648342215071535279od_b_c @ Xy ) @ ( set_Product_prod_b_c2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_352_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_l8907847357763382004od_b_c] :
      ( ( set_Pr9038908268211613900_b_c_a
        @ ( concat8247407584477306488_b_c_a
          @ ( map_li2020277641456335763_b_c_a
            @ ^ [X: list_P903359562653991662od_b_c] : ( map_a_4856183684908223869_b_c_a @ ( produc595631027891595047_b_c_a @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec7502767205067501474_b_c_a
        @ ^ [Xy: produc2267687419657828279_b_c_a] :
            ( ( member6330420149250801815od_b_c @ ( produc1715878850288909075_b_c_a @ Xy ) @ ( set_li4480668622519654659od_b_c @ Xs ) )
            & ( member_a @ ( produc4803134064720617301_b_c_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_353_concat__pair__set,axiom,
    ! [Ys: list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( set_Pr292865031892111420od_b_c
        @ ( concat5338053325402172392od_b_c
          @ ( map_li3082895796472359171od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] : ( map_li1896284803375742973od_b_c @ ( produc3076307724246486423od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec4655644190068631058od_b_c
        @ ^ [Xy: produc5878855887281848999od_b_c] :
            ( ( member6330420149250801815od_b_c @ ( produc1807356612421382019od_b_c @ Xy ) @ ( set_li4480668622519654659od_b_c @ Xs ) )
            & ( member6330420149250801815od_b_c @ ( produc8256383748742478021od_b_c @ Xy ) @ ( set_li4480668622519654659od_b_c @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_354_concat__pair__set,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( set_Pr6339120575877180470od_b_c
        @ ( concat8285913752383688162od_b_c
          @ ( map_li994475539454223869od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] : ( map_Pr3717219041064202493od_b_c @ ( produc106421975150051729od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec3949805069543384332od_b_c
        @ ^ [Xy: produc4003095055046660129od_b_c] :
            ( ( member6330420149250801815od_b_c @ ( produc7960711977528723837od_b_c @ Xy ) @ ( set_li4480668622519654659od_b_c @ Xs ) )
            & ( member7862447941013992593od_b_c @ ( produc6654658060879993791od_b_c @ Xy ) @ ( set_Product_prod_b_c2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_355_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_P903359562653991662od_b_c] :
      ( ( set_Pr1869262346523236050_b_c_a
        @ ( concat749497074916346494_b_c_a
          @ ( map_Pr1798716480651293471_b_c_a
            @ ^ [X: product_prod_b_c] : ( map_a_2551615777404289027_b_c_a @ ( produc794085425710893613_b_c_a @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec9129496073346562216_b_c_a
        @ ^ [Xy: produc8127812663931047741_b_c_a] :
            ( ( member7862447941013992593od_b_c @ ( produc6878540623311373337_b_c_a @ Xy ) @ ( set_Product_prod_b_c2 @ Xs ) )
            & ( member_a @ ( produc2574665835870597467_b_c_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_356_concat__pair__set,axiom,
    ! [Ys: list_l8907847357763382004od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( set_Pr4145341638095603010od_b_c
        @ ( concat6092134814602110702od_b_c
          @ ( map_Pr16825066796629647od_b_c
            @ ^ [X: product_prod_b_c] : ( map_li68304194280966531od_b_c @ ( produc1467100693395308445od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec1756026131761806872od_b_c
        @ ^ [Xy: produc1809316117265082669od_b_c] :
            ( ( member7862447941013992593od_b_c @ ( produc98018658919204745od_b_c @ Xy ) @ ( set_Product_prod_b_c2 @ Xs ) )
            & ( member6330420149250801815od_b_c @ ( produc8015336779125250507od_b_c @ Xy ) @ ( set_li4480668622519654659od_b_c @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_357_concat__pair__set,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( set_Pr7267323205303737404od_b_c
        @ ( concat9136675030996222952od_b_c
          @ ( map_Pr7701655480119177609od_b_c
            @ ^ [X: product_prod_b_c] : ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec784383060099741714od_b_c
        @ ^ [Xy: produc4272613772517093799od_b_c] :
            ( ( member7862447941013992593od_b_c @ ( produc4830044058343513219od_b_c @ Xy ) @ ( set_Product_prod_b_c2 @ Xs ) )
            & ( member7862447941013992593od_b_c @ ( produc8692230813408409541od_b_c @ Xy ) @ ( set_Product_prod_b_c2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_358_concat__pair__set,axiom,
    ! [Ys: list_c,Xs: list_b] :
      ( ( set_Product_prod_b_c2
        @ ( concat3284542956341965481od_b_c
          @ ( map_b_608464085996867187od_b_c
            @ ^ [X: b] : ( map_c_3793715988870728620od_b_c @ ( product_Pair_b_c @ X ) @ Ys )
            @ Xs ) ) )
      = ( collec548942224018234067od_b_c
        @ ^ [Xy: product_prod_b_c] :
            ( ( member_b @ ( product_fst_b_c @ Xy ) @ ( set_b2 @ Xs ) )
            & ( member_c @ ( product_snd_b_c @ Xy ) @ ( set_c2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_359_is__in__language_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c] :
      ( ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
     => ( ( Xb != nil_Product_prod_b_c )
       => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
             => ~ ( case_option_o_a @ $false
                  @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
                  @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ) ).

% is_in_language.elims(2)
thf(fact_360_is__in__language_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: $o] :
      ( ( ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Product_prod_b_c )
         => ~ Y )
       => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
             => ( Y
                = ( ~ ( case_option_o_a @ $false
                      @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
                      @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ) ) ) ).

% is_in_language.elims(1)
thf(fact_361_after_Opelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: a] :
      ( ( ( after_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P4000836841735168996od_b_c @ after_rel_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ( ( Y = Xa )
             => ~ ( accp_P4000836841735168996od_b_c @ after_rel_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) ) )
         => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
               => ( ( Y
                    = ( after_a_b_c @ X2 @ ( the_a @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) @ Io4 ) )
                 => ~ ( accp_P4000836841735168996od_b_c @ after_rel_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) ) ) ) ) ) ) ) ) ).

% after.pelims
thf(fact_362_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_c] :
      ( ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_363_fsm__transition__target,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) ) ) ).

% fsm_transition_target
thf(fact_364_option_Ocollapse,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( ( some_a @ ( the_a @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_365_sndI,axiom,
    ! [X2: product_prod_b_c,Y: b,Z5: c] :
      ( ( X2
        = ( product_Pair_b_c @ Y @ Z5 ) )
     => ( ( product_snd_b_c @ X2 )
        = Z5 ) ) ).

% sndI
thf(fact_366_snd__eqD,axiom,
    ! [X2: b,Y: c,A: c] :
      ( ( ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_367_snd__conv,axiom,
    ! [X1: b,X23: c] :
      ( ( product_snd_b_c @ ( product_Pair_b_c @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_368_eq__snd__iff,axiom,
    ! [B3: c,P2: product_prod_b_c] :
      ( ( B3
        = ( product_snd_b_c @ P2 ) )
      = ( ? [A4: b] :
            ( P2
            = ( product_Pair_b_c @ A4 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_369_prod_Oexpand,axiom,
    ! [Prod: product_prod_b_c,Prod2: product_prod_b_c] :
      ( ( ( ( product_fst_b_c @ Prod )
          = ( product_fst_b_c @ Prod2 ) )
        & ( ( product_snd_b_c @ Prod )
          = ( product_snd_b_c @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_370_prod__eqI,axiom,
    ! [P2: product_prod_b_c,Q: product_prod_b_c] :
      ( ( ( product_fst_b_c @ P2 )
        = ( product_fst_b_c @ Q ) )
     => ( ( ( product_snd_b_c @ P2 )
          = ( product_snd_b_c @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_371_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_b_c,Z4: product_prod_b_c] : ( Y4 = Z4 ) )
    = ( ^ [S3: product_prod_b_c,T2: product_prod_b_c] :
          ( ( ( product_fst_b_c @ S3 )
            = ( product_fst_b_c @ T2 ) )
          & ( ( product_snd_b_c @ S3 )
            = ( product_snd_b_c @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_372_option_Osel,axiom,
    ! [X23: a] :
      ( ( the_a @ ( some_a @ X23 ) )
      = X23 ) ).

% option.sel
thf(fact_373_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X2 )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X2 ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X2 ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                 => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_374_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ( observable_a_b_c @ X2 )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X2 ) )
         => ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xa3 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_375_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Y: $o] :
      ( ( ( observable_a_b_c @ X2 )
        = Y )
     => ( Y
        = ( ! [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X2 ) )
             => ! [Y2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y2 @ ( transitions_a_b_c @ X2 ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X )
                        = ( produc4225943634317630866od_c_a @ Y2 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) )
                   => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_376_observable_Osimps,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M2 ) )
         => ! [Y2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y2 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X )
                    = ( produc4225943634317630866od_c_a @ Y2 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_377_observable__transition__target__language__eq,axiom,
    ! [M: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T1 ) )
        = ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) )
             => ( ( observable_a_b_c @ M )
               => ( ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) )
                  = ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_378_surjective__pairing,axiom,
    ! [T: product_prod_b_c] :
      ( T
      = ( product_Pair_b_c @ ( product_fst_b_c @ T ) @ ( product_snd_b_c @ T ) ) ) ).

% surjective_pairing
thf(fact_379_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_c] :
      ( Prod
      = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_380_pair__list__eqI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ys ) )
     => ( ( ( map_Pr8222292247188198876_b_c_c @ product_snd_b_c @ Xs )
          = ( map_Pr8222292247188198876_b_c_c @ product_snd_b_c @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_381_language__next__transition__ob,axiom,
    ! [X2: b,Y: c,Ios: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Ios ) @ ( lS_a_b_c @ M @ Q ) )
     => ~ ! [T4: produc2669932201312201782od_c_a] :
            ( ( ( produc4225943634317630866od_c_a @ T4 )
              = Q )
           => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M ) )
             => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) )
                  = X2 )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                    = Y )
                 => ~ ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_382_LS__prepend__transition,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io3 ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_383_option_Oexhaust__sel,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
     => ( Option
        = ( some_a @ ( the_a @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_384_observable__transition__target__language__subset,axiom,
    ! [M: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T1 ) ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) )
             => ( ( observable_a_b_c @ M )
               => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) ) @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_385_observable__transition__unique,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M ) )
            & ( ( produc4225943634317630866od_c_a @ X3 )
              = ( produc4225943634317630866od_c_a @ T ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            & ! [Y5: produc2669932201312201782od_c_a] :
                ( ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ Y5 )
                    = ( produc4225943634317630866od_c_a @ T ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_386_observable__language__transition__target,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io3 ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_387_after__transition__exhaust,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
          = ( after_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_388_LS__single__transition,axiom,
    ! [X2: b,Y: c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) @ ( lS_a_b_c @ M @ Q ) )
      = ( ? [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M ) )
            & ( ( produc4225943634317630866od_c_a @ X )
              = Q )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
              = X2 )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
              = Y ) ) ) ) ).

% LS_single_transition
thf(fact_389_after_Osimps_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Io3: list_P903359562653991662od_b_c] :
      ( ( after_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Io3 ) )
      = ( after_a_b_c @ M @ ( the_a @ ( h_obs_a_b_c @ M @ Q @ X2 @ Y ) ) @ Io3 ) ) ).

% after.simps(2)
thf(fact_390_is__in__language__iff,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
       => ( ( is_in_7104650932667917939_c_b_c @ M @ Q @ Io3 )
          = ( member6330420149250801815od_b_c @ Io3 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_391_is__in__language__iff,axiom,
    ! [M: fsm_a_b_c,Q: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( is_in_language_a_b_c @ M @ Q @ Io3 )
          = ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_392_is__in__language_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c] :
      ( ~ ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
     => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
            ( ( Xb
              = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
           => ( case_option_o_a @ $false
              @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
              @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ).

% is_in_language.elims(3)
thf(fact_393_is__in__language_Osimps_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Io3: list_P903359562653991662od_b_c] :
      ( ( is_in_language_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Io3 ) )
      = ( case_option_o_a @ $false
        @ ^ [Q5: a] : ( is_in_language_a_b_c @ M @ Q5 @ Io3 )
        @ ( h_obs_a_b_c @ M @ Q @ X2 @ Y ) ) ) ).

% is_in_language.simps(2)
thf(fact_394_after_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: a] :
      ( ( ( after_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Product_prod_b_c )
         => ( Y != Xa ) )
       => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
             => ( Y
               != ( after_a_b_c @ X2 @ ( the_a @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) @ Io4 ) ) ) ) ) ).

% after.elims
thf(fact_395_is__in__language_Opelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c,Y: $o] :
      ( ( ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ( Y
             => ~ ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) ) )
         => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
               => ( ( Y
                    = ( case_option_o_a @ $false
                      @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
                      @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) )
                 => ~ ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) ) ) ) ) ) ) ) ) ).

% is_in_language.pelims(1)
thf(fact_396_is__in__language_Opelims_I2_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c] :
      ( ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
     => ( ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ~ ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) )
         => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
               => ( ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) ) ) )
                 => ~ ( case_option_o_a @ $false
                      @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
                      @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ) ) ) ).

% is_in_language.pelims(2)
thf(fact_397_is__in__language_Opelims_I3_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P903359562653991662od_b_c] :
      ( ~ ( is_in_language_a_b_c @ X2 @ Xa @ Xb )
     => ( ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ Xb ) ) )
       => ~ ! [X3: b,Y3: c,Io4: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) )
             => ( ( accp_P4000836841735168996od_b_c @ is_in_4296126181398531264_a_b_c @ ( produc603523481723729227od_b_c @ X2 @ ( produc6078649392988585351od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ Io4 ) ) ) )
               => ( case_option_o_a @ $false
                  @ ^ [Q5: a] : ( is_in_language_a_b_c @ X2 @ Q5 @ Io4 )
                  @ ( h_obs_a_b_c @ X2 @ Xa @ X3 @ Y3 ) ) ) ) ) ) ).

% is_in_language.pelims(3)
thf(fact_398_observable_Opelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Y: $o] :
      ( ( ( observable_a_b_c @ X2 )
        = Y )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 )
       => ~ ( ( Y
              = ( ! [X: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X2 ) )
                   => ! [Y2: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y2 @ ( transitions_a_b_c @ X2 ) )
                       => ( ( ( ( produc4225943634317630866od_c_a @ X )
                              = ( produc4225943634317630866od_c_a @ Y2 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) )
                            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) )
                         => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                            = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y2 ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 ) ) ) ) ).

% observable.pelims(1)
thf(fact_399_observable_Opelims_I2_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ( observable_a_b_c @ X2 )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 )
           => ~ ! [X5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X2 ) )
                 => ! [Xa3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X2 ) )
                     => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                            = ( produc4225943634317630866od_c_a @ Xa3 ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                          & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                            = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
                       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                          = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_400_observable_Opelims_I3_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X2 )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X2 )
           => ! [X3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X2 ) )
               => ! [Xa2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X2 ) )
                   => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                          = ( produc4225943634317630866od_c_a @ Xa2 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                        & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                          = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                     => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_401_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: b > c > $o,X2: b,Y: c,A: product_prod_b_c] :
      ( ( P @ X2 @ Y )
     => ( ( A
          = ( product_Pair_b_c @ X2 @ Y ) )
       => ( P @ ( product_fst_b_c @ A ) @ ( product_snd_b_c @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_402_conjI__realizer,axiom,
    ! [P: b > $o,P2: b,Q4: c > $o,Q: c] :
      ( ( P @ P2 )
     => ( ( Q4 @ Q )
       => ( ( P @ ( product_fst_b_c @ ( product_Pair_b_c @ P2 @ Q ) ) )
          & ( Q4 @ ( product_snd_b_c @ ( product_Pair_b_c @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_403_exI__realizer,axiom,
    ! [P: c > b > $o,Y: c,X2: b] :
      ( ( P @ Y @ X2 )
     => ( P @ ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y ) ) @ ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_404_map__default_Opelims,axiom,
    ! [X2: b,Xa: c,Xb: c > c,Xc: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( map_default_b_c @ X2 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X2 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Product_prod_b_c )
           => ( ( Y
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Xa ) @ nil_Product_prod_b_c ) )
             => ~ ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X2 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ nil_Product_prod_b_c ) ) ) ) ) )
         => ~ ! [P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
                ( ( Xc
                  = ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) )
               => ( ( ( ( ( product_fst_b_c @ P4 )
                        = X2 )
                     => ( Y
                        = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ ( Xb @ ( product_snd_b_c @ P4 ) ) ) @ Ps ) ) )
                    & ( ( ( product_fst_b_c @ P4 )
                       != X2 )
                     => ( Y
                        = ( cons_P4529483553340347422od_b_c @ P4 @ ( map_default_b_c @ X2 @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X2 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_405_subset__Collect__iff,axiom,
    ! [B: set_a,A2: set_a,P: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( ord_less_eq_set_a @ B
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: a] :
              ( ( member_a @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_406_subset__Collect__iff,axiom,
    ! [B: set_nat,A2: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( ord_less_eq_set_nat @ B
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_407_subset__Collect__iff,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ B
          @ ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_408_subset__CollectI,axiom,
    ! [B: set_a,A2: set_a,Q4: a > $o,P: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ B )
           => ( ( Q4 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ B )
                & ( Q4 @ X ) ) )
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_409_subset__CollectI,axiom,
    ! [B: set_nat,A2: set_nat,Q4: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B )
           => ( ( Q4 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B )
                & ( Q4 @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_410_subset__CollectI,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,Q4: list_P903359562653991662od_b_c > $o,P: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B @ A2 )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ B )
           => ( ( Q4 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X @ B )
                & ( Q4 @ X ) ) )
          @ ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_411_map__default_Osimps_I1_J,axiom,
    ! [K3: b,V2: c,F: c > c] :
      ( ( map_default_b_c @ K3 @ V2 @ F @ nil_Product_prod_b_c )
      = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K3 @ V2 ) @ nil_Product_prod_b_c ) ) ).

% map_default.simps(1)
thf(fact_412_exE__realizer_H,axiom,
    ! [P: c > b > $o,P2: product_prod_b_c] :
      ( ( P @ ( product_snd_b_c @ P2 ) @ ( product_fst_b_c @ P2 ) )
     => ~ ! [X3: b,Y3: c] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_413_map__default_Osimps_I2_J,axiom,
    ! [P2: product_prod_b_c,K3: b,V2: c,F: c > c,Ps2: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P2 )
          = K3 )
       => ( ( map_default_b_c @ K3 @ V2 @ F @ ( cons_P4529483553340347422od_b_c @ P2 @ Ps2 ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K3 @ ( F @ ( product_snd_b_c @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( product_fst_b_c @ P2 )
         != K3 )
       => ( ( map_default_b_c @ K3 @ V2 @ F @ ( cons_P4529483553340347422od_b_c @ P2 @ Ps2 ) )
          = ( cons_P4529483553340347422od_b_c @ P2 @ ( map_default_b_c @ K3 @ V2 @ F @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_414_map__default_Oelims,axiom,
    ! [X2: b,Xa: c,Xb: c > c,Xc: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( map_default_b_c @ X2 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( Xc = nil_Product_prod_b_c )
         => ( Y
           != ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Xa ) @ nil_Product_prod_b_c ) ) )
       => ~ ! [P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
              ( ( Xc
                = ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) )
             => ~ ( ( ( ( product_fst_b_c @ P4 )
                      = X2 )
                   => ( Y
                      = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ ( Xb @ ( product_snd_b_c @ P4 ) ) ) @ Ps ) ) )
                  & ( ( ( product_fst_b_c @ P4 )
                     != X2 )
                   => ( Y
                      = ( cons_P4529483553340347422od_b_c @ P4 @ ( map_default_b_c @ X2 @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_415_map__entry_Oelims,axiom,
    ! [X2: b,Xa: c > c,Xb: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( map_entry_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Product_prod_b_c )
         => ( Y != nil_Product_prod_b_c ) )
       => ~ ! [P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) )
             => ~ ( ( ( ( product_fst_b_c @ P4 )
                      = X2 )
                   => ( Y
                      = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ ( Xa @ ( product_snd_b_c @ P4 ) ) ) @ Ps ) ) )
                  & ( ( ( product_fst_b_c @ P4 )
                     != X2 )
                   => ( Y
                      = ( cons_P4529483553340347422od_b_c @ P4 @ ( map_entry_b_c @ X2 @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_416_map__entry_Opelims,axiom,
    ! [X2: b,Xa: c > c,Xb: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( ( map_entry_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X2 @ ( produc1101678828523717360od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ( ( Y = nil_Product_prod_b_c )
             => ~ ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X2 @ ( produc1101678828523717360od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) ) )
         => ~ ! [P4: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) )
               => ( ( ( ( ( product_fst_b_c @ P4 )
                        = X2 )
                     => ( Y
                        = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ ( Xa @ ( product_snd_b_c @ P4 ) ) ) @ Ps ) ) )
                    & ( ( ( product_fst_b_c @ P4 )
                       != X2 )
                     => ( Y
                        = ( cons_P4529483553340347422od_b_c @ P4 @ ( map_entry_b_c @ X2 @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X2 @ ( produc1101678828523717360od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ P4 @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_417_path_Ocases,axiom,
    ! [A1: fsm_a_b_c,A22: a,A32: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ A1 @ A22 @ A32 )
     => ( ( ( A32 = nil_Pr1342775757158464060od_c_a )
         => ~ ( member_a @ A22 @ ( states_a_b_c @ A1 ) ) )
       => ~ ! [T4: produc2669932201312201782od_c_a] :
              ( ( A22
                = ( produc4225943634317630866od_c_a @ T4 ) )
             => ! [Ts: list_P6327159017948738492od_c_a] :
                  ( ( A32
                    = ( cons_P8659537020926305772od_c_a @ T4 @ Ts ) )
                 => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ A1 ) )
                   => ~ ( path_a_b_c @ A1 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) @ Ts ) ) ) ) ) ) ).

% path.cases
thf(fact_418_path_Osimps,axiom,
    ( path_a_b_c
    = ( ^ [A12: fsm_a_b_c,A23: a,A33: list_P6327159017948738492od_c_a] :
          ( ? [Q6: a,M2: fsm_a_b_c] :
              ( ( A12 = M2 )
              & ( A23 = Q6 )
              & ( A33 = nil_Pr1342775757158464060od_c_a )
              & ( member_a @ Q6 @ ( states_a_b_c @ M2 ) ) )
          | ? [T2: produc2669932201312201782od_c_a,M2: fsm_a_b_c,Ts2: list_P6327159017948738492od_c_a] :
              ( ( A12 = M2 )
              & ( A23
                = ( produc4225943634317630866od_c_a @ T2 ) )
              & ( A33
                = ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) )
              & ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M2 ) )
              & ( path_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Ts2 ) ) ) ) ) ).

% path.simps
thf(fact_419_path__target__is__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P2 )
     => ( member_a @ ( target_a_b_c @ Q @ P2 ) @ ( states_a_b_c @ M ) ) ) ).

% path_target_is_state
thf(fact_420_path__begin__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P2 )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% path_begin_state
thf(fact_421_transition__subset__path,axiom,
    ! [A2: fsm_a_b_c,B: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ A2 ) @ ( transitions_a_b_c @ B ) )
     => ( ( path_a_b_c @ A2 @ Q @ P2 )
       => ( ( member_a @ Q @ ( states_a_b_c @ B ) )
         => ( path_a_b_c @ B @ Q @ P2 ) ) ) ) ).

% transition_subset_path
thf(fact_422_nil,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( path_a_b_c @ M @ Q @ nil_Pr1342775757158464060od_c_a ) ) ).

% nil
thf(fact_423_path__nil__elim,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( path_a_b_c @ M @ Q @ nil_Pr1342775757158464060od_c_a )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% path_nil_elim
thf(fact_424_after__path,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( after_a_b_c @ M @ Q
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
              @ P2 ) )
          = ( target_a_b_c @ Q @ P2 ) ) ) ) ).

% after_path
thf(fact_425_observable__after__path,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P4 )
             => ( ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                    @ P4 )
                  = Io3 )
               => ( ( target_a_b_c @ Q @ P4 )
                 != ( after_a_b_c @ M @ Q @ Io3 ) ) ) ) ) ) ).

% observable_after_path
thf(fact_426_language__intro,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P2 )
     => ( member6330420149250801815od_b_c
        @ ( map_Pr776894537227705078od_b_c
          @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
          @ P2 )
        @ ( lS_a_b_c @ M @ Q ) ) ) ).

% language_intro
thf(fact_427_language__state__containment,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P2 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
            @ P2 )
          = Io3 )
       => ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_428_observable__path__unique,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( path_a_b_c @ M @ Q @ P7 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                @ P2 )
              = ( map_Pr776894537227705078od_b_c
                @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                @ P7 ) )
           => ( P2 = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_429_map__entry_Osimps_I2_J,axiom,
    ! [P2: product_prod_b_c,K3: b,F: c > c,Ps2: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P2 )
          = K3 )
       => ( ( map_entry_b_c @ K3 @ F @ ( cons_P4529483553340347422od_b_c @ P2 @ Ps2 ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K3 @ ( F @ ( product_snd_b_c @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( product_fst_b_c @ P2 )
         != K3 )
       => ( ( map_entry_b_c @ K3 @ F @ ( cons_P4529483553340347422od_b_c @ P2 @ Ps2 ) )
          = ( cons_P4529483553340347422od_b_c @ P2 @ ( map_entry_b_c @ K3 @ F @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_430_observable__path__language__step,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,X2: b,Y: c] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ~ ? [X3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M ) )
                & ( ( produc4225943634317630866od_c_a @ X3 )
                  = ( target_a_b_c @ Q @ P2 ) )
                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                  = X2 )
                & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                  = Y ) )
         => ~ ( member6330420149250801815od_b_c
              @ ( append2547753245680614915od_b_c
                @ ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P2 )
                @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) )
              @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_path_language_step
thf(fact_431_observable__after__target,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Io5: list_P903359562653991662od_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ Io5 ) @ ( lS_a_b_c @ M @ Q ) )
       => ( ( path_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io3 ) @ P2 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                @ P2 )
              = Io5 )
           => ( ( target_a_b_c @ ( after_a_b_c @ M @ Q @ Io3 ) @ P2 )
              = ( after_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Io3 @ Io5 ) ) ) ) ) ) ) ).

% observable_after_target
thf(fact_432_observable__path__suffix,axiom,
    ! [P2: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
            @ P2 )
          @ Io3 )
        @ ( lS_a_b_c @ M @ Q ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( observable_a_b_c @ M )
         => ~ ! [P8: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P2 ) @ P8 )
               => ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_433_map__append,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( append_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_434_append1__eq__conv,axiom,
    ! [Xs: list_b,X2: b,Ys: list_b,Y: b] :
      ( ( ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) )
        = ( append_b @ Ys @ ( cons_b @ Y @ nil_b ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_435_append1__eq__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X2: product_prod_b_c,Ys: list_P903359562653991662od_b_c,Y: product_prod_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) )
        = ( append2547753245680614915od_b_c @ Ys @ ( cons_P4529483553340347422od_b_c @ Y @ nil_Product_prod_b_c ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_436_prefixes__snoc,axiom,
    ! [Xs: list_b,X2: b] :
      ( ( prefixes_b @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) )
      = ( append_list_b @ ( prefixes_b @ Xs ) @ ( cons_list_b @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) @ nil_list_b ) ) ) ).

% prefixes_snoc
thf(fact_437_prefixes__snoc,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X2: product_prod_b_c] :
      ( ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) )
      = ( append6992831555715836169od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) @ ( cons_l342771267123639716od_b_c @ ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) @ nil_li8071162985807626740od_b_c ) ) ) ).

% prefixes_snoc
thf(fact_438_Cons__eq__appendI,axiom,
    ! [X2: b,Xs1: list_b,Ys: list_b,Xs: list_b,Zs3: list_b] :
      ( ( ( cons_b @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_b @ Xs1 @ Zs3 ) )
       => ( ( cons_b @ X2 @ Xs )
          = ( append_b @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_439_Cons__eq__appendI,axiom,
    ! [X2: product_prod_b_c,Xs1: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append2547753245680614915od_b_c @ Xs1 @ Zs3 ) )
       => ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
          = ( append2547753245680614915od_b_c @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_440_append__Cons,axiom,
    ! [X2: b,Xs: list_b,Ys: list_b] :
      ( ( append_b @ ( cons_b @ X2 @ Xs ) @ Ys )
      = ( cons_b @ X2 @ ( append_b @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_441_append__Cons,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ Ys )
      = ( cons_P4529483553340347422od_b_c @ X2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_442_list__set__sym,axiom,
    ! [X2: list_l8907847357763382004od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ X2 @ Y ) )
      = ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_443_list__set__sym,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ X2 @ Y ) )
      = ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_444_list__prefix__elem,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_445_list__prefix__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs @ Ys ) ) )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_446_list__prefix__elem,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) )
     => ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys ) )
       => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_447_list__concat__non__elem,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ~ ( member_a @ X2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_448_list__concat__non__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_449_list__concat__non__elem,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys ) )
       => ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_450_map__eq__append__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr8222292247188198875_b_c_b @ F @ Us ) )
            & ( Zs3
              = ( map_Pr8222292247188198875_b_c_b @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_451_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr8222292247188198875_b_c_b @ F @ Us ) )
            & ( Zs3
              = ( map_Pr8222292247188198875_b_c_b @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_452_language__prefix,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q ) )
     => ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) ) ) ).

% language_prefix
thf(fact_453_rev__nonempty__induct,axiom,
    ! [Xs: list_b,P: list_b > $o] :
      ( ( Xs != nil_b )
     => ( ! [X3: b] : ( P @ ( cons_b @ X3 @ nil_b ) )
       => ( ! [X3: b,Xs3: list_b] :
              ( ( Xs3 != nil_b )
             => ( ( P @ Xs3 )
               => ( P @ ( append_b @ Xs3 @ ( cons_b @ X3 @ nil_b ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_454_rev__nonempty__induct,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( Xs3 != nil_Product_prod_b_c )
             => ( ( P @ Xs3 )
               => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_455_append__eq__Cons__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,X2: b,Xs: list_b] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( cons_b @ X2 @ Xs ) )
      = ( ( ( Ys = nil_b )
          & ( Zs3
            = ( cons_b @ X2 @ Xs ) ) )
        | ? [Ys4: list_b] :
            ( ( Ys
              = ( cons_b @ X2 @ Ys4 ) )
            & ( ( append_b @ Ys4 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_456_append__eq__Cons__conv,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c,X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Zs3 )
        = ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( ( ( Ys = nil_Product_prod_b_c )
          & ( Zs3
            = ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) ) )
        | ? [Ys4: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ X2 @ Ys4 ) )
            & ( ( append2547753245680614915od_b_c @ Ys4 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_457_Cons__eq__append__conv,axiom,
    ! [X2: b,Xs: list_b,Ys: list_b,Zs3: list_b] :
      ( ( ( cons_b @ X2 @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_b )
          & ( ( cons_b @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys4: list_b] :
            ( ( ( cons_b @ X2 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_b @ Ys4 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_458_Cons__eq__append__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( append2547753245680614915od_b_c @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_Product_prod_b_c )
          & ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys4: list_P903359562653991662od_b_c] :
            ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append2547753245680614915od_b_c @ Ys4 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_459_rev__exhaust,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ~ ! [Ys2: list_b,Y3: b] :
            ( Xs
           != ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) ) ) ).

% rev_exhaust
thf(fact_460_rev__exhaust,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,Y3: product_prod_b_c] :
            ( Xs
           != ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) ) ) ).

% rev_exhaust
thf(fact_461_rev__induct,axiom,
    ! [P: list_b > $o,Xs: list_b] :
      ( ( P @ nil_b )
     => ( ! [X3: b,Xs3: list_b] :
            ( ( P @ Xs3 )
           => ( P @ ( append_b @ Xs3 @ ( cons_b @ X3 @ nil_b ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_462_rev__induct,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( ( P @ Xs3 )
           => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_463_split__list,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs: list_a] :
          ( Xs
          = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs ) ) ) ) ).

% split_list
thf(fact_464_split__list,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs: list_l8907847357763382004od_b_c] :
          ( Xs
          = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs ) ) ) ) ).

% split_list
thf(fact_465_split__list,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ? [Ys2: list_b,Zs: list_b] :
          ( Xs
          = ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs ) ) ) ) ).

% split_list
thf(fact_466_split__list,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
          ( Xs
          = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs ) ) ) ) ).

% split_list
thf(fact_467_split__list__last,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs ) ) )
          & ~ ( member_a @ X2 @ ( set_a2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_468_split__list__last,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs: list_l8907847357763382004od_b_c] :
          ( ( Xs
            = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs ) ) )
          & ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Zs ) ) ) ) ).

% split_list_last
thf(fact_469_split__list__last,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ? [Ys2: list_b,Zs: list_b] :
          ( ( Xs
            = ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs ) ) )
          & ~ ( member_b @ X2 @ ( set_b2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_470_split__list__last,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs ) ) )
          & ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Zs ) ) ) ) ).

% split_list_last
thf(fact_471_split__list__prop,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c] :
          ( ? [Zs: list_l8907847357763382004od_b_c] :
              ( Xs
              = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_472_split__list__prop,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_b,X3: b] :
          ( ? [Zs: list_b] :
              ( Xs
              = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_473_split__list__prop,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
          ( ? [Zs: list_P903359562653991662od_b_c] :
              ( Xs
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_474_split__list__first,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs ) ) )
          & ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_475_split__list__first,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs: list_l8907847357763382004od_b_c] :
          ( ( Xs
            = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs ) ) )
          & ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_476_split__list__first,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ? [Ys2: list_b,Zs: list_b] :
          ( ( Xs
            = ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs ) ) )
          & ~ ( member_b @ X2 @ ( set_b2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_477_split__list__first,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs ) ) )
          & ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_478_split__list__propE,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c] :
            ( ? [Zs: list_l8907847357763382004od_b_c] :
                ( Xs
                = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_479_split__list__propE,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_b,X3: b] :
            ( ? [Zs: list_b] :
                ( Xs
                = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_480_split__list__propE,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
            ( ? [Zs: list_P903359562653991662od_b_c] :
                ( Xs
                = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_481_append__Cons__eq__iff,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
            = ( append_a @ Xs4 @ ( cons_a @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_482_append__Cons__eq__iff,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c,Xs4: list_l8907847357763382004od_b_c,Ys5: list_l8907847357763382004od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ( ( ( append6992831555715836169od_b_c @ Xs @ ( cons_l342771267123639716od_b_c @ X2 @ Ys ) )
            = ( append6992831555715836169od_b_c @ Xs4 @ ( cons_l342771267123639716od_b_c @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_483_append__Cons__eq__iff,axiom,
    ! [X2: b,Xs: list_b,Ys: list_b,Xs4: list_b,Ys5: list_b] :
      ( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
     => ( ~ ( member_b @ X2 @ ( set_b2 @ Ys ) )
       => ( ( ( append_b @ Xs @ ( cons_b @ X2 @ Ys ) )
            = ( append_b @ Xs4 @ ( cons_b @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_484_append__Cons__eq__iff,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Xs4: list_P903359562653991662od_b_c,Ys5: list_P903359562653991662od_b_c] :
      ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys ) )
       => ( ( ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X2 @ Ys ) )
            = ( append2547753245680614915od_b_c @ Xs4 @ ( cons_P4529483553340347422od_b_c @ X2 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_485_in__set__conv__decomp,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_486_in__set__conv__decomp,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( Xs
            = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_487_in__set__conv__decomp,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
      = ( ? [Ys3: list_b,Zs2: list_b] :
            ( Xs
            = ( append_b @ Ys3 @ ( cons_b @ X2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_488_in__set__conv__decomp,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( Xs
            = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_489_split__list__last__prop,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c,Zs: list_l8907847357763382004od_b_c] :
          ( ( Xs
            = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa3 @ ( set_li4480668622519654659od_b_c @ Zs ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_490_split__list__last__prop,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_b,X3: b,Zs: list_b] :
          ( ( Xs
            = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: b] :
              ( ( member_b @ Xa3 @ ( set_b2 @ Zs ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_491_split__list__last__prop,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ Xa3 @ ( set_Product_prod_b_c2 @ Zs ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_492_split__list__first__prop,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c] :
          ( ? [Zs: list_l8907847357763382004od_b_c] :
              ( Xs
              = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa3 @ ( set_li4480668622519654659od_b_c @ Ys2 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_493_split__list__first__prop,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_b,X3: b] :
          ( ? [Zs: list_b] :
              ( Xs
              = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: b] :
              ( ( member_b @ Xa3 @ ( set_b2 @ Ys2 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_494_split__list__first__prop,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
          ( ? [Zs: list_P903359562653991662od_b_c] :
              ( Xs
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
          & ( P @ X3 )
          & ! [Xa3: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ Xa3 @ ( set_Product_prod_b_c2 @ Ys2 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_495_split__list__last__propE,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c,Zs: list_l8907847357763382004od_b_c] :
            ( ( Xs
              = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: list_P903359562653991662od_b_c] :
                    ( ( member6330420149250801815od_b_c @ Xa3 @ ( set_li4480668622519654659od_b_c @ Zs ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_496_split__list__last__propE,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_b,X3: b,Zs: list_b] :
            ( ( Xs
              = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: b] :
                    ( ( member_b @ Xa3 @ ( set_b2 @ Zs ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_497_split__list__last__propE,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: product_prod_b_c] :
                    ( ( member7862447941013992593od_b_c @ Xa3 @ ( set_Product_prod_b_c2 @ Zs ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_498_split__list__first__propE,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_l8907847357763382004od_b_c,X3: list_P903359562653991662od_b_c] :
            ( ? [Zs: list_l8907847357763382004od_b_c] :
                ( Xs
                = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: list_P903359562653991662od_b_c] :
                    ( ( member6330420149250801815od_b_c @ Xa3 @ ( set_li4480668622519654659od_b_c @ Ys2 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_499_split__list__first__propE,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_b,X3: b] :
            ( ? [Zs: list_b] :
                ( Xs
                = ( append_b @ Ys2 @ ( cons_b @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: b] :
                    ( ( member_b @ Xa3 @ ( set_b2 @ Ys2 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_500_split__list__first__propE,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
            ( ? [Zs: list_P903359562653991662od_b_c] :
                ( Xs
                = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa3: product_prod_b_c] :
                    ( ( member7862447941013992593od_b_c @ Xa3 @ ( set_Product_prod_b_c2 @ Ys2 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_501_in__set__conv__decomp__last,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
            & ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_502_in__set__conv__decomp__last,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( ( Xs
              = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs2 ) ) )
            & ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_503_in__set__conv__decomp__last,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
      = ( ? [Ys3: list_b,Zs2: list_b] :
            ( ( Xs
              = ( append_b @ Ys3 @ ( cons_b @ X2 @ Zs2 ) ) )
            & ~ ( member_b @ X2 @ ( set_b2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_504_in__set__conv__decomp__last,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs2 ) ) )
            & ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_505_in__set__conv__decomp__first,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
            & ~ ( member_a @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_506_in__set__conv__decomp__first,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( ( Xs
              = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X2 @ Zs2 ) ) )
            & ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_507_in__set__conv__decomp__first,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
      = ( ? [Ys3: list_b,Zs2: list_b] :
            ( ( Xs
              = ( append_b @ Ys3 @ ( cons_b @ X2 @ Zs2 ) ) )
            & ~ ( member_b @ X2 @ ( set_b2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_508_in__set__conv__decomp__first,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X2 @ Zs2 ) ) )
            & ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_509_split__list__last__prop__iff,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( ? [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,X: list_P903359562653991662od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( ( Xs
              = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y2 @ ( set_li4480668622519654659od_b_c @ Zs2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_510_split__list__last__prop__iff,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ( ? [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_b,X: b,Zs2: list_b] :
            ( ( Xs
              = ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: b] :
                ( ( member_b @ Y2 @ ( set_b2 @ Zs2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_511_split__list__last__prop__iff,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ( ? [X: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,X: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: product_prod_b_c] :
                ( ( member7862447941013992593od_b_c @ Y2 @ ( set_Product_prod_b_c2 @ Zs2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_512_split__list__first__prop__iff,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( ? [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,X: list_P903359562653991662od_b_c] :
            ( ? [Zs2: list_l8907847357763382004od_b_c] :
                ( Xs
                = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y2 @ ( set_li4480668622519654659od_b_c @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_513_split__list__first__prop__iff,axiom,
    ! [Xs: list_b,P: b > $o] :
      ( ( ? [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_b,X: b] :
            ( ? [Zs2: list_b] :
                ( Xs
                = ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: b] :
                ( ( member_b @ Y2 @ ( set_b2 @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_514_split__list__first__prop__iff,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ( ? [X: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,X: product_prod_b_c] :
            ( ? [Zs2: list_P903359562653991662od_b_c] :
                ( Xs
                = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y2: product_prod_b_c] :
                ( ( member7862447941013992593od_b_c @ Y2 @ ( set_Product_prod_b_c2 @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_515_list__append__subset3,axiom,
    ! [Xs1: list_P903359562653991662od_b_c,Ys1: list_P903359562653991662od_b_c,Xs22: list_P903359562653991662od_b_c,Ys22: list_P903359562653991662od_b_c,Xs32: list_P903359562653991662od_b_c,Ys32: list_P903359562653991662od_b_c] :
      ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs1 ) @ ( set_Product_prod_b_c2 @ Ys1 ) )
     => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs22 ) @ ( set_Product_prod_b_c2 @ Ys22 ) )
       => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs32 ) @ ( set_Product_prod_b_c2 @ Ys32 ) )
         => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs1 @ ( append2547753245680614915od_b_c @ Xs22 @ Xs32 ) ) ) @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Ys1 @ ( append2547753245680614915od_b_c @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_516_list__append__subset3,axiom,
    ! [Xs1: list_l8907847357763382004od_b_c,Ys1: list_l8907847357763382004od_b_c,Xs22: list_l8907847357763382004od_b_c,Ys22: list_l8907847357763382004od_b_c,Xs32: list_l8907847357763382004od_b_c,Ys32: list_l8907847357763382004od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs1 ) @ ( set_li4480668622519654659od_b_c @ Ys1 ) )
     => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs22 ) @ ( set_li4480668622519654659od_b_c @ Ys22 ) )
       => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs32 ) @ ( set_li4480668622519654659od_b_c @ Ys32 ) )
         => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs1 @ ( append6992831555715836169od_b_c @ Xs22 @ Xs32 ) ) ) @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Ys1 @ ( append6992831555715836169od_b_c @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_517_list__prefix__subset,axiom,
    ! [Ts3: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ? [Ys6: list_P903359562653991662od_b_c] :
          ( Ts3
          = ( append2547753245680614915od_b_c @ Xs @ Ys6 ) )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ts3 ) ) ) ).

% list_prefix_subset
thf(fact_518_list__prefix__subset,axiom,
    ! [Ts3: list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ? [Ys6: list_l8907847357763382004od_b_c] :
          ( Ts3
          = ( append6992831555715836169od_b_c @ Xs @ Ys6 ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Ts3 ) ) ) ).

% list_prefix_subset
thf(fact_519_prefixes__set__ob,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xss2: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xss2 ) ) )
     => ~ ! [Xs5: list_P903359562653991662od_b_c] :
            ( Xss2
           != ( append2547753245680614915od_b_c @ Xs @ Xs5 ) ) ) ).

% prefixes_set_ob
thf(fact_520_prefixes__prepend,axiom,
    ! [Xs4: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs4 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ys @ Xs4 ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ Ys @ Xs ) ) ) ) ) ).

% prefixes_prepend
thf(fact_521_prefixes__eq__snoc,axiom,
    ! [Ys: list_b,Xs: list_list_b,X2: list_b] :
      ( ( ( prefixes_b @ Ys )
        = ( append_list_b @ Xs @ ( cons_list_b @ X2 @ nil_list_b ) ) )
      = ( ( ( ( Ys = nil_b )
            & ( Xs = nil_list_b ) )
          | ? [Z3: b,Zs2: list_b] :
              ( ( Ys
                = ( append_b @ Zs2 @ ( cons_b @ Z3 @ nil_b ) ) )
              & ( Xs
                = ( prefixes_b @ Zs2 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_522_prefixes__eq__snoc,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( ( prefix1131979855692807669od_b_c @ Ys )
        = ( append6992831555715836169od_b_c @ Xs @ ( cons_l342771267123639716od_b_c @ X2 @ nil_li8071162985807626740od_b_c ) ) )
      = ( ( ( ( Ys = nil_Product_prod_b_c )
            & ( Xs = nil_li8071162985807626740od_b_c ) )
          | ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
              ( ( Ys
                = ( append2547753245680614915od_b_c @ Zs2 @ ( cons_P4529483553340347422od_b_c @ Z3 @ nil_Product_prod_b_c ) ) )
              & ( Xs
                = ( prefix1131979855692807669od_b_c @ Zs2 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_523_prefixes__prefix__suffix__ob,axiom,
    ! [A: list_P903359562653991662od_b_c,B3: list_P903359562653991662od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ B3 @ C ) ) ) )
     => ( ~ ( member6330420149250801815od_b_c @ A @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ B3 ) ) )
       => ~ ! [C3: list_P903359562653991662od_b_c] :
              ( ? [C4: list_P903359562653991662od_b_c] :
                  ( C
                  = ( append2547753245680614915od_b_c @ C3 @ C4 ) )
             => ( ( A
                  = ( append2547753245680614915od_b_c @ B3 @ C3 ) )
               => ( C3 = nil_Product_prod_b_c ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_524_prefixes__set__subset,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) ) ) ).

% prefixes_set_subset
thf(fact_525_observable__after__language__append,axiom,
    ! [M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io22: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_526_observable__after__language__none,axiom,
    ! [M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io22: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_527_observable__after__eq,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( ( after_a_b_c @ M @ Q @ Io1 )
          = ( after_a_b_c @ M @ Q @ Io22 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io3 ) @ ( lS_a_b_c @ M @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io22 @ Io3 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_528_after__language__iff,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Q: a,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_529_after__split,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ Q ) )
       => ( ( after_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Alpha ) @ Gamma )
          = ( after_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_530_List_Oproduct_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b,Ys: list_c] :
      ( ( product_b_c @ ( cons_b @ X2 @ Xs ) @ Ys )
      = ( append2547753245680614915od_b_c @ ( map_c_3793715988870728620od_b_c @ ( product_Pair_b_c @ X2 ) @ Ys ) @ ( product_b_c @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_531_language__path__append__transition,axiom,
    ! [Ios: list_P903359562653991662od_b_c,Io3: product_prod_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io3 @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ Q ) )
     => ~ ! [P4: list_P6327159017948738492od_c_a,T4: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P4 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( map_Pr776894537227705078od_b_c
                @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                @ ( append3884164603879732689od_c_a @ P4 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
             != ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io3 @ nil_Product_prod_b_c ) ) ) ) ) ).

% language_path_append_transition
thf(fact_532_language__maximal__contained__prefix__ob,axiom,
    ! [Io3: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ Io3 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
       => ( ( observ6293852833591064631_c_b_c @ M )
         => ~ ! [Io2: list_P903359562653991662od_b_c,X3: b,Y3: c] :
                ( ? [Io6: list_P903359562653991662od_b_c] :
                    ( Io3
                    = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ nil_Product_prod_b_c ) @ Io6 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ nil_Product_prod_b_c ) ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_533_language__maximal__contained__prefix__ob,axiom,
    ! [Io3: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ~ ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( observable_a_b_c @ M )
         => ~ ! [Io2: list_P903359562653991662od_b_c,X3: b,Y3: c] :
                ( ? [Io6: list_P903359562653991662od_b_c] :
                    ( Io3
                    = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ nil_Product_prod_b_c ) @ Io6 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_534_language__path__append__transition__observable,axiom,
    ! [P2: list_P6327159017948738492od_c_a,X2: b,Y: c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
            @ P2 )
          @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) )
        @ ( lS_a_b_c @ M @ Q ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( observable_a_b_c @ M )
         => ~ ! [T4: produc2669932201312201782od_c_a] :
                ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P2 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) )
                    = X2 )
                 => ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                   != Y ) ) ) ) ) ) ).

% language_path_append_transition_observable
thf(fact_535_language__path__append,axiom,
    ! [M1: fsm_a_b_c,Q14: a,P1: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M1 @ Q14 @ P1 )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M1 @ ( target_a_b_c @ Q14 @ P1 ) ) )
       => ( member6330420149250801815od_b_c
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_a_b_c @ M1 @ Q14 ) ) ) ) ).

% language_path_append
thf(fact_536_language__state__split,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q14: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q14 ) )
     => ~ ! [P12: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ M @ Q14 @ P12 )
           => ! [P22: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( target_a_b_c @ Q14 @ P12 ) @ P22 )
               => ( ( ( map_Pr776894537227705078od_b_c
                      @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr776894537227705078od_b_c
                      @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                      @ P22 )
                   != Io22 ) ) ) ) ) ).

% language_state_split
thf(fact_537_observable__io__targets__next,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( io_targets_a_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr776894537227705078od_b_c
                @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) )
              @ Io3 )
            @ ( produc4225943634317630866od_c_a @ T ) )
          = ( io_targets_a_b_c @ M @ Io3 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_538_io__targets__next,axiom,
    ! [T: produc596031391177498678od_b_c,M: fsm_li6801133765522507155_c_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( member8124909232574984543od_b_c @ T @ ( transi3161963839906981764_c_b_c @ M ) )
     => ( ord_le282488521294790766od_b_c @ ( io_tar7088883562557314310_c_b_c @ M @ Io3 @ ( produc8475728678562477363od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T ) ) ) )
        @ ( io_tar7088883562557314310_c_b_c @ M
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr4814182253694984182od_b_c
              @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
              @ ( cons_P3567095617774512108od_b_c @ T @ nil_Pr7618980518134713404od_b_c ) )
            @ Io3 )
          @ ( produc5084291339869529746od_b_c @ T ) ) ) ) ).

% io_targets_next
thf(fact_539_language__append__path__ob,axiom,
    ! [Io3: list_P903359562653991662od_b_c,X2: b,Y: c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P4: list_P6327159017948738492od_c_a,T4: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P4 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( ( map_Pr776894537227705078od_b_c
                  @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                  @ P4 )
                = Io3 )
             => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) )
                  = X2 )
               => ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                 != Y ) ) ) ) ) ).

% language_append_path_ob
thf(fact_540_language__initial__path__append__transition,axiom,
    ! [Ios: list_P903359562653991662od_b_c,Io3: product_prod_b_c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io3 @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P4: list_P6327159017948738492od_c_a,T4: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P4 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( map_Pr776894537227705078od_b_c
                @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                @ ( append3884164603879732689od_c_a @ P4 @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) )
             != ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io3 @ nil_Product_prod_b_c ) ) ) ) ) ).

% language_initial_path_append_transition
thf(fact_541_Diff__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( minus_5375846548645262261od_b_c @ A2 @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A2 )
        & ~ ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Diff_iff
thf(fact_542_Diff__iff,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
      = ( ( member_a @ C @ A2 )
        & ~ ( member_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_543_DiffI,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A2 )
     => ( ~ ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ ( minus_5375846548645262261od_b_c @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_544_DiffI,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ A2 )
     => ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_545_fsm__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( states_a_b_c @ M ) ) ).

% fsm_initial
thf(fact_546_set__diff__eq,axiom,
    ( minus_minus_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A3 )
              & ~ ( member_a @ X @ B2 ) ) ) ) ) ).

% set_diff_eq
thf(fact_547_set__diff__eq,axiom,
    ( minus_5375846548645262261od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ A3 )
              & ~ ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ) ).

% set_diff_eq
thf(fact_548_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ~ ( member_nat @ X @ B2 ) ) ) ) ) ).

% set_diff_eq
thf(fact_549_DiffD2,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( minus_5375846548645262261od_b_c @ A2 @ B ) )
     => ~ ( member6330420149250801815od_b_c @ C @ B ) ) ).

% DiffD2
thf(fact_550_DiffD2,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ~ ( member_a @ C @ B ) ) ).

% DiffD2
thf(fact_551_DiffD1,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( minus_5375846548645262261od_b_c @ A2 @ B ) )
     => ( member6330420149250801815od_b_c @ C @ A2 ) ) ).

% DiffD1
thf(fact_552_DiffD1,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ( member_a @ C @ A2 ) ) ).

% DiffD1
thf(fact_553_DiffE,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( minus_5375846548645262261od_b_c @ A2 @ B ) )
     => ~ ( ( member6330420149250801815od_b_c @ C @ A2 )
         => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% DiffE
thf(fact_554_DiffE,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ~ ( ( member_a @ C @ A2 )
         => ( member_a @ C @ B ) ) ) ).

% DiffE
thf(fact_555_Diff__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ D @ B )
       => ( ord_le282488521294790766od_b_c @ ( minus_5375846548645262261od_b_c @ A2 @ B ) @ ( minus_5375846548645262261od_b_c @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_556_Diff__subset,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( minus_5375846548645262261od_b_c @ A2 @ B ) @ A2 ) ).

% Diff_subset
thf(fact_557_double__diff,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ C2 )
       => ( ( minus_5375846548645262261od_b_c @ B @ ( minus_5375846548645262261od_b_c @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_558_io__targets__language,axiom,
    ! [Q3: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Io3: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q3 @ ( io_tar7088883562557314310_c_b_c @ M @ Io3 @ Q ) )
     => ( member6330420149250801815od_b_c @ Io3 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ).

% io_targets_language
thf(fact_559_io__targets__language,axiom,
    ! [Q3: a,M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( member_a @ Q3 @ ( io_targets_a_b_c @ M @ Io3 @ Q ) )
     => ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) ) ) ).

% io_targets_language
thf(fact_560_observable__language__target,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io1: list_P903359562653991662od_b_c,T: list_P903359562653991662od_b_c,T3: fsm_li6801133765522507155_c_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ ( initia3567573336347591134_c_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ T @ ( io_tar7088883562557314310_c_b_c @ T3 @ Io1 @ ( initia3567573336347591134_c_b_c @ T3 ) ) )
         => ( ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ ( initia3567573336347591134_c_b_c @ T3 ) ) @ ( lS_lis2930931384350476499_c_b_c @ M @ ( initia3567573336347591134_c_b_c @ M ) ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ T ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ) ) ).

% observable_language_target
thf(fact_561_observable__language__target,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io1: list_P903359562653991662od_b_c,T: a,T3: fsm_a_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ ( initia3567573336347591134_c_b_c @ M ) ) )
       => ( ( member_a @ T @ ( io_targets_a_b_c @ T3 @ Io1 @ ( initial_a_b_c @ T3 ) ) )
         => ( ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ ( initial_a_b_c @ T3 ) ) @ ( lS_lis2930931384350476499_c_b_c @ M @ ( initia3567573336347591134_c_b_c @ M ) ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ T ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ) ) ).

% observable_language_target
thf(fact_562_observable__language__target,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,T: list_P903359562653991662od_b_c,T3: fsm_li6801133765522507155_c_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( io_targets_a_b_c @ M @ Io1 @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ T @ ( io_tar7088883562557314310_c_b_c @ T3 @ Io1 @ ( initia3567573336347591134_c_b_c @ T3 ) ) )
         => ( ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ ( initia3567573336347591134_c_b_c @ T3 ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ T ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ).

% observable_language_target
thf(fact_563_observable__language__target,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,T: a,T3: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( io_targets_a_b_c @ M @ Io1 @ ( initial_a_b_c @ M ) ) )
       => ( ( member_a @ T @ ( io_targets_a_b_c @ T3 @ Io1 @ ( initial_a_b_c @ T3 ) ) )
         => ( ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ ( initial_a_b_c @ T3 ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ T ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ).

% observable_language_target
thf(fact_564_observable__language__target__failure,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io1: list_P903359562653991662od_b_c,T: list_P903359562653991662od_b_c,T3: fsm_li6801133765522507155_c_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ ( initia3567573336347591134_c_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ T @ ( io_tar7088883562557314310_c_b_c @ T3 @ Io1 @ ( initia3567573336347591134_c_b_c @ T3 ) ) )
         => ( ~ ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ T ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) )
           => ~ ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ ( initia3567573336347591134_c_b_c @ T3 ) ) @ ( lS_lis2930931384350476499_c_b_c @ M @ ( initia3567573336347591134_c_b_c @ M ) ) ) ) ) ) ) ).

% observable_language_target_failure
thf(fact_565_observable__language__target__failure,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io1: list_P903359562653991662od_b_c,T: a,T3: fsm_a_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Q @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ ( initia3567573336347591134_c_b_c @ M ) ) )
       => ( ( member_a @ T @ ( io_targets_a_b_c @ T3 @ Io1 @ ( initial_a_b_c @ T3 ) ) )
         => ( ~ ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ T ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) )
           => ~ ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ ( initial_a_b_c @ T3 ) ) @ ( lS_lis2930931384350476499_c_b_c @ M @ ( initia3567573336347591134_c_b_c @ M ) ) ) ) ) ) ) ).

% observable_language_target_failure
thf(fact_566_observable__language__target__failure,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,T: list_P903359562653991662od_b_c,T3: fsm_li6801133765522507155_c_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( io_targets_a_b_c @ M @ Io1 @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ T @ ( io_tar7088883562557314310_c_b_c @ T3 @ Io1 @ ( initia3567573336347591134_c_b_c @ T3 ) ) )
         => ( ~ ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ T ) @ ( lS_a_b_c @ M @ Q ) )
           => ~ ( ord_le282488521294790766od_b_c @ ( lS_lis2930931384350476499_c_b_c @ T3 @ ( initia3567573336347591134_c_b_c @ T3 ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ) ) ) ).

% observable_language_target_failure
thf(fact_567_observable__language__target__failure,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,T: a,T3: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( io_targets_a_b_c @ M @ Io1 @ ( initial_a_b_c @ M ) ) )
       => ( ( member_a @ T @ ( io_targets_a_b_c @ T3 @ Io1 @ ( initial_a_b_c @ T3 ) ) )
         => ( ~ ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ T ) @ ( lS_a_b_c @ M @ Q ) )
           => ~ ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ T3 @ ( initial_a_b_c @ T3 ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ) ) ) ).

% observable_language_target_failure
thf(fact_568_language__contains__empty__sequence,axiom,
    ! [M: fsm_a_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ).

% language_contains_empty_sequence
thf(fact_569_io__targets__states,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] : ( ord_less_eq_set_a @ ( io_targets_a_b_c @ M @ Io3 @ Q ) @ ( states_a_b_c @ M ) ) ).

% io_targets_states
thf(fact_570_io__targets__language__append,axiom,
    ! [Q14: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Io1: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q14 @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ Q ) )
     => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q14 ) )
       => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ).

% io_targets_language_append
thf(fact_571_io__targets__language__append,axiom,
    ! [Q14: a,M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io22: list_P903359562653991662od_b_c] :
      ( ( member_a @ Q14 @ ( io_targets_a_b_c @ M @ Io1 @ Q ) )
     => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ Q14 ) )
       => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% io_targets_language_append
thf(fact_572_language__io__target__append,axiom,
    ! [Q3: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Io1: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q3 @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ Q ) )
     => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q3 ) )
       => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q ) ) ) ) ).

% language_io_target_append
thf(fact_573_language__io__target__append,axiom,
    ! [Q3: a,M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io22: list_P903359562653991662od_b_c] :
      ( ( member_a @ Q3 @ ( io_targets_a_b_c @ M @ Io1 @ Q ) )
     => ( ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ Q3 ) )
       => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% language_io_target_append
thf(fact_574_after__language__subset,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_575_after__language__append__iff,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_576_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Q14: list_P903359562653991662od_b_c,Q22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_lis2930931384350476499_c_b_c @ M @ Q14 ) )
     => ( ( observ6293852833591064631_c_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ Q22 @ ( io_tar7088883562557314310_c_b_c @ M @ Io1 @ Q14 ) )
         => ( member6330420149250801815od_b_c @ Io22 @ ( lS_lis2930931384350476499_c_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_577_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q14: a,Q22: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ Q14 ) )
     => ( ( observable_a_b_c @ M )
       => ( ( member_a @ Q22 @ ( io_targets_a_b_c @ M @ Io1 @ Q14 ) )
         => ( member6330420149250801815od_b_c @ Io22 @ ( lS_a_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_578_defined__inputs__language__diff,axiom,
    ! [X2: b,M1: fsm_a_b_c,Q14: a,M22: fsm_a_b_c,Q22: a] :
      ( ( member_b @ X2 @ ( defined_inputs_a_b_c @ M1 @ Q14 ) )
     => ( ~ ( member_b @ X2 @ ( defined_inputs_a_b_c @ M22 @ Q22 ) )
       => ~ ! [Y3: c] :
              ~ ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y3 ) @ nil_Product_prod_b_c ) @ ( minus_5375846548645262261od_b_c @ ( lS_a_b_c @ M1 @ Q14 ) @ ( lS_a_b_c @ M22 @ Q22 ) ) ) ) ) ).

% defined_inputs_language_diff
thf(fact_579_language__split,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io22 ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P12: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P12 @ P22 ) )
           => ( ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P22 )
               != Io22 ) ) ) ) ).

% language_split
thf(fact_580_h__obs__language__append,axiom,
    ! [M: fsm_a_b_c,U2: list_P903359562653991662od_b_c,X2: b,Y: c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ U2 @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( ( h_obs_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ U2 ) @ X2 @ Y )
           != none_a )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U2 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ) ) ).

% h_obs_language_append
thf(fact_581_pass__io__set__maximal__def,axiom,
    ( pass_i5868551379204665570_a_b_c
    = ( ^ [M2: fsm_a_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io: list_P903359562653991662od_b_c,X: b,Y2: c,Io7: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ nil_Product_prod_b_c ) @ Io7 ) ) @ Ios2 )
         => ! [Y6: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y6 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
             => ? [Io8: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y6 ) @ nil_Product_prod_b_c ) @ Io8 ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_maximal_def
thf(fact_582_pass__io__set__def,axiom,
    ( pass_io_set_a_b_c
    = ( ^ [M2: fsm_a_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io: list_P903359562653991662od_b_c,X: b,Y2: c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ nil_Product_prod_b_c ) ) @ Ios2 )
         => ! [Y6: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y6 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
             => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y6 ) @ nil_Product_prod_b_c ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_def
thf(fact_583_language__state__prepend__transition,axiom,
    ! [Io3: list_P903359562653991662od_b_c,A2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ ( from_FSM_a_b_c @ A2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ A2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ A2 ) )
       => ( member6330420149250801815od_b_c
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
              @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) )
            @ Io3 )
          @ ( lS_a_b_c @ A2 @ ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ).

% language_state_prepend_transition
thf(fact_584_from__from,axiom,
    ! [Q14: a,M: fsm_a_b_c,Q15: a] :
      ( ( member_a @ Q14 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q15 @ ( states_a_b_c @ M ) )
       => ( ( from_FSM_a_b_c @ ( from_FSM_a_b_c @ M @ Q14 ) @ Q15 )
          = ( from_FSM_a_b_c @ M @ Q15 ) ) ) ) ).

% from_from
thf(fact_585_from__FSM__simps_I5_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( states_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( states_a_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_586_from__FSM__simps_I4_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( transitions_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( transitions_a_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_587_from__FSM__simps_I1_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_588_from__FSM__observable,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( observable_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_589_from__FSM__path,axiom,
    ! [Q: a,M: fsm_a_b_c,Q3: a,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ Q3 @ P2 )
       => ( path_a_b_c @ M @ Q3 @ P2 ) ) ) ).

% from_FSM_path
thf(fact_590_from__FSM__language,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( lS_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) )
        = ( lS_a_b_c @ M @ Q ) ) ) ).

% from_FSM_language
thf(fact_591_from__FSM__path__initial,axiom,
    ! [Q: a,M: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
        = ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) @ P2 ) ) ) ).

% from_FSM_path_initial
thf(fact_592_observable__language__next,axiom,
    ! [Io3: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io3 @ Ios ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) )
     => ( ( observable_a_b_c @ M )
       => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) )
              = ( product_fst_b_c @ Io3 ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                = ( product_snd_b_c @ Io3 ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ ( from_FSM_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_593_pow__list_Oelims,axiom,
    ! [X2: list_b,Y: list_list_b] :
      ( ( ( pow_list_b @ X2 )
        = Y )
     => ( ( ( X2 = nil_b )
         => ( Y
           != ( cons_list_b @ nil_b @ nil_list_b ) ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( X2
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( append_list_b @ ( pow_list_b @ Xs3 ) @ ( map_list_b_list_b @ ( cons_b @ X3 ) @ ( pow_list_b @ Xs3 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_594_pow__list_Oelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( pow_li4341120398627903455od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_Product_prod_b_c )
         => ( Y
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( X2
                = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
             => ( Y
               != ( append6992831555715836169od_b_c @ ( pow_li4341120398627903455od_b_c @ Xs3 ) @ ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 ) @ ( pow_li4341120398627903455od_b_c @ Xs3 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_595_linear__order__from__list__position_H_Oelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( linear6964191089970808489tion_b @ X2 )
        = Y )
     => ( ( ( X2 = nil_b )
         => ( Y != nil_Product_prod_b_b ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( X2
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( cons_P4529483549037118621od_b_b @ ( product_Pair_b_b @ X3 @ X3 ) @ ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( linear6964191089970808489tion_b @ Xs3 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_596_linear__order__from__list__position_H_Oelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( linear5647500195961616402od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_Product_prod_b_c )
         => ( Y != nil_Pr2945457328363356077od_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( X2
                = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
             => ( Y
               != ( cons_P1038846555276421981od_b_c @ ( produc6366901981060558487od_b_c @ X3 @ X3 ) @ ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( linear5647500195961616402od_b_c @ Xs3 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_597_list__ordered__pairs_Oelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( list_ordered_pairs_b @ X2 )
        = Y )
     => ( ( ( X2 = nil_b )
         => ( Y != nil_Product_prod_b_b ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( X2
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( list_ordered_pairs_b @ Xs3 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_598_list__ordered__pairs_Oelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( list_o804723041229831855od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_Product_prod_b_c )
         => ( Y != nil_Pr2945457328363356077od_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( X2
                = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
             => ( Y
               != ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( list_o804723041229831855od_b_c @ Xs3 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_599_member__filter,axiom,
    ! [X2: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( filter4546979855996260255od_b_c @ P @ A2 ) )
      = ( ( member6330420149250801815od_b_c @ X2 @ A2 )
        & ( P @ X2 ) ) ) ).

% member_filter
thf(fact_600_member__filter,axiom,
    ! [X2: a,P: a > $o,A2: set_a] :
      ( ( member_a @ X2 @ ( filter_a @ P @ A2 ) )
      = ( ( member_a @ X2 @ A2 )
        & ( P @ X2 ) ) ) ).

% member_filter
thf(fact_601_Set_Ofilter__def,axiom,
    ( filter_a
    = ( ^ [P6: a > $o,A3: set_a] :
          ( collect_a
          @ ^ [A4: a] :
              ( ( member_a @ A4 @ A3 )
              & ( P6 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_602_Set_Ofilter__def,axiom,
    ( filter4546979855996260255od_b_c
    = ( ^ [P6: list_P903359562653991662od_b_c > $o,A3: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [A4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ A4 @ A3 )
              & ( P6 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_603_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P6: nat > $o,A3: set_nat] :
          ( collect_nat
          @ ^ [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
              & ( P6 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_604_list__ordered__pairs__set__containment,axiom,
    ! [X2: a,Xs: list_a,Y: a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y @ ( set_a2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_605_list__ordered__pairs__set__containment,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( list_o8363625500407931829od_b_c @ Xs ) ) )
            | ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ Y @ X2 ) @ ( set_Pr292865031892111420od_b_c @ ( list_o8363625500407931829od_b_c @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_606_list__ordered__pairs__set__containment,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y: product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member7104721839413753296od_b_c @ ( produc6366901981060558487od_b_c @ X2 @ Y ) @ ( set_Pr7267323205303737404od_b_c @ ( list_o804723041229831855od_b_c @ Xs ) ) )
            | ( member7104721839413753296od_b_c @ ( produc6366901981060558487od_b_c @ Y @ X2 ) @ ( set_Pr7267323205303737404od_b_c @ ( list_o804723041229831855od_b_c @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_607_pow__list_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( pow_list_b @ ( cons_b @ X2 @ Xs ) )
      = ( append_list_b @ ( pow_list_b @ Xs ) @ ( map_list_b_list_b @ ( cons_b @ X2 ) @ ( pow_list_b @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_608_pow__list_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( pow_li4341120398627903455od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append6992831555715836169od_b_c @ ( pow_li4341120398627903455od_b_c @ Xs ) @ ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 ) @ ( pow_li4341120398627903455od_b_c @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_609_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( list_ordered_pairs_b @ ( cons_b @ X2 @ Xs ) )
      = ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X2 ) @ Xs ) @ ( list_ordered_pairs_b @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_610_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( list_o804723041229831855od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X2 ) @ Xs ) @ ( list_o804723041229831855od_b_c @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_611_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( linear6964191089970808489tion_b @ ( cons_b @ X2 @ Xs ) )
      = ( cons_P4529483549037118621od_b_b @ ( product_Pair_b_b @ X2 @ X2 ) @ ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X2 ) @ Xs ) @ ( linear6964191089970808489tion_b @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_612_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( linear5647500195961616402od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( cons_P1038846555276421981od_b_c @ ( produc6366901981060558487od_b_c @ X2 @ X2 ) @ ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X2 ) @ Xs ) @ ( linear5647500195961616402od_b_c @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_613_linear__order__from__list__position_H_Opelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( linear6964191089970808489tion_b @ X2 )
        = Y )
     => ( ( accp_list_b @ linear4418539559401215898_rel_b @ X2 )
       => ( ( ( X2 = nil_b )
           => ( ( Y = nil_Product_prod_b_b )
             => ~ ( accp_list_b @ linear4418539559401215898_rel_b @ nil_b ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( X2
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( cons_P4529483549037118621od_b_b @ ( product_Pair_b_b @ X3 @ X3 ) @ ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( linear6964191089970808489tion_b @ Xs3 ) ) ) )
                 => ~ ( accp_list_b @ linear4418539559401215898_rel_b @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_614_linear__order__from__list__position_H_Opelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( linear5647500195961616402od_b_c @ X2 )
        = Y )
     => ( ( accp_l4654477130474446199od_b_c @ linear4366178405326712835od_b_c @ X2 )
       => ( ( ( X2 = nil_Product_prod_b_c )
           => ( ( Y = nil_Pr2945457328363356077od_b_c )
             => ~ ( accp_l4654477130474446199od_b_c @ linear4366178405326712835od_b_c @ nil_Product_prod_b_c ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
                ( ( X2
                  = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( cons_P1038846555276421981od_b_c @ ( produc6366901981060558487od_b_c @ X3 @ X3 ) @ ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( linear5647500195961616402od_b_c @ Xs3 ) ) ) )
                 => ~ ( accp_l4654477130474446199od_b_c @ linear4366178405326712835od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_615_pow__list_Opelims,axiom,
    ! [X2: list_b,Y: list_list_b] :
      ( ( ( pow_list_b @ X2 )
        = Y )
     => ( ( accp_list_b @ pow_list_rel_b @ X2 )
       => ( ( ( X2 = nil_b )
           => ( ( Y
                = ( cons_list_b @ nil_b @ nil_list_b ) )
             => ~ ( accp_list_b @ pow_list_rel_b @ nil_b ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( X2
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append_list_b @ ( pow_list_b @ Xs3 ) @ ( map_list_b_list_b @ ( cons_b @ X3 ) @ ( pow_list_b @ Xs3 ) ) ) )
                 => ~ ( accp_list_b @ pow_list_rel_b @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_616_pow__list_Opelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( pow_li4341120398627903455od_b_c @ X2 )
        = Y )
     => ( ( accp_l4654477130474446199od_b_c @ pow_li3598863925982933110od_b_c @ X2 )
       => ( ( ( X2 = nil_Product_prod_b_c )
           => ( ( Y
                = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) )
             => ~ ( accp_l4654477130474446199od_b_c @ pow_li3598863925982933110od_b_c @ nil_Product_prod_b_c ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
                ( ( X2
                  = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append6992831555715836169od_b_c @ ( pow_li4341120398627903455od_b_c @ Xs3 ) @ ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 ) @ ( pow_li4341120398627903455od_b_c @ Xs3 ) ) ) )
                 => ~ ( accp_l4654477130474446199od_b_c @ pow_li3598863925982933110od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_617_list__ordered__pairs_Opelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( list_ordered_pairs_b @ X2 )
        = Y )
     => ( ( accp_list_b @ list_o4545026682469208125_rel_b @ X2 )
       => ( ( ( X2 = nil_b )
           => ( ( Y = nil_Product_prod_b_b )
             => ~ ( accp_list_b @ list_o4545026682469208125_rel_b @ nil_b ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( X2
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( list_ordered_pairs_b @ Xs3 ) ) )
                 => ~ ( accp_list_b @ list_o4545026682469208125_rel_b @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_618_list__ordered__pairs_Opelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( list_o804723041229831855od_b_c @ X2 )
        = Y )
     => ( ( accp_l4654477130474446199od_b_c @ list_o3318826711941139878od_b_c @ X2 )
       => ( ( ( X2 = nil_Product_prod_b_c )
           => ( ( Y = nil_Pr2945457328363356077od_b_c )
             => ~ ( accp_l4654477130474446199od_b_c @ list_o3318826711941139878od_b_c @ nil_Product_prod_b_c ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
                ( ( X2
                  = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( list_o804723041229831855od_b_c @ Xs3 ) ) )
                 => ~ ( accp_l4654477130474446199od_b_c @ list_o3318826711941139878od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_619_LS__from__LS__acyclic,axiom,
    ! [M: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M )
     => ( ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) )
        = ( lS_acyclic_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ).

% LS_from_LS_acyclic
thf(fact_620_extract__SomeE,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c,Y: list_P903359562653991662od_b_c,Zs3: list_l8907847357763382004od_b_c] :
      ( ( ( extrac8538413086128177910od_b_c @ P @ Xs )
        = ( some_P4558499654748856901od_b_c @ ( produc1106521052449266896od_b_c @ Ys @ ( produc169793319869648157od_b_c @ Y @ Zs3 ) ) ) )
     => ( ( Xs
          = ( append6992831555715836169od_b_c @ Ys @ ( cons_l342771267123639716od_b_c @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X5 @ ( set_li4480668622519654659od_b_c @ Ys ) )
              & ( P @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_621_extract__SomeE,axiom,
    ! [P: b > $o,Xs: list_b,Ys: list_b,Y: b,Zs3: list_b] :
      ( ( ( extract_b @ P @ Xs )
        = ( some_P7780201447483665076list_b @ ( produc5759544327908914751list_b @ Ys @ ( produc8158425114075849053list_b @ Y @ Zs3 ) ) ) )
     => ( ( Xs
          = ( append_b @ Ys @ ( cons_b @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X5: b] :
              ( ( member_b @ X5 @ ( set_b2 @ Ys ) )
              & ( P @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_622_extract__SomeE,axiom,
    ! [P: product_prod_b_c > $o,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Y: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
      ( ( ( extrac2790361661052662384od_b_c @ P @ Xs )
        = ( some_P278523132111576139od_b_c @ ( produc1861375773765947862od_b_c @ Ys @ ( produc1467100693395308445od_b_c @ Y @ Zs3 ) ) ) )
     => ( ( Xs
          = ( append2547753245680614915od_b_c @ Ys @ ( cons_P4529483553340347422od_b_c @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X5: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Ys ) )
              & ( P @ X5 ) ) ) ) ).

% extract_SomeE
thf(fact_623_extract__Some__iff,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c,Y: list_P903359562653991662od_b_c,Zs3: list_l8907847357763382004od_b_c] :
      ( ( ( extrac8538413086128177910od_b_c @ P @ Xs )
        = ( some_P4558499654748856901od_b_c @ ( produc1106521052449266896od_b_c @ Ys @ ( produc169793319869648157od_b_c @ Y @ Zs3 ) ) ) )
      = ( ( Xs
          = ( append6992831555715836169od_b_c @ Ys @ ( cons_l342771267123639716od_b_c @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys ) )
              & ( P @ X ) ) ) ) ).

% extract_Some_iff
thf(fact_624_extract__Some__iff,axiom,
    ! [P: b > $o,Xs: list_b,Ys: list_b,Y: b,Zs3: list_b] :
      ( ( ( extract_b @ P @ Xs )
        = ( some_P7780201447483665076list_b @ ( produc5759544327908914751list_b @ Ys @ ( produc8158425114075849053list_b @ Y @ Zs3 ) ) ) )
      = ( ( Xs
          = ( append_b @ Ys @ ( cons_b @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X: b] :
              ( ( member_b @ X @ ( set_b2 @ Ys ) )
              & ( P @ X ) ) ) ) ).

% extract_Some_iff
thf(fact_625_extract__Some__iff,axiom,
    ! [P: product_prod_b_c > $o,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Y: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
      ( ( ( extrac2790361661052662384od_b_c @ P @ Xs )
        = ( some_P278523132111576139od_b_c @ ( produc1861375773765947862od_b_c @ Ys @ ( produc1467100693395308445od_b_c @ Y @ Zs3 ) ) ) )
      = ( ( Xs
          = ( append2547753245680614915od_b_c @ Ys @ ( cons_P4529483553340347422od_b_c @ Y @ Zs3 ) ) )
        & ( P @ Y )
        & ~ ? [X: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys ) )
              & ( P @ X ) ) ) ) ).

% extract_Some_iff
thf(fact_626_extract__None__iff,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs: list_l8907847357763382004od_b_c] :
      ( ( ( extrac8538413086128177910od_b_c @ P @ Xs )
        = none_P4896853209150303425od_b_c )
      = ( ~ ? [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
              & ( P @ X ) ) ) ) ).

% extract_None_iff
thf(fact_627_extract__None__iff,axiom,
    ! [P: product_prod_b_c > $o,Xs: list_P903359562653991662od_b_c] :
      ( ( ( extrac2790361661052662384od_b_c @ P @ Xs )
        = none_P8002138689083699399od_b_c )
      = ( ~ ? [X: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% extract_None_iff
thf(fact_628_prefixes__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Xs6: list_P903359562653991662od_b_c] :
          ? [Xs7: list_P903359562653991662od_b_c] :
            ( ( append2547753245680614915od_b_c @ Xs6 @ Xs7 )
            = Xs ) ) ) ).

% prefixes_set
thf(fact_629_LS_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Y: set_li6436108459499378894od_b_c] :
      ( ( ( lS_a_b_c @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [P9: list_P6327159017948738492od_c_a] :
              ( ( Uu2
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P9 ) )
              & ( path_a_b_c @ X2 @ Xa @ P9 ) ) ) ) ) ).

% LS.elims
thf(fact_630_LS_Osimps,axiom,
    ( lS_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [P9: list_P6327159017948738492od_c_a] :
              ( ( Uu2
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P9 ) )
              & ( path_a_b_c @ M2 @ Q6 @ P9 ) ) ) ) ) ).

% LS.simps
thf(fact_631_io__targets_Osimps,axiom,
    ( io_tar7088883562557314310_c_b_c
    = ( ^ [M2: fsm_li6801133765522507155_c_b_c,Io: list_P903359562653991662od_b_c,Q6: list_P903359562653991662od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [P9: list_P8364599791545511612od_b_c] :
              ( ( Uu2
                = ( target5379785874326849321_c_b_c @ Q6 @ P9 ) )
              & ( path_l6977963524759401717_c_b_c @ M2 @ Q6 @ P9 )
              & ( ( map_Pr4814182253694984182od_b_c
                  @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
                  @ P9 )
                = Io ) ) ) ) ) ).

% io_targets.simps
thf(fact_632_io__targets_Osimps,axiom,
    ( io_targets_nat_b_c
    = ( ^ [M2: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q6: nat] :
          ( collect_nat
          @ ^ [Uu2: nat] :
            ? [P9: list_P1412026030649758308_c_nat] :
              ( ( Uu2
                = ( target_nat_b_c @ Q6 @ P9 ) )
              & ( path_nat_b_c @ M2 @ Q6 @ P9 )
              & ( ( map_Pr6527000103508470350od_b_c
                  @ ^ [T2: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) ) )
                  @ P9 )
                = Io ) ) ) ) ) ).

% io_targets.simps
thf(fact_633_io__targets_Oelims,axiom,
    ! [X2: fsm_li6801133765522507155_c_b_c,Xa: list_P903359562653991662od_b_c,Xb: list_P903359562653991662od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( io_tar7088883562557314310_c_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [P9: list_P8364599791545511612od_b_c] :
              ( ( Uu2
                = ( target5379785874326849321_c_b_c @ Xb @ P9 ) )
              & ( path_l6977963524759401717_c_b_c @ X2 @ Xb @ P9 )
              & ( ( map_Pr4814182253694984182od_b_c
                  @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
                  @ P9 )
                = Xa ) ) ) ) ) ).

% io_targets.elims
thf(fact_634_io__targets_Oelims,axiom,
    ! [X2: fsm_nat_b_c,Xa: list_P903359562653991662od_b_c,Xb: nat,Y: set_nat] :
      ( ( ( io_targets_nat_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collect_nat
          @ ^ [Uu2: nat] :
            ? [P9: list_P1412026030649758308_c_nat] :
              ( ( Uu2
                = ( target_nat_b_c @ Xb @ P9 ) )
              & ( path_nat_b_c @ X2 @ Xb @ P9 )
              & ( ( map_Pr6527000103508470350od_b_c
                  @ ^ [T2: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) ) )
                  @ P9 )
                = Xa ) ) ) ) ) ).

% io_targets.elims
thf(fact_635_io__targets_Opelims,axiom,
    ! [X2: fsm_li6801133765522507155_c_b_c,Xa: list_P903359562653991662od_b_c,Xb: list_P903359562653991662od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( io_tar7088883562557314310_c_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P8649962251078022660od_b_c @ io_tar514468572546654301_c_b_c @ ( produc464129659894981483od_b_c @ X2 @ ( produc3076307724246486423od_b_c @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( collec2280997390073109977od_b_c
                @ ^ [Uu2: list_P903359562653991662od_b_c] :
                  ? [P9: list_P8364599791545511612od_b_c] :
                    ( ( Uu2
                      = ( target5379785874326849321_c_b_c @ Xb @ P9 ) )
                    & ( path_l6977963524759401717_c_b_c @ X2 @ Xb @ P9 )
                    & ( ( map_Pr4814182253694984182od_b_c
                        @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
                        @ P9 )
                      = Xa ) ) ) )
           => ~ ( accp_P8649962251078022660od_b_c @ io_tar514468572546654301_c_b_c @ ( produc464129659894981483od_b_c @ X2 @ ( produc3076307724246486423od_b_c @ Xa @ Xb ) ) ) ) ) ) ).

% io_targets.pelims
thf(fact_636_io__targets_Opelims,axiom,
    ! [X2: fsm_nat_b_c,Xa: list_P903359562653991662od_b_c,Xb: nat,Y: set_nat] :
      ( ( ( io_targets_nat_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P1120439291432126234_c_nat @ io_tar8316814253418191331at_b_c @ ( produc4036149245550939465_c_nat @ X2 @ ( produc1142203413938751335_c_nat @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( collect_nat
                @ ^ [Uu2: nat] :
                  ? [P9: list_P1412026030649758308_c_nat] :
                    ( ( Uu2
                      = ( target_nat_b_c @ Xb @ P9 ) )
                    & ( path_nat_b_c @ X2 @ Xb @ P9 )
                    & ( ( map_Pr6527000103508470350od_b_c
                        @ ^ [T2: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) ) )
                        @ P9 )
                      = Xa ) ) ) )
           => ~ ( accp_P1120439291432126234_c_nat @ io_tar8316814253418191331at_b_c @ ( produc4036149245550939465_c_nat @ X2 @ ( produc1142203413938751335_c_nat @ Xa @ Xb ) ) ) ) ) ) ).

% io_targets.pelims
thf(fact_637_LS_Opelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Y: set_li6436108459499378894od_b_c] :
      ( ( ( lS_a_b_c @ X2 @ Xa )
        = Y )
     => ( ( accp_P8775296850406592587_b_c_a @ lS_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X2 @ Xa ) )
       => ~ ( ( Y
              = ( collec2280997390073109977od_b_c
                @ ^ [Uu2: list_P903359562653991662od_b_c] :
                  ? [P9: list_P6327159017948738492od_c_a] :
                    ( ( Uu2
                      = ( map_Pr776894537227705078od_b_c
                        @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                        @ P9 ) )
                    & ( path_a_b_c @ X2 @ Xa @ P9 ) ) ) )
           => ~ ( accp_P8775296850406592587_b_c_a @ lS_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X2 @ Xa ) ) ) ) ) ).

% LS.pelims
thf(fact_638_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_c_6732129604652128891_b_c_b,Xa: c,Xb: set_li6436108459499378894od_b_c,Y: set_li7348942078743790372_b_c_b] :
      ( ( ( lS_i_n6458876116888321724_b_c_b @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3754252191680048431_b_c_b
          @ ^ [Uu2: list_P7742544443125151044_b_c_b] :
            ? [P9: list_P5750542147746484308od_b_c] :
              ( ( Uu2
                = ( map_Pr4449925462467644084_b_c_b
                  @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                  @ P9 ) )
              & ( path_c982548547860747997_b_c_b @ X2 @ Xa @ P9 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr3481561618413555566od_b_c @ produc6878540623311373338_b_c_b
                  @ ( map_Pr4449925462467644084_b_c_b
                    @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                    @ P9 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_639_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n6458876116888321724_b_c_b
    = ( ^ [M2: fsm_c_6732129604652128891_b_c_b,Q6: c,Xss3: set_li6436108459499378894od_b_c] :
          ( collec3754252191680048431_b_c_b
          @ ^ [Uu2: list_P7742544443125151044_b_c_b] :
            ? [P9: list_P5750542147746484308od_b_c] :
              ( ( Uu2
                = ( map_Pr4449925462467644084_b_c_b
                  @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                  @ P9 ) )
              & ( path_c982548547860747997_b_c_b @ M2 @ Q6 @ P9 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr3481561618413555566od_b_c @ produc6878540623311373338_b_c_b
                  @ ( map_Pr4449925462467644084_b_c_b
                    @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                    @ P9 ) )
                @ Xss3 ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_640_LS_092_060_094sub_062i_092_060_094sub_062n_Opelims,axiom,
    ! [X2: fsm_c_6732129604652128891_b_c_b,Xa: c,Xb: set_li6436108459499378894od_b_c,Y: set_li7348942078743790372_b_c_b] :
      ( ( ( lS_i_n6458876116888321724_b_c_b @ X2 @ Xa @ Xb )
        = Y )
     => ( ( accp_P7989004692349800170od_b_c @ lS_i_n5679008109505604727_b_c_b @ ( produc7056996068739793233od_b_c @ X2 @ ( produc4529774005887432421od_b_c @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( collec3754252191680048431_b_c_b
                @ ^ [Uu2: list_P7742544443125151044_b_c_b] :
                  ? [P9: list_P5750542147746484308od_b_c] :
                    ( ( Uu2
                      = ( map_Pr4449925462467644084_b_c_b
                        @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                        @ P9 ) )
                    & ( path_c982548547860747997_b_c_b @ X2 @ Xa @ P9 )
                    & ( member6330420149250801815od_b_c
                      @ ( map_Pr3481561618413555566od_b_c @ produc6878540623311373338_b_c_b
                        @ ( map_Pr4449925462467644084_b_c_b
                          @ ^ [T2: produc6564011220051093454od_b_c] : ( produc794085425710893614_b_c_b @ ( produc4830044058343513219od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) @ ( product_fst_b_c @ ( produc8692230813408409541od_b_c @ ( produc6001597805492596844od_b_c @ T2 ) ) ) )
                          @ P9 ) )
                      @ Xb ) ) ) )
           => ~ ( accp_P7989004692349800170od_b_c @ lS_i_n5679008109505604727_b_c_b @ ( produc7056996068739793233od_b_c @ X2 @ ( produc4529774005887432421od_b_c @ Xa @ Xb ) ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.pelims
thf(fact_641_remove__proper__prefixes__def,axiom,
    ( remove5339202614008355685od_b_c
    = ( ^ [Xs2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ Xs2 )
              & ~ ? [X6: list_P903359562653991662od_b_c] :
                    ( ( X6 != nil_Product_prod_b_c )
                    & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ X @ X6 ) @ Xs2 ) ) ) ) ) ) ).

% remove_proper_prefixes_def
thf(fact_642_h__out_Oelims,axiom,
    ! [X2: fsm_b_2727426217447713361od_b_c,Xa: product_prod_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( h_out_6535201742914350802od_b_c @ X2 @ Xa )
        = Y )
     => ~ ! [Q7: b,X3: c] :
            ( ( Xa
              = ( product_Pair_b_c @ Q7 @ X3 ) )
           => ( Y
             != ( collec2280997390073109977od_b_c
                @ ^ [Y2: list_P903359562653991662od_b_c] :
                  ? [Q5: b] : ( member5243421482830010352_b_c_b @ ( produc8206783688450391351_b_c_b @ Q7 @ ( produc8730802830273647183_b_c_b @ X3 @ ( produc595631027891595048_b_c_b @ Y2 @ Q5 ) ) ) @ ( transi6377170199346834114od_b_c @ X2 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_643_h__out_Oelims,axiom,
    ! [X2: fsm_b_c_nat,Xa: product_prod_b_c,Y: set_nat] :
      ( ( ( h_out_b_c_nat @ X2 @ Xa )
        = Y )
     => ~ ! [Q7: b,X3: c] :
            ( ( Xa
              = ( product_Pair_b_c @ Q7 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y2: nat] :
                  ? [Q5: b] : ( member7205605469418551960_nat_b @ ( produc2952021607952343931_nat_b @ Q7 @ ( produc1482701325287381453_nat_b @ X3 @ ( product_Pair_nat_b @ Y2 @ Q5 ) ) ) @ ( transitions_b_c_nat @ X2 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_644_h__out_Osimps,axiom,
    ! [M: fsm_b_2727426217447713361od_b_c,Q: b,X2: c] :
      ( ( h_out_6535201742914350802od_b_c @ M @ ( product_Pair_b_c @ Q @ X2 ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Y2: list_P903359562653991662od_b_c] :
          ? [Q5: b] : ( member5243421482830010352_b_c_b @ ( produc8206783688450391351_b_c_b @ Q @ ( produc8730802830273647183_b_c_b @ X2 @ ( produc595631027891595048_b_c_b @ Y2 @ Q5 ) ) ) @ ( transi6377170199346834114od_b_c @ M ) ) ) ) ).

% h_out.simps
thf(fact_645_h__out_Osimps,axiom,
    ! [M: fsm_b_c_nat,Q: b,X2: c] :
      ( ( h_out_b_c_nat @ M @ ( product_Pair_b_c @ Q @ X2 ) )
      = ( collect_nat
        @ ^ [Y2: nat] :
          ? [Q5: b] : ( member7205605469418551960_nat_b @ ( produc2952021607952343931_nat_b @ Q @ ( produc1482701325287381453_nat_b @ X2 @ ( product_Pair_nat_b @ Y2 @ Q5 ) ) ) @ ( transitions_b_c_nat @ M ) ) ) ) ).

% h_out.simps
thf(fact_646_h__out__alt__def,axiom,
    ! [M: fsm_b_2727426217447713361od_b_c,Q: b,X2: c] :
      ( ( h_out_6535201742914350802od_b_c @ M @ ( product_Pair_b_c @ Q @ X2 ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Uu2: list_P903359562653991662od_b_c] :
          ? [T2: produc4979789236725255495_b_c_b] :
            ( ( Uu2
              = ( produc1715878850288909076_b_c_b @ ( produc6055666879148813437_b_c_b @ ( produc930338527285197157_b_c_b @ T2 ) ) ) )
            & ( member5243421482830010352_b_c_b @ T2 @ ( transi6377170199346834114od_b_c @ M ) )
            & ( ( produc40664336333682979_b_c_b @ T2 )
              = Q )
            & ( ( produc7361720795797543483_b_c_b @ ( produc930338527285197157_b_c_b @ T2 ) )
              = X2 ) ) ) ) ).

% h_out_alt_def
thf(fact_647_h__out__alt__def,axiom,
    ! [M: fsm_b_c_nat,Q: b,X2: c] :
      ( ( h_out_b_c_nat @ M @ ( product_Pair_b_c @ Q @ X2 ) )
      = ( collect_nat
        @ ^ [Uu2: nat] :
          ? [T2: produc897840418342192897_nat_b] :
            ( ( Uu2
              = ( product_fst_nat_b @ ( produc936460378501341983_nat_b @ ( produc6092916665544297165_nat_b @ T2 ) ) ) )
            & ( member7205605469418551960_nat_b @ T2 @ ( transitions_b_c_nat @ M ) )
            & ( ( produc6660495173513958031_nat_b @ T2 )
              = Q )
            & ( ( produc383478219494620385_nat_b @ ( produc6092916665544297165_nat_b @ T2 ) )
              = X2 ) ) ) ) ).

% h_out_alt_def
thf(fact_648_h__out_Opelims,axiom,
    ! [X2: fsm_b_2727426217447713361od_b_c,Xa: product_prod_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( h_out_6535201742914350802od_b_c @ X2 @ Xa )
        = Y )
     => ( ( accp_P8566271851537230855od_b_c @ h_out_631537543757430413od_b_c @ ( produc2566182017150619630od_b_c @ X2 @ Xa ) )
       => ~ ! [Q7: b,X3: c] :
              ( ( Xa
                = ( product_Pair_b_c @ Q7 @ X3 ) )
             => ( ( Y
                  = ( collec2280997390073109977od_b_c
                    @ ^ [Y2: list_P903359562653991662od_b_c] :
                      ? [Q5: b] : ( member5243421482830010352_b_c_b @ ( produc8206783688450391351_b_c_b @ Q7 @ ( produc8730802830273647183_b_c_b @ X3 @ ( produc595631027891595048_b_c_b @ Y2 @ Q5 ) ) ) @ ( transi6377170199346834114od_b_c @ X2 ) ) ) )
               => ~ ( accp_P8566271851537230855od_b_c @ h_out_631537543757430413od_b_c @ ( produc2566182017150619630od_b_c @ X2 @ ( product_Pair_b_c @ Q7 @ X3 ) ) ) ) ) ) ) ).

% h_out.pelims
thf(fact_649_h__out_Opelims,axiom,
    ! [X2: fsm_b_c_nat,Xa: product_prod_b_c,Y: set_nat] :
      ( ( ( h_out_b_c_nat @ X2 @ Xa )
        = Y )
     => ( ( accp_P3075213358698714001od_b_c @ h_out_rel_b_c_nat @ ( produc5709871137852095500od_b_c @ X2 @ Xa ) )
       => ~ ! [Q7: b,X3: c] :
              ( ( Xa
                = ( product_Pair_b_c @ Q7 @ X3 ) )
             => ( ( Y
                  = ( collect_nat
                    @ ^ [Y2: nat] :
                      ? [Q5: b] : ( member7205605469418551960_nat_b @ ( produc2952021607952343931_nat_b @ Q7 @ ( produc1482701325287381453_nat_b @ X3 @ ( product_Pair_nat_b @ Y2 @ Q5 ) ) ) @ ( transitions_b_c_nat @ X2 ) ) ) )
               => ~ ( accp_P3075213358698714001od_b_c @ h_out_rel_b_c_nat @ ( produc5709871137852095500od_b_c @ X2 @ ( product_Pair_b_c @ Q7 @ X3 ) ) ) ) ) ) ) ).

% h_out.pelims
thf(fact_650_paths__for__input_H__set,axiom,
    ! [Q: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Xs: list_b,Prev2: list_P8364599791545511612od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
     => ( ( paths_2402314502658583498_c_b_c @ ( h_from2122548697928753168_c_b_c @ M ) @ Xs @ Q @ Prev2 )
        = ( collec1242728128799169191od_b_c
          @ ^ [Uu2: list_P8364599791545511612od_b_c] :
            ? [P9: list_P8364599791545511612od_b_c] :
              ( ( Uu2
                = ( append8574118600258580689od_b_c @ Prev2 @ P9 ) )
              & ( path_l6977963524759401717_c_b_c @ M @ Q @ P9 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr4814182253694984182od_b_c
                    @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
                    @ P9 ) )
                = Xs ) ) ) ) ) ).

% paths_for_input'_set
thf(fact_651_paths__for__input_H__set,axiom,
    ! [Q: a,M: fsm_a_b_c,Xs: list_b,Prev2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( paths_6276656153717473370_a_b_c @ ( h_from_a_b_c @ M ) @ Xs @ Q @ Prev2 )
        = ( collec6273869032445462695od_c_a
          @ ^ [Uu2: list_P6327159017948738492od_c_a] :
            ? [P9: list_P6327159017948738492od_c_a] :
              ( ( Uu2
                = ( append3884164603879732689od_c_a @ Prev2 @ P9 ) )
              & ( path_a_b_c @ M @ Q @ P9 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                    @ P9 ) )
                = Xs ) ) ) ) ) ).

% paths_for_input'_set
thf(fact_652_suffixes_Oelims,axiom,
    ! [X2: list_b,Y: list_list_b] :
      ( ( ( suffixes_b2 @ X2 )
        = Y )
     => ( ( ( X2 = nil_b )
         => ( Y
           != ( cons_list_b @ nil_b @ nil_list_b ) ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( X2
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( append_list_b @ ( suffixes_b2 @ Xs3 ) @ ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ nil_list_b ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_653_suffixes_Oelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( suffix2151784340763841114od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_Product_prod_b_c )
         => ( Y
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( X2
                = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
             => ( Y
               != ( append6992831555715836169od_b_c @ ( suffix2151784340763841114od_b_c @ Xs3 ) @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_li8071162985807626740od_b_c ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_654_add__prefixes__set,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( add_pr7303551824293825364od_b_c @ Xs ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Xs6: list_P903359562653991662od_b_c] :
          ? [Xs7: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Xs6 @ Xs7 ) @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% add_prefixes_set
thf(fact_655_suffixes__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( suffix2151784340763841114od_b_c @ Xs ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Zs2: list_P903359562653991662od_b_c] :
          ? [Ys3: list_P903359562653991662od_b_c] :
            ( ( append2547753245680614915od_b_c @ Ys3 @ Zs2 )
            = Xs ) ) ) ).

% suffixes_set
thf(fact_656_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( suffixes_b2 @ ( cons_b @ X2 @ Xs ) )
      = ( append_list_b @ ( suffixes_b2 @ Xs ) @ ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ nil_list_b ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_657_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( suffix2151784340763841114od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append6992831555715836169od_b_c @ ( suffix2151784340763841114od_b_c @ Xs ) @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ nil_li8071162985807626740od_b_c ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_658_suffixes_Opelims,axiom,
    ! [X2: list_b,Y: list_list_b] :
      ( ( ( suffixes_b2 @ X2 )
        = Y )
     => ( ( accp_list_b @ suffixes_rel_b @ X2 )
       => ( ( ( X2 = nil_b )
           => ( ( Y
                = ( cons_list_b @ nil_b @ nil_list_b ) )
             => ~ ( accp_list_b @ suffixes_rel_b @ nil_b ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( X2
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append_list_b @ ( suffixes_b2 @ Xs3 ) @ ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ nil_list_b ) ) )
                 => ~ ( accp_list_b @ suffixes_rel_b @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_659_suffixes_Opelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( suffix2151784340763841114od_b_c @ X2 )
        = Y )
     => ( ( accp_l4654477130474446199od_b_c @ suffix5297622422209064891od_b_c @ X2 )
       => ( ( ( X2 = nil_Product_prod_b_c )
           => ( ( Y
                = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ nil_li8071162985807626740od_b_c ) )
             => ~ ( accp_l4654477130474446199od_b_c @ suffix5297622422209064891od_b_c @ nil_Product_prod_b_c ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
                ( ( X2
                  = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append6992831555715836169od_b_c @ ( suffix2151784340763841114od_b_c @ Xs3 ) @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_li8071162985807626740od_b_c ) ) )
                 => ~ ( accp_l4654477130474446199od_b_c @ suffix5297622422209064891od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_660_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( non_sym_dist_pairs_b @ X2 )
        = Y )
     => ( ( ( X2 = nil_b )
         => ( Y != nil_Product_prod_b_b ) )
       => ~ ! [X3: b,Xs3: list_b] :
              ( ( X2
                = ( cons_b @ X3 @ Xs3 ) )
             => ( Y
               != ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( non_sym_dist_pairs_b @ Xs3 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_661_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( non_sy4916969364292876019od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_Product_prod_b_c )
         => ( Y != nil_Pr2945457328363356077od_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( X2
                = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
             => ( Y
               != ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( non_sy4916969364292876019od_b_c @ Xs3 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_662_in__set__takeD,axiom,
    ! [X2: a,N: nat,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( take_a @ N @ Xs ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_663_in__set__takeD,axiom,
    ! [X2: list_P903359562653991662od_b_c,N: nat,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( take_l2014012074268970460od_b_c @ N @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ).

% in_set_takeD
thf(fact_664_in__set__takeD,axiom,
    ! [X2: product_prod_b_c,N: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( take_P8422700459012106326od_b_c @ N @ Xs ) ) )
     => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_665_take__map,axiom,
    ! [N: nat,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( take_b @ N @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ F @ ( take_P8422700459012106326od_b_c @ N @ Xs ) ) ) ).

% take_map
thf(fact_666_set__take__subset,axiom,
    ! [N: nat,Xs: list_P903359562653991662od_b_c] : ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( take_P8422700459012106326od_b_c @ N @ Xs ) ) @ ( set_Product_prod_b_c2 @ Xs ) ) ).

% set_take_subset
thf(fact_667_set__take__subset,axiom,
    ! [N: nat,Xs: list_l8907847357763382004od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( take_l2014012074268970460od_b_c @ N @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ Xs ) ) ).

% set_take_subset
thf(fact_668_set__take__subset__set__take,axiom,
    ! [M3: nat,N: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( take_P8422700459012106326od_b_c @ M3 @ Xs ) ) @ ( set_Product_prod_b_c2 @ ( take_P8422700459012106326od_b_c @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_669_set__take__subset__set__take,axiom,
    ! [M3: nat,N: nat,Xs: list_l8907847357763382004od_b_c] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( take_l2014012074268970460od_b_c @ M3 @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( take_l2014012074268970460od_b_c @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_670_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( non_sym_dist_pairs_b @ ( cons_b @ X2 @ Xs ) )
      = ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X2 ) @ Xs ) @ ( non_sym_dist_pairs_b @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_671_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( non_sy4916969364292876019od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X2 ) @ Xs ) @ ( non_sy4916969364292876019od_b_c @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_672_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X2: list_b,Y: list_P832325522607645677od_b_b] :
      ( ( ( non_sym_dist_pairs_b @ X2 )
        = Y )
     => ( ( accp_list_b @ non_sy8358513589392214265_rel_b @ X2 )
       => ( ( ( X2 = nil_b )
           => ( ( Y = nil_Product_prod_b_b )
             => ~ ( accp_list_b @ non_sy8358513589392214265_rel_b @ nil_b ) ) )
         => ~ ! [X3: b,Xs3: list_b] :
                ( ( X2
                  = ( cons_b @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append2547753241377386114od_b_b @ ( map_b_4433156284899315948od_b_b @ ( product_Pair_b_b @ X3 ) @ Xs3 ) @ ( non_sym_dist_pairs_b @ Xs3 ) ) )
                 => ~ ( accp_list_b @ non_sy8358513589392214265_rel_b @ ( cons_b @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_673_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P7987531494101635245od_b_c] :
      ( ( ( non_sy4916969364292876019od_b_c @ X2 )
        = Y )
     => ( ( accp_l4654477130474446199od_b_c @ non_sy5902841407094123106od_b_c @ X2 )
       => ( ( ( X2 = nil_Product_prod_b_c )
           => ( ( Y = nil_Pr2945457328363356077od_b_c )
             => ~ ( accp_l4654477130474446199od_b_c @ non_sy5902841407094123106od_b_c @ nil_Product_prod_b_c ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
                ( ( X2
                  = ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( append5486846175084624706od_b_c @ ( map_Pr8917322328026853507od_b_c @ ( produc6366901981060558487od_b_c @ X3 ) @ Xs3 ) @ ( non_sy4916969364292876019od_b_c @ Xs3 ) ) )
                 => ~ ( accp_l4654477130474446199od_b_c @ non_sy5902841407094123106od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_674_suffixes__snoc,axiom,
    ! [Xs: list_b,X2: b] :
      ( ( suffixes_b @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) )
      = ( cons_list_b @ nil_b
        @ ( map_list_b_list_b
          @ ^ [Ys3: list_b] : ( append_b @ Ys3 @ ( cons_b @ X2 @ nil_b ) )
          @ ( suffixes_b @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_675_suffixes__snoc,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X2: product_prod_b_c] :
      ( ( suffix3867853578027978934od_b_c @ ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) ) )
      = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c
        @ ( map_li8081432502061035332od_b_c
          @ ^ [Ys3: list_P903359562653991662od_b_c] : ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X2 @ nil_Product_prod_b_c ) )
          @ ( suffix3867853578027978934od_b_c @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_676_length__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( size_size_list_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( size_s3392097710323735898od_b_c @ Xs ) ) ).

% length_map
thf(fact_677_map__eq__imp__length__eq,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ G @ Ys ) )
     => ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_678_impossible__Cons,axiom,
    ! [Xs: list_b,Ys: list_b,X2: b] :
      ( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) )
     => ( Xs
       != ( cons_b @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_679_impossible__Cons,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,X2: product_prod_b_c] :
      ( ( ord_less_eq_nat @ ( size_s3392097710323735898od_b_c @ Xs ) @ ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( Xs
       != ( cons_P4529483553340347422od_b_c @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_680_in__set__product__lists__length,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xss2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( produc3104363228908302167od_b_c @ Xss2 ) ) )
     => ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s4150533072628932448od_b_c @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_681_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_b,Ws: list_b,P: list_b > list_b > list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_b @ nil_b @ nil_b @ nil_b )
           => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: b,Zs: list_b,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_682_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_b,Ws: list_P903359562653991662od_b_c,P: list_b > list_b > list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_b @ nil_b @ nil_b @ nil_Product_prod_b_c )
           => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: b,Zs: list_b,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_683_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_P903359562653991662od_b_c,Ws: list_b,P: list_b > list_b > list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_b @ nil_b @ nil_Product_prod_b_c @ nil_b )
           => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_s3392097710323735898od_b_c @ Zs ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_684_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,Zs3: list_b,Ws: list_b,P: list_b > list_P903359562653991662od_b_c > list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_b @ nil_Product_prod_b_c @ nil_b @ nil_b )
           => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: b,Zs: list_b,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_s3392097710323735898od_b_c @ Ys2 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_685_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_b,Ws: list_b,P: list_P903359562653991662od_b_c > list_b > list_b > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_b @ nil_b @ nil_b )
           => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b,Z: b,Zs: list_b,W: b,Ws2: list_b] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_686_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_P903359562653991662od_b_c,Ws: list_P903359562653991662od_b_c,P: list_b > list_b > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs3 )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_b @ nil_b @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
           => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_s3392097710323735898od_b_c @ Zs ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_687_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,Zs3: list_b,Ws: list_P903359562653991662od_b_c,P: list_b > list_P903359562653991662od_b_c > list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_b @ nil_Product_prod_b_c @ nil_b @ nil_Product_prod_b_c )
           => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: b,Zs: list_b,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_s3392097710323735898od_b_c @ Ys2 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_688_list__induct4,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c,Ws: list_b,P: list_b > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_b @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_b )
           => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c,W: b,Ws2: list_b] :
                  ( ( ( size_size_list_b @ Xs3 )
                    = ( size_s3392097710323735898od_b_c @ Ys2 ) )
                 => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                      = ( size_s3392097710323735898od_b_c @ Zs ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_689_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_b,Ws: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_b > list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_s3392097710323735898od_b_c @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_b @ nil_b @ nil_Product_prod_b_c )
           => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b,Z: b,Zs: list_b,W: product_prod_b_c,Ws2: list_P903359562653991662od_b_c] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_size_list_b @ Zs ) )
                   => ( ( ( size_size_list_b @ Zs )
                        = ( size_s3392097710323735898od_b_c @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) @ ( cons_P4529483553340347422od_b_c @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_690_list__induct4,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_P903359562653991662od_b_c,Ws: list_b,P: list_P903359562653991662od_b_c > list_b > list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( ( size_s3392097710323735898od_b_c @ Zs3 )
            = ( size_size_list_b @ Ws ) )
         => ( ( P @ nil_Product_prod_b_c @ nil_b @ nil_Product_prod_b_c @ nil_b )
           => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c,W: b,Ws2: list_b] :
                  ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                    = ( size_size_list_b @ Ys2 ) )
                 => ( ( ( size_size_list_b @ Ys2 )
                      = ( size_s3392097710323735898od_b_c @ Zs ) )
                   => ( ( ( size_s3392097710323735898od_b_c @ Zs )
                        = ( size_size_list_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
                       => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_691_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_b,P: list_b > list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_b @ nil_b @ nil_b )
         => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: b,Zs: list_b] :
                ( ( ( size_size_list_b @ Xs3 )
                  = ( size_size_list_b @ Ys2 ) )
               => ( ( ( size_size_list_b @ Ys2 )
                    = ( size_size_list_b @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_692_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_b,Zs3: list_P903359562653991662od_b_c,P: list_b > list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( P @ nil_b @ nil_b @ nil_Product_prod_b_c )
         => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
                ( ( ( size_size_list_b @ Xs3 )
                  = ( size_size_list_b @ Ys2 ) )
               => ( ( ( size_size_list_b @ Ys2 )
                    = ( size_s3392097710323735898od_b_c @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_693_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,Zs3: list_b,P: list_b > list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_b @ nil_Product_prod_b_c @ nil_b )
         => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: b,Zs: list_b] :
                ( ( ( size_size_list_b @ Xs3 )
                  = ( size_s3392097710323735898od_b_c @ Ys2 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                    = ( size_size_list_b @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_694_list__induct3,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c,P: list_b > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( P @ nil_b @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
         => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
                ( ( ( size_size_list_b @ Xs3 )
                  = ( size_s3392097710323735898od_b_c @ Ys2 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                    = ( size_s3392097710323735898od_b_c @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_695_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_b,P: list_P903359562653991662od_b_c > list_b > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_b @ nil_b )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b,Z: b,Zs: list_b] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                  = ( size_size_list_b @ Ys2 ) )
               => ( ( ( size_size_list_b @ Ys2 )
                    = ( size_size_list_b @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_696_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs3: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( size_size_list_b @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_b @ nil_Product_prod_b_c )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                  = ( size_size_list_b @ Ys2 ) )
               => ( ( ( size_size_list_b @ Ys2 )
                    = ( size_s3392097710323735898od_b_c @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_697_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs3: list_b,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_b )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: b,Zs: list_b] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                  = ( size_s3392097710323735898od_b_c @ Ys2 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                    = ( size_size_list_b @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_698_list__induct3,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Ys )
          = ( size_s3392097710323735898od_b_c @ Zs3 ) )
       => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c,Z: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
                ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                  = ( size_s3392097710323735898od_b_c @ Ys2 ) )
               => ( ( ( size_s3392097710323735898od_b_c @ Ys2 )
                    = ( size_s3392097710323735898od_b_c @ Zs ) )
                 => ( ( P @ Xs3 @ Ys2 @ Zs )
                   => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) @ ( cons_P4529483553340347422od_b_c @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_699_list__induct2,axiom,
    ! [Xs: list_b,Ys: list_b,P: list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_b @ nil_b )
       => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
              ( ( ( size_size_list_b @ Xs3 )
                = ( size_size_list_b @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_700_list__induct2,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,P: list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( P @ nil_b @ nil_Product_prod_b_c )
       => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( ( ( size_size_list_b @ Xs3 )
                = ( size_s3392097710323735898od_b_c @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_b @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_701_list__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,P: list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_b )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                = ( size_size_list_b @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_702_list__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                = ( size_s3392097710323735898od_b_c @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_703_lists__of__length__length,axiom,
    ! [Xs: list_P903359562653991662od_b_c,T3: list_P903359562653991662od_b_c,N: nat] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( lists_2361477089028216131od_b_c @ T3 @ N ) ) )
     => ( ( size_s3392097710323735898od_b_c @ Xs )
        = N ) ) ).

% lists_of_length_length
thf(fact_704_same__length__different,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_b @ Xs )
          = ( size_size_list_b @ Ys ) )
       => ? [Pre: list_b,X3: b,Xs5: list_b,Y3: b,Ys7: list_b] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_b @ Pre @ ( append_b @ ( cons_b @ X3 @ nil_b ) @ Xs5 ) ) )
            & ( Ys
              = ( append_b @ Pre @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ Ys7 ) ) ) ) ) ) ).

% same_length_different
thf(fact_705_same__length__different,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Xs != Ys )
     => ( ( ( size_s3392097710323735898od_b_c @ Xs )
          = ( size_s3392097710323735898od_b_c @ Ys ) )
       => ? [Pre: list_P903359562653991662od_b_c,X3: product_prod_b_c,Xs5: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys7: list_P903359562653991662od_b_c] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append2547753245680614915od_b_c @ Pre @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) @ Xs5 ) ) )
            & ( Ys
              = ( append2547753245680614915od_b_c @ Pre @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) @ Ys7 ) ) ) ) ) ) ).

% same_length_different
thf(fact_706_rev__induct2,axiom,
    ! [Xs: list_b,Ys: list_b,P: list_b > list_b > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_b @ nil_b )
       => ( ! [X3: b,Xs3: list_b,Y3: b,Ys2: list_b] :
              ( ( ( size_size_list_b @ Xs3 )
                = ( size_size_list_b @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( append_b @ Xs3 @ ( cons_b @ X3 @ nil_b ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_707_rev__induct2,axiom,
    ! [Xs: list_b,Ys: list_P903359562653991662od_b_c,P: list_b > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( P @ nil_b @ nil_Product_prod_b_c )
       => ( ! [X3: b,Xs3: list_b,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( ( ( size_size_list_b @ Xs3 )
                = ( size_s3392097710323735898od_b_c @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( append_b @ Xs3 @ ( cons_b @ X3 @ nil_b ) ) @ ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_708_rev__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_b,P: list_P903359562653991662od_b_c > list_b > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_b )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: b,Ys2: list_b] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                = ( size_size_list_b @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_709_rev__induct2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( ( ( size_s3392097710323735898od_b_c @ Xs3 )
                = ( size_s3392097710323735898od_b_c @ Ys2 ) )
             => ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) @ ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_710_prefixes__take__iff,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) )
      = ( ( take_P8422700459012106326od_b_c @ ( size_s3392097710323735898od_b_c @ Xs ) @ Ys )
        = Xs ) ) ).

% prefixes_take_iff
thf(fact_711_lists__of__length__containment,axiom,
    ! [Xs: list_P903359562653991662od_b_c,T3: list_P903359562653991662od_b_c,N: nat] :
      ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ T3 ) )
     => ( ( ( size_s3392097710323735898od_b_c @ Xs )
          = N )
       => ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( lists_2361477089028216131od_b_c @ T3 @ N ) ) ) ) ) ).

% lists_of_length_containment
thf(fact_712_lists__of__length__containment,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,T3: list_l8907847357763382004od_b_c,N: nat] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ T3 ) )
     => ( ( ( size_s4150533072628932448od_b_c @ Xs )
          = N )
       => ( member7934871825613078557od_b_c @ Xs @ ( set_li2649297040355570313od_b_c @ ( lists_3751910231668219977od_b_c @ T3 @ N ) ) ) ) ) ).

% lists_of_length_containment
thf(fact_713_lists__of__length__list__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K3: nat] :
      ( ( set_li4480668622519654659od_b_c @ ( lists_2361477089028216131od_b_c @ Xs @ K3 ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Xs6: list_P903359562653991662od_b_c] :
            ( ( ( size_s3392097710323735898od_b_c @ Xs6 )
              = K3 )
            & ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs6 ) @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ) ).

% lists_of_length_list_set
thf(fact_714_lists__of__length__list__set,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,K3: nat] :
      ( ( set_li2649297040355570313od_b_c @ ( lists_3751910231668219977od_b_c @ Xs @ K3 ) )
      = ( collec686158730324120671od_b_c
        @ ^ [Xs6: list_l8907847357763382004od_b_c] :
            ( ( ( size_s4150533072628932448od_b_c @ Xs6 )
              = K3 )
            & ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs6 ) @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ) ).

% lists_of_length_list_set
thf(fact_715_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( suffixes_b @ ( cons_b @ X2 @ Xs ) )
      = ( append_list_b @ ( suffixes_b @ Xs ) @ ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ nil_list_b ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_716_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( suffix3867853578027978934od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append6992831555715836169od_b_c @ ( suffix3867853578027978934od_b_c @ Xs ) @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ nil_li8071162985807626740od_b_c ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_717_suffixes__eq__snoc,axiom,
    ! [Ys: list_b,Xs: list_list_b,X2: list_b] :
      ( ( ( suffixes_b @ Ys )
        = ( append_list_b @ Xs @ ( cons_list_b @ X2 @ nil_list_b ) ) )
      = ( ( ( ( Ys = nil_b )
            & ( Xs = nil_list_b ) )
          | ? [Z3: b,Zs2: list_b] :
              ( ( Ys
                = ( cons_b @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_b @ Zs2 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_718_suffixes__eq__snoc,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( ( suffix3867853578027978934od_b_c @ Ys )
        = ( append6992831555715836169od_b_c @ Xs @ ( cons_l342771267123639716od_b_c @ X2 @ nil_li8071162985807626740od_b_c ) ) )
      = ( ( ( ( Ys = nil_Product_prod_b_c )
            & ( Xs = nil_li8071162985807626740od_b_c ) )
          | ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
              ( ( Ys
                = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffix3867853578027978934od_b_c @ Zs2 ) ) ) )
        & ( X2 = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_719_in__set__dropD,axiom,
    ! [X2: a,N: nat,Xs: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_720_in__set__dropD,axiom,
    ! [X2: list_P903359562653991662od_b_c,N: nat,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( drop_l3568355655282140980od_b_c @ N @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ).

% in_set_dropD
thf(fact_721_in__set__dropD,axiom,
    ! [X2: product_prod_b_c,N: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( drop_P5669314423915875758od_b_c @ N @ Xs ) ) )
     => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_722_drop__map,axiom,
    ! [N: nat,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( drop_b @ N @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ F @ ( drop_P5669314423915875758od_b_c @ N @ Xs ) ) ) ).

% drop_map
thf(fact_723_set__drop__subset,axiom,
    ! [N: nat,Xs: list_P903359562653991662od_b_c] : ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( drop_P5669314423915875758od_b_c @ N @ Xs ) ) @ ( set_Product_prod_b_c2 @ Xs ) ) ).

% set_drop_subset
thf(fact_724_set__drop__subset,axiom,
    ! [N: nat,Xs: list_l8907847357763382004od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( drop_l3568355655282140980od_b_c @ N @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ Xs ) ) ).

% set_drop_subset
thf(fact_725_set__drop__subset__set__drop,axiom,
    ! [N: nat,M3: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( ord_less_eq_nat @ N @ M3 )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( drop_P5669314423915875758od_b_c @ M3 @ Xs ) ) @ ( set_Product_prod_b_c2 @ ( drop_P5669314423915875758od_b_c @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_726_set__drop__subset__set__drop,axiom,
    ! [N: nat,M3: nat,Xs: list_l8907847357763382004od_b_c] :
      ( ( ord_less_eq_nat @ N @ M3 )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( drop_l3568355655282140980od_b_c @ M3 @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( drop_l3568355655282140980od_b_c @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_727_paths__up__to__length__set,axiom,
    ! [Q: a,M: fsm_a_b_c,K3: nat] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( paths_5655032219139660648_a_b_c @ M @ Q @ K3 )
        = ( collec6273869032445462695od_c_a
          @ ^ [P9: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P9 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P9 ) @ K3 ) ) ) ) ) ).

% paths_up_to_length_set
thf(fact_728_set__n__lists,axiom,
    ! [N: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( n_list5224936925692304502od_b_c @ N @ Xs ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Ys3: list_P903359562653991662od_b_c] :
            ( ( ( size_s3392097710323735898od_b_c @ Ys3 )
              = N )
            & ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Ys3 ) @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_729_set__n__lists,axiom,
    ! [N: nat,Xs: list_l8907847357763382004od_b_c] :
      ( ( set_li2649297040355570313od_b_c @ ( n_list1942462909063115260od_b_c @ N @ Xs ) )
      = ( collec686158730324120671od_b_c
        @ ^ [Ys3: list_l8907847357763382004od_b_c] :
            ( ( ( size_s4150533072628932448od_b_c @ Ys3 )
              = N )
            & ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Ys3 ) @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_730_length__n__lists__elem,axiom,
    ! [Ys: list_P903359562653991662od_b_c,N: nat,Xs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Ys @ ( set_li4480668622519654659od_b_c @ ( n_list5224936925692304502od_b_c @ N @ Xs ) ) )
     => ( ( size_s3392097710323735898od_b_c @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_731_sublists_Osimps_I2_J,axiom,
    ! [X2: b,Xs: list_b] :
      ( ( sublists_b @ ( cons_b @ X2 @ Xs ) )
      = ( append_list_b @ ( sublists_b @ Xs ) @ ( map_list_b_list_b @ ( cons_b @ X2 ) @ ( prefixes_b @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_732_sublists_Osimps_I2_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( sublis8706766506341057792od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) )
      = ( append6992831555715836169od_b_c @ ( sublis8706766506341057792od_b_c @ Xs ) @ ( map_li8081432502061035332od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 ) @ ( prefix1131979855692807669od_b_c @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_733_transpose_Opinduct,axiom,
    ! [A0: list_list_b,P: list_list_b > $o] :
      ( ( accp_list_list_b @ transpose_rel_b @ A0 )
     => ( ( ( accp_list_list_b @ transpose_rel_b @ nil_list_b )
         => ( P @ nil_list_b ) )
       => ( ! [Xss: list_list_b] :
              ( ( accp_list_list_b @ transpose_rel_b @ ( cons_list_b @ nil_b @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_b @ nil_b @ Xss ) ) ) )
         => ( ! [X3: b,Xs3: list_b,Xss: list_list_b] :
                ( ( accp_list_list_b @ transpose_rel_b @ ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_b @ Xs3
                      @ ( concat_list_b
                        @ ( map_li4464339650582430090list_b
                          @ ( case_l1835703025255241754st_b_b @ nil_list_b
                            @ ^ [H: b,T2: list_b] : ( cons_list_b @ T2 @ nil_list_b ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_734_transpose_Opinduct,axiom,
    ! [A0: list_l8907847357763382004od_b_c,P: list_l8907847357763382004od_b_c > $o] :
      ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ A0 )
     => ( ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ nil_li8071162985807626740od_b_c )
         => ( P @ nil_li8071162985807626740od_b_c ) )
       => ( ! [Xss: list_l8907847357763382004od_b_c] :
              ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) ) ) )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
                ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) )
               => ( ( P
                    @ ( cons_l342771267123639716od_b_c @ Xs3
                      @ ( concat8136016674533020335od_b_c
                        @ ( map_li6701939492004672586od_b_c
                          @ ( case_l5988721774173819482od_b_c @ nil_li8071162985807626740od_b_c
                            @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_l342771267123639716od_b_c @ T2 @ nil_li8071162985807626740od_b_c ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_735_transpose_Osimps_I3_J,axiom,
    ! [X2: b,Xs: list_b,Xss2: list_list_b] :
      ( ( transpose_b @ ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ Xss2 ) )
      = ( cons_list_b
        @ ( cons_b @ X2
          @ ( concat_b
            @ ( map_list_b_list_b
              @ ( case_list_list_b_b @ nil_b
                @ ^ [H: b,T2: list_b] : ( cons_b @ H @ nil_b ) )
              @ Xss2 ) ) )
        @ ( transpose_b
          @ ( cons_list_b @ Xs
            @ ( concat_list_b
              @ ( map_li4464339650582430090list_b
                @ ( case_l1835703025255241754st_b_b @ nil_list_b
                  @ ^ [H: b,T2: list_b] : ( cons_list_b @ T2 @ nil_list_b ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_736_transpose_Osimps_I3_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Xss2: list_l8907847357763382004od_b_c] :
      ( ( transp1488944918686653944od_b_c @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ Xss2 ) )
      = ( cons_l342771267123639716od_b_c
        @ ( cons_P4529483553340347422od_b_c @ X2
          @ ( concat3284542956341965481od_b_c
            @ ( map_li8081432502061035332od_b_c
              @ ( case_l4454746133744813664od_b_c @ nil_Product_prod_b_c
                @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_P4529483553340347422od_b_c @ H @ nil_Product_prod_b_c ) )
              @ Xss2 ) ) )
        @ ( transp1488944918686653944od_b_c
          @ ( cons_l342771267123639716od_b_c @ Xs
            @ ( concat8136016674533020335od_b_c
              @ ( map_li6701939492004672586od_b_c
                @ ( case_l5988721774173819482od_b_c @ nil_li8071162985807626740od_b_c
                  @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_l342771267123639716od_b_c @ T2 @ nil_li8071162985807626740od_b_c ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_737_transpose_Oelims,axiom,
    ! [X2: list_list_b,Y: list_list_b] :
      ( ( ( transpose_b @ X2 )
        = Y )
     => ( ( ( X2 = nil_list_b )
         => ( Y != nil_list_b ) )
       => ( ! [Xss: list_list_b] :
              ( ( X2
                = ( cons_list_b @ nil_b @ Xss ) )
             => ( Y
               != ( transpose_b @ Xss ) ) )
         => ~ ! [X3: b,Xs3: list_b,Xss: list_list_b] :
                ( ( X2
                  = ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) )
               => ( Y
                 != ( cons_list_b
                    @ ( cons_b @ X3
                      @ ( concat_b
                        @ ( map_list_b_list_b
                          @ ( case_list_list_b_b @ nil_b
                            @ ^ [H: b,T2: list_b] : ( cons_b @ H @ nil_b ) )
                          @ Xss ) ) )
                    @ ( transpose_b
                      @ ( cons_list_b @ Xs3
                        @ ( concat_list_b
                          @ ( map_li4464339650582430090list_b
                            @ ( case_l1835703025255241754st_b_b @ nil_list_b
                              @ ^ [H: b,T2: list_b] : ( cons_list_b @ T2 @ nil_list_b ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_738_transpose_Oelims,axiom,
    ! [X2: list_l8907847357763382004od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( transp1488944918686653944od_b_c @ X2 )
        = Y )
     => ( ( ( X2 = nil_li8071162985807626740od_b_c )
         => ( Y != nil_li8071162985807626740od_b_c ) )
       => ( ! [Xss: list_l8907847357763382004od_b_c] :
              ( ( X2
                = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
             => ( Y
               != ( transp1488944918686653944od_b_c @ Xss ) ) )
         => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
                ( ( X2
                  = ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) )
               => ( Y
                 != ( cons_l342771267123639716od_b_c
                    @ ( cons_P4529483553340347422od_b_c @ X3
                      @ ( concat3284542956341965481od_b_c
                        @ ( map_li8081432502061035332od_b_c
                          @ ( case_l4454746133744813664od_b_c @ nil_Product_prod_b_c
                            @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_P4529483553340347422od_b_c @ H @ nil_Product_prod_b_c ) )
                          @ Xss ) ) )
                    @ ( transp1488944918686653944od_b_c
                      @ ( cons_l342771267123639716od_b_c @ Xs3
                        @ ( concat8136016674533020335od_b_c
                          @ ( map_li6701939492004672586od_b_c
                            @ ( case_l5988721774173819482od_b_c @ nil_li8071162985807626740od_b_c
                              @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_l342771267123639716od_b_c @ T2 @ nil_li8071162985807626740od_b_c ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_739_transpose__map__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_l8907847357763382004od_b_c] :
      ( ( transpose_b @ ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ Xs ) )
      = ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( transp1488944918686653944od_b_c @ Xs ) ) ) ).

% transpose_map_map
thf(fact_740_transpose__empty,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] :
      ( ( ( transp1488944918686653944od_b_c @ Xs )
        = nil_li8071162985807626740od_b_c )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ( X = nil_Product_prod_b_c ) ) ) ) ).

% transpose_empty
thf(fact_741_transpose_Opelims,axiom,
    ! [X2: list_list_b,Y: list_list_b] :
      ( ( ( transpose_b @ X2 )
        = Y )
     => ( ( accp_list_list_b @ transpose_rel_b @ X2 )
       => ( ( ( X2 = nil_list_b )
           => ( ( Y = nil_list_b )
             => ~ ( accp_list_list_b @ transpose_rel_b @ nil_list_b ) ) )
         => ( ! [Xss: list_list_b] :
                ( ( X2
                  = ( cons_list_b @ nil_b @ Xss ) )
               => ( ( Y
                    = ( transpose_b @ Xss ) )
                 => ~ ( accp_list_list_b @ transpose_rel_b @ ( cons_list_b @ nil_b @ Xss ) ) ) )
           => ~ ! [X3: b,Xs3: list_b,Xss: list_list_b] :
                  ( ( X2
                    = ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_b
                        @ ( cons_b @ X3
                          @ ( concat_b
                            @ ( map_list_b_list_b
                              @ ( case_list_list_b_b @ nil_b
                                @ ^ [H: b,T2: list_b] : ( cons_b @ H @ nil_b ) )
                              @ Xss ) ) )
                        @ ( transpose_b
                          @ ( cons_list_b @ Xs3
                            @ ( concat_list_b
                              @ ( map_li4464339650582430090list_b
                                @ ( case_l1835703025255241754st_b_b @ nil_list_b
                                  @ ^ [H: b,T2: list_b] : ( cons_list_b @ T2 @ nil_list_b ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_list_list_b @ transpose_rel_b @ ( cons_list_b @ ( cons_b @ X3 @ Xs3 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_742_transpose_Opelims,axiom,
    ! [X2: list_l8907847357763382004od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( ( transp1488944918686653944od_b_c @ X2 )
        = Y )
     => ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ X2 )
       => ( ( ( X2 = nil_li8071162985807626740od_b_c )
           => ( ( Y = nil_li8071162985807626740od_b_c )
             => ~ ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ nil_li8071162985807626740od_b_c ) ) )
         => ( ! [Xss: list_l8907847357763382004od_b_c] :
                ( ( X2
                  = ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
               => ( ( Y
                    = ( transp1488944918686653944od_b_c @ Xss ) )
                 => ~ ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) ) ) )
           => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
                  ( ( X2
                    = ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_l342771267123639716od_b_c
                        @ ( cons_P4529483553340347422od_b_c @ X3
                          @ ( concat3284542956341965481od_b_c
                            @ ( map_li8081432502061035332od_b_c
                              @ ( case_l4454746133744813664od_b_c @ nil_Product_prod_b_c
                                @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_P4529483553340347422od_b_c @ H @ nil_Product_prod_b_c ) )
                              @ Xss ) ) )
                        @ ( transp1488944918686653944od_b_c
                          @ ( cons_l342771267123639716od_b_c @ Xs3
                            @ ( concat8136016674533020335od_b_c
                              @ ( map_li6701939492004672586od_b_c
                                @ ( case_l5988721774173819482od_b_c @ nil_li8071162985807626740od_b_c
                                  @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_l342771267123639716od_b_c @ T2 @ nil_li8071162985807626740od_b_c ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_743_transpose_Opsimps_I3_J,axiom,
    ! [X2: b,Xs: list_b,Xss2: list_list_b] :
      ( ( accp_list_list_b @ transpose_rel_b @ ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ Xss2 ) )
     => ( ( transpose_b @ ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ Xss2 ) )
        = ( cons_list_b
          @ ( cons_b @ X2
            @ ( concat_b
              @ ( map_list_b_list_b
                @ ( case_list_list_b_b @ nil_b
                  @ ^ [H: b,T2: list_b] : ( cons_b @ H @ nil_b ) )
                @ Xss2 ) ) )
          @ ( transpose_b
            @ ( cons_list_b @ Xs
              @ ( concat_list_b
                @ ( map_li4464339650582430090list_b
                  @ ( case_l1835703025255241754st_b_b @ nil_list_b
                    @ ^ [H: b,T2: list_b] : ( cons_list_b @ T2 @ nil_list_b ) )
                  @ Xss2 ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_744_transpose_Opsimps_I3_J,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Xss2: list_l8907847357763382004od_b_c] :
      ( ( accp_l7120146224913739005od_b_c @ transp6505999919020861853od_b_c @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ Xss2 ) )
     => ( ( transp1488944918686653944od_b_c @ ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X2 @ Xs ) @ Xss2 ) )
        = ( cons_l342771267123639716od_b_c
          @ ( cons_P4529483553340347422od_b_c @ X2
            @ ( concat3284542956341965481od_b_c
              @ ( map_li8081432502061035332od_b_c
                @ ( case_l4454746133744813664od_b_c @ nil_Product_prod_b_c
                  @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_P4529483553340347422od_b_c @ H @ nil_Product_prod_b_c ) )
                @ Xss2 ) ) )
          @ ( transp1488944918686653944od_b_c
            @ ( cons_l342771267123639716od_b_c @ Xs
              @ ( concat8136016674533020335od_b_c
                @ ( map_li6701939492004672586od_b_c
                  @ ( case_l5988721774173819482od_b_c @ nil_li8071162985807626740od_b_c
                    @ ^ [H: product_prod_b_c,T2: list_P903359562653991662od_b_c] : ( cons_l342771267123639716od_b_c @ T2 @ nil_li8071162985807626740od_b_c ) )
                  @ Xss2 ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_745_minimal__failure__prefix__ob,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,I: fsm_li6801133765522507155_c_b_c,QM: list_P903359562653991662od_b_c,QI: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( observ6293852833591064631_c_b_c @ I )
       => ( ( member6330420149250801815od_b_c @ QM @ ( states7681702920031268536_c_b_c @ M ) )
         => ( ( member6330420149250801815od_b_c @ QI @ ( states7681702920031268536_c_b_c @ I ) )
           => ( ( member6330420149250801815od_b_c @ Io3 @ ( minus_5375846548645262261od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) )
             => ~ ! [Io2: list_P903359562653991662od_b_c,Xy2: product_prod_b_c] :
                    ( ? [Io6: list_P903359562653991662od_b_c] :
                        ( Io3
                        = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) @ Io6 ) ) )
                   => ( ( member6330420149250801815od_b_c @ Io2 @ ( inf_in4978071631833541052od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) )
                     => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) ) @ ( minus_5375846548645262261od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) ) ) ) ) ) ) ) ) ).

% minimal_failure_prefix_ob
thf(fact_746_minimal__failure__prefix__ob,axiom,
    ! [M: fsm_li6801133765522507155_c_b_c,I: fsm_a_b_c,QM: list_P903359562653991662od_b_c,QI: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M )
     => ( ( observable_a_b_c @ I )
       => ( ( member6330420149250801815od_b_c @ QM @ ( states7681702920031268536_c_b_c @ M ) )
         => ( ( member_a @ QI @ ( states_a_b_c @ I ) )
           => ( ( member6330420149250801815od_b_c @ Io3 @ ( minus_5375846548645262261od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) )
             => ~ ! [Io2: list_P903359562653991662od_b_c,Xy2: product_prod_b_c] :
                    ( ? [Io6: list_P903359562653991662od_b_c] :
                        ( Io3
                        = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) @ Io6 ) ) )
                   => ( ( member6330420149250801815od_b_c @ Io2 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) )
                     => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) ) @ ( minus_5375846548645262261od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_lis2930931384350476499_c_b_c @ M @ QM ) ) ) ) ) ) ) ) ) ) ).

% minimal_failure_prefix_ob
thf(fact_747_minimal__failure__prefix__ob,axiom,
    ! [M: fsm_a_b_c,I: fsm_li6801133765522507155_c_b_c,QM: a,QI: list_P903359562653991662od_b_c,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( observ6293852833591064631_c_b_c @ I )
       => ( ( member_a @ QM @ ( states_a_b_c @ M ) )
         => ( ( member6330420149250801815od_b_c @ QI @ ( states7681702920031268536_c_b_c @ I ) )
           => ( ( member6330420149250801815od_b_c @ Io3 @ ( minus_5375846548645262261od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) )
             => ~ ! [Io2: list_P903359562653991662od_b_c,Xy2: product_prod_b_c] :
                    ( ? [Io6: list_P903359562653991662od_b_c] :
                        ( Io3
                        = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) @ Io6 ) ) )
                   => ( ( member6330420149250801815od_b_c @ Io2 @ ( inf_in4978071631833541052od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) )
                     => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) ) @ ( minus_5375846548645262261od_b_c @ ( lS_lis2930931384350476499_c_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) ) ) ) ) ) ) ) ) ).

% minimal_failure_prefix_ob
thf(fact_748_minimal__failure__prefix__ob,axiom,
    ! [M: fsm_a_b_c,I: fsm_a_b_c,QM: a,QI: a,Io3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( observable_a_b_c @ I )
       => ( ( member_a @ QM @ ( states_a_b_c @ M ) )
         => ( ( member_a @ QI @ ( states_a_b_c @ I ) )
           => ( ( member6330420149250801815od_b_c @ Io3 @ ( minus_5375846548645262261od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) )
             => ~ ! [Io2: list_P903359562653991662od_b_c,Xy2: product_prod_b_c] :
                    ( ? [Io6: list_P903359562653991662od_b_c] :
                        ( Io3
                        = ( append2547753245680614915od_b_c @ Io2 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) @ Io6 ) ) )
                   => ( ( member6330420149250801815od_b_c @ Io2 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) )
                     => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ ( cons_P4529483553340347422od_b_c @ Xy2 @ nil_Product_prod_b_c ) ) @ ( minus_5375846548645262261od_b_c @ ( lS_a_b_c @ I @ QI ) @ ( lS_a_b_c @ M @ QM ) ) ) ) ) ) ) ) ) ) ).

% minimal_failure_prefix_ob
thf(fact_749_paths__for__input__set__code,axiom,
    ( paths_6942934977415288259_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,Xs2: list_b] : ( if_set5301400340663253090od_c_a @ ( member_a @ Q6 @ ( states_a_b_c @ M2 ) ) @ ( paths_6276656153717473370_a_b_c @ ( h_from_a_b_c @ M2 ) @ Xs2 @ Q6 @ nil_Pr1342775757158464060od_c_a ) @ bot_bo6236370880139903240od_c_a ) ) ) ).

% paths_for_input_set_code
thf(fact_750_set__Cons__def,axiom,
    ( set_Co4997685901046503076od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,XS: set_li336664667694595924od_b_c] :
          ( collec686158730324120671od_b_c
          @ ^ [Z3: list_l8907847357763382004od_b_c] :
            ? [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
              ( ( Z3
                = ( cons_l342771267123639716od_b_c @ X @ Xs2 ) )
              & ( member6330420149250801815od_b_c @ X @ A3 )
              & ( member7934871825613078557od_b_c @ Xs2 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_751_set__Cons__def,axiom,
    ( set_Cons_a
    = ( ^ [A3: set_a,XS: set_list_a] :
          ( collect_list_a
          @ ^ [Z3: list_a] :
            ? [X: a,Xs2: list_a] :
              ( ( Z3
                = ( cons_a @ X @ Xs2 ) )
              & ( member_a @ X @ A3 )
              & ( member_list_a @ Xs2 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_752_set__Cons__def,axiom,
    ( set_Cons_b
    = ( ^ [A3: set_b,XS: set_list_b] :
          ( collect_list_b
          @ ^ [Z3: list_b] :
            ? [X: b,Xs2: list_b] :
              ( ( Z3
                = ( cons_b @ X @ Xs2 ) )
              & ( member_b @ X @ A3 )
              & ( member_list_b @ Xs2 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_753_set__Cons__def,axiom,
    ( set_Co7706362885935805726od_b_c
    = ( ^ [A3: set_Product_prod_b_c,XS: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [Z3: list_P903359562653991662od_b_c] :
            ? [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( ( Z3
                = ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) )
              & ( member7862447941013992593od_b_c @ X @ A3 )
              & ( member6330420149250801815od_b_c @ Xs2 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_754_empty__iff,axiom,
    ! [C: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ C @ bot_bo4166481423041325370od_b_c ) ).

% empty_iff
thf(fact_755_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_756_all__not__in__conv,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( ! [X: list_P903359562653991662od_b_c] :
            ~ ( member6330420149250801815od_b_c @ X @ A2 ) )
      = ( A2 = bot_bo4166481423041325370od_b_c ) ) ).

% all_not_in_conv
thf(fact_757_all__not__in__conv,axiom,
    ! [A2: set_a] :
      ( ( ! [X: a] :
            ~ ( member_a @ X @ A2 ) )
      = ( A2 = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_758_Collect__empty__eq,axiom,
    ! [P: list_P903359562653991662od_b_c > $o] :
      ( ( ( collec2280997390073109977od_b_c @ P )
        = bot_bo4166481423041325370od_b_c )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_759_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_760_empty__Collect__eq,axiom,
    ! [P: list_P903359562653991662od_b_c > $o] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( collec2280997390073109977od_b_c @ P ) )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_761_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_762_IntI,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A2 )
     => ( ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% IntI
thf(fact_763_IntI,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ A2 )
     => ( ( member_a @ C @ B )
       => ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% IntI
thf(fact_764_Int__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A2 )
        & ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Int_iff
thf(fact_765_Int__iff,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
      = ( ( member_a @ C @ A2 )
        & ( member_a @ C @ B ) ) ) ).

% Int_iff
thf(fact_766_empty__subsetI,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ bot_bo4166481423041325370od_b_c @ A2 ) ).

% empty_subsetI
thf(fact_767_subset__empty,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ bot_bo4166481423041325370od_b_c )
      = ( A2 = bot_bo4166481423041325370od_b_c ) ) ).

% subset_empty
thf(fact_768_Int__subset__iff,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) )
      = ( ( ord_le282488521294790766od_b_c @ C2 @ A2 )
        & ( ord_le282488521294790766od_b_c @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_769_set__empty2,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( Xs = nil_li8071162985807626740od_b_c ) ) ).

% set_empty2
thf(fact_770_set__empty2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( bot_bo2863795366942399540od_b_c
        = ( set_Product_prod_b_c2 @ Xs ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% set_empty2
thf(fact_771_set__empty,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] :
      ( ( ( set_li4480668622519654659od_b_c @ Xs )
        = bot_bo4166481423041325370od_b_c )
      = ( Xs = nil_li8071162985807626740od_b_c ) ) ).

% set_empty
thf(fact_772_set__empty,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( ( set_Product_prod_b_c2 @ Xs )
        = bot_bo2863795366942399540od_b_c )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% set_empty
thf(fact_773_Diff__eq__empty__iff,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( minus_5375846548645262261od_b_c @ A2 @ B )
        = bot_bo4166481423041325370od_b_c )
      = ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_774_IntE,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) )
     => ~ ( ( member6330420149250801815od_b_c @ C @ A2 )
         => ~ ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% IntE
thf(fact_775_IntE,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
     => ~ ( ( member_a @ C @ A2 )
         => ~ ( member_a @ C @ B ) ) ) ).

% IntE
thf(fact_776_IntD1,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) )
     => ( member6330420149250801815od_b_c @ C @ A2 ) ) ).

% IntD1
thf(fact_777_IntD1,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
     => ( member_a @ C @ A2 ) ) ).

% IntD1
thf(fact_778_IntD2,axiom,
    ! [C: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) )
     => ( member6330420149250801815od_b_c @ C @ B ) ) ).

% IntD2
thf(fact_779_IntD2,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
     => ( member_a @ C @ B ) ) ).

% IntD2
thf(fact_780_emptyE,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ).

% emptyE
thf(fact_781_emptyE,axiom,
    ! [A: a] :
      ~ ( member_a @ A @ bot_bot_set_a ) ).

% emptyE
thf(fact_782_equals0D,axiom,
    ! [A2: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( A2 = bot_bo4166481423041325370od_b_c )
     => ~ ( member6330420149250801815od_b_c @ A @ A2 ) ) ).

% equals0D
thf(fact_783_equals0D,axiom,
    ! [A2: set_a,A: a] :
      ( ( A2 = bot_bot_set_a )
     => ~ ( member_a @ A @ A2 ) ) ).

% equals0D
thf(fact_784_equals0I,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ! [Y3: list_P903359562653991662od_b_c] :
          ~ ( member6330420149250801815od_b_c @ Y3 @ A2 )
     => ( A2 = bot_bo4166481423041325370od_b_c ) ) ).

% equals0I
thf(fact_785_equals0I,axiom,
    ! [A2: set_a] :
      ( ! [Y3: a] :
          ~ ( member_a @ Y3 @ A2 )
     => ( A2 = bot_bot_set_a ) ) ).

% equals0I
thf(fact_786_Int__emptyI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A2 )
         => ~ ( member6330420149250801815od_b_c @ X3 @ B ) )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
        = bot_bo4166481423041325370od_b_c ) ) ).

% Int_emptyI
thf(fact_787_Int__emptyI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ~ ( member_a @ X3 @ B ) )
     => ( ( inf_inf_set_a @ A2 @ B )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_788_ex__in__conv,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( ? [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A2 ) )
      = ( A2 != bot_bo4166481423041325370od_b_c ) ) ).

% ex_in_conv
thf(fact_789_ex__in__conv,axiom,
    ! [A2: set_a] :
      ( ( ? [X: a] : ( member_a @ X @ A2 ) )
      = ( A2 != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_790_disjoint__iff,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
        = bot_bo4166481423041325370od_b_c )
      = ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ A2 )
           => ~ ( member6330420149250801815od_b_c @ X @ B ) ) ) ) ).

% disjoint_iff
thf(fact_791_disjoint__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A2 @ B )
        = bot_bot_set_a )
      = ( ! [X: a] :
            ( ( member_a @ X @ A2 )
           => ~ ( member_a @ X @ B ) ) ) ) ).

% disjoint_iff
thf(fact_792_Set_Oempty__def,axiom,
    ( bot_bo4166481423041325370od_b_c
    = ( collec2280997390073109977od_b_c
      @ ^ [X: list_P903359562653991662od_b_c] : $false ) ) ).

% Set.empty_def
thf(fact_793_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_794_Collect__conj__eq,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( P @ X )
            & ( Q4 @ X ) ) )
      = ( inf_in4978071631833541052od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ).

% Collect_conj_eq
thf(fact_795_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q4 @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q4 ) ) ) ).

% Collect_conj_eq
thf(fact_796_Int__Collect,axiom,
    ! [X2: a,A2: set_a,P: a > $o] :
      ( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
      = ( ( member_a @ X2 @ A2 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_797_Int__Collect,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ A2 @ ( collec2280997390073109977od_b_c @ P ) ) )
      = ( ( member6330420149250801815od_b_c @ X2 @ A2 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_798_Int__Collect,axiom,
    ! [X2: nat,A2: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X2 @ A2 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_799_Int__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A3 )
              & ( member_a @ X @ B2 ) ) ) ) ) ).

% Int_def
thf(fact_800_Int__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ A3 )
              & ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ) ).

% Int_def
thf(fact_801_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( member_nat @ X @ B2 ) ) ) ) ) ).

% Int_def
thf(fact_802_bot_Oextremum,axiom,
    ! [A: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ bot_bo4166481423041325370od_b_c @ A ) ).

% bot.extremum
thf(fact_803_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_804_bot_Oextremum__unique,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ bot_bo4166481423041325370od_b_c )
      = ( A = bot_bo4166481423041325370od_b_c ) ) ).

% bot.extremum_unique
thf(fact_805_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_806_bot_Oextremum__uniqueI,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ bot_bo4166481423041325370od_b_c )
     => ( A = bot_bo4166481423041325370od_b_c ) ) ).

% bot.extremum_uniqueI
thf(fact_807_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_808_Int__Collect__mono,axiom,
    ! [A2: set_a,B: set_a,P: a > $o,Q4: a > $o] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ( P @ X3 )
             => ( Q4 @ X3 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q4 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_809_Int__Collect__mono,axiom,
    ! [A2: set_nat,B: set_nat,P: nat > $o,Q4: nat > $o] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ( P @ X3 )
             => ( Q4 @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q4 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_810_Int__Collect__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ( ( P @ X3 )
             => ( Q4 @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ ( collec2280997390073109977od_b_c @ P ) ) @ ( inf_in4978071631833541052od_b_c @ B @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_811_Int__greatest,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ C2 @ B )
       => ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% Int_greatest
thf(fact_812_Int__absorb2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
        = A2 ) ) ).

% Int_absorb2
thf(fact_813_Int__absorb1,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ A2 )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_814_Int__lower2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) @ B ) ).

% Int_lower2
thf(fact_815_Int__lower1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) @ A2 ) ).

% Int_lower1
thf(fact_816_Int__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ B @ D )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) @ ( inf_in4978071631833541052od_b_c @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_817_empty__set,axiom,
    ( bot_bo4166481423041325370od_b_c
    = ( set_li4480668622519654659od_b_c @ nil_li8071162985807626740od_b_c ) ) ).

% empty_set
thf(fact_818_empty__set,axiom,
    ( bot_bo2863795366942399540od_b_c
    = ( set_Product_prod_b_c2 @ nil_Product_prod_b_c ) ) ).

% empty_set
thf(fact_819_le__inf__iff,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z5: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ Y @ Z5 ) )
      = ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
        & ( ord_le282488521294790766od_b_c @ X2 @ Z5 ) ) ) ).

% le_inf_iff
thf(fact_820_le__inf__iff,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z5 ) )
      = ( ( ord_less_eq_nat @ X2 @ Y )
        & ( ord_less_eq_nat @ X2 @ Z5 ) ) ) ).

% le_inf_iff
thf(fact_821_inf_Obounded__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B3 @ C ) )
      = ( ( ord_le282488521294790766od_b_c @ A @ B3 )
        & ( ord_le282488521294790766od_b_c @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_822_inf_Obounded__iff,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) )
      = ( ( ord_less_eq_nat @ A @ B3 )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_823_observable__path__io__target,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( io_targets_a_b_c @ M
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
              @ P2 )
            @ Q )
          = ( insert_a @ ( target_a_b_c @ Q @ P2 ) @ bot_bot_set_a ) ) ) ) ).

% observable_path_io_target
thf(fact_824_insert__iff,axiom,
    ! [A: list_P903359562653991662od_b_c,B3: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B3 @ A2 ) )
      = ( ( A = B3 )
        | ( member6330420149250801815od_b_c @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_825_insert__iff,axiom,
    ! [A: a,B3: a,A2: set_a] :
      ( ( member_a @ A @ ( insert_a @ B3 @ A2 ) )
      = ( ( A = B3 )
        | ( member_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_826_insertCI,axiom,
    ! [A: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c] :
      ( ( ~ ( member6330420149250801815od_b_c @ A @ B )
       => ( A = B3 ) )
     => ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B3 @ B ) ) ) ).

% insertCI
thf(fact_827_insertCI,axiom,
    ! [A: a,B: set_a,B3: a] :
      ( ( ~ ( member_a @ A @ B )
       => ( A = B3 ) )
     => ( member_a @ A @ ( insert_a @ B3 @ B ) ) ) ).

% insertCI
thf(fact_828_singletonI,axiom,
    ! [A: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ).

% singletonI
thf(fact_829_singletonI,axiom,
    ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_830_insert__subset,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B )
      = ( ( member_a @ X2 @ B )
        & ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_831_insert__subset,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ X2 @ A2 ) @ B )
      = ( ( member6330420149250801815od_b_c @ X2 @ B )
        & ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_832_Int__insert__left__if0,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B ) @ C2 )
        = ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_833_Int__insert__left__if0,axiom,
    ! [A: a,C2: set_a,B: set_a] :
      ( ~ ( member_a @ A @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
        = ( inf_inf_set_a @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_834_Int__insert__left__if1,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B ) @ C2 )
        = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_835_Int__insert__left__if1,axiom,
    ! [A: a,C2: set_a,B: set_a] :
      ( ( member_a @ A @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
        = ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_836_Int__insert__right__if0,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A @ B ) )
        = ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_837_Int__insert__right__if0,axiom,
    ! [A: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ A @ A2 )
     => ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
        = ( inf_inf_set_a @ A2 @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_838_Int__insert__right__if1,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A @ B ) )
        = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_839_Int__insert__right__if1,axiom,
    ! [A: a,A2: set_a,B: set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
        = ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_840_Diff__insert0,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ( minus_5375846548645262261od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ B ) )
        = ( minus_5375846548645262261od_b_c @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_841_Diff__insert0,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B ) )
        = ( minus_minus_set_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_842_insert__Diff1,axiom,
    ! [X2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ B )
     => ( ( minus_5375846548645262261od_b_c @ ( insert6227932334100060350od_b_c @ X2 @ A2 ) @ B )
        = ( minus_5375846548645262261od_b_c @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_843_insert__Diff1,axiom,
    ! [X2: a,B: set_a,A2: set_a] :
      ( ( member_a @ X2 @ B )
     => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
        = ( minus_minus_set_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_844_singleton__conv2,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ( ^ [Y4: list_P903359562653991662od_b_c,Z4: list_P903359562653991662od_b_c] : ( Y4 = Z4 )
          @ A ) )
      = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ).

% singleton_conv2
thf(fact_845_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_846_singleton__conv,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] : ( X = A ) )
      = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ).

% singleton_conv
thf(fact_847_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_848_list_Osimps_I15_J,axiom,
    ! [X21: list_P903359562653991662od_b_c,X22: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X21 @ X22 ) )
      = ( insert6227932334100060350od_b_c @ X21 @ ( set_li4480668622519654659od_b_c @ X22 ) ) ) ).

% list.simps(15)
thf(fact_849_list_Osimps_I15_J,axiom,
    ! [X21: b,X22: list_b] :
      ( ( set_b2 @ ( cons_b @ X21 @ X22 ) )
      = ( insert_b @ X21 @ ( set_b2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_850_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( insert1747480808674937912od_b_c @ X21 @ ( set_Product_prod_b_c2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_851_singleton__insert__inj__eq,axiom,
    ! [B3: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ B3 @ bot_bo4166481423041325370od_b_c )
        = ( insert6227932334100060350od_b_c @ A @ A2 ) )
      = ( ( A = B3 )
        & ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B3 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_852_singleton__insert__inj__eq_H,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ A @ A2 )
        = ( insert6227932334100060350od_b_c @ B3 @ bot_bo4166481423041325370od_b_c ) )
      = ( ( A = B3 )
        & ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B3 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_853_insert__disjoint_I1_J,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ A2 ) @ B )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B )
        & ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% insert_disjoint(1)
thf(fact_854_insert__disjoint_I1_J,axiom,
    ! [A: a,A2: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A @ B )
        & ( ( inf_inf_set_a @ A2 @ B )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_855_insert__disjoint_I2_J,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ A2 ) @ B ) )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_856_insert__disjoint_I2_J,axiom,
    ! [A: a,A2: set_a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
      = ( ~ ( member_a @ A @ B )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_857_disjoint__insert_I1_J,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ B @ ( insert6227932334100060350od_b_c @ A @ A2 ) )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B )
        & ( ( inf_in4978071631833541052od_b_c @ B @ A2 )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% disjoint_insert(1)
thf(fact_858_disjoint__insert_I1_J,axiom,
    ! [B: set_a,A: a,A2: set_a] :
      ( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A @ B )
        & ( ( inf_inf_set_a @ B @ A2 )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_859_disjoint__insert_I2_J,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B3 @ B ) ) )
      = ( ~ ( member6330420149250801815od_b_c @ B3 @ A2 )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_860_disjoint__insert_I2_J,axiom,
    ! [A2: set_a,B3: a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A2 @ ( insert_a @ B3 @ B ) ) )
      = ( ~ ( member_a @ B3 @ A2 )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_861_singletonD,axiom,
    ! [B3: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_862_singletonD,axiom,
    ! [B3: a,A: a] :
      ( ( member_a @ B3 @ ( insert_a @ A @ bot_bot_set_a ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_863_bot__set__def,axiom,
    ( bot_bo4166481423041325370od_b_c
    = ( collec2280997390073109977od_b_c @ bot_bo5496101219168594979_b_c_o ) ) ).

% bot_set_def
thf(fact_864_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_865_singleton__iff,axiom,
    ! [B3: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_866_singleton__iff,axiom,
    ! [B3: a,A: a] :
      ( ( member_a @ B3 @ ( insert_a @ A @ bot_bot_set_a ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_867_Int__insert__left,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B ) @ C2 )
          = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B ) @ C2 )
          = ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_868_Int__insert__left,axiom,
    ! [A: a,C2: set_a,B: set_a] :
      ( ( ( member_a @ A @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
          = ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
      & ( ~ ( member_a @ A @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
          = ( inf_inf_set_a @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_869_Int__insert__right,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A @ A2 )
       => ( ( inf_in4978071631833541052od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A @ B ) )
          = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A @ A2 )
       => ( ( inf_in4978071631833541052od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A @ B ) )
          = ( inf_in4978071631833541052od_b_c @ A2 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_870_Int__insert__right,axiom,
    ! [A: a,A2: set_a,B: set_a] :
      ( ( ( member_a @ A @ A2 )
       => ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
          = ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
      & ( ~ ( member_a @ A @ A2 )
       => ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
          = ( inf_inf_set_a @ A2 @ B ) ) ) ) ).

% Int_insert_right
thf(fact_871_Collect__conv__if2,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,A: list_P903359562653991662od_b_c] :
      ( ( ( P @ A )
       => ( ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) )
      & ( ~ ( P @ A )
       => ( ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% Collect_conv_if2
thf(fact_872_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_873_Collect__conv__if,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,A: list_P903359562653991662od_b_c] :
      ( ( ( P @ A )
       => ( ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) )
      & ( ~ ( P @ A )
       => ( ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% Collect_conv_if
thf(fact_874_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_875_inf__set__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( collect_a
          @ ( inf_inf_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 )
            @ ^ [X: a] : ( member_a @ X @ B2 ) ) ) ) ) ).

% inf_set_def
thf(fact_876_inf__set__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ( inf_in3324677931821484577_b_c_o
            @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A3 )
            @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ) ).

% inf_set_def
thf(fact_877_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A3 )
            @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).

% inf_set_def
thf(fact_878_bot__empty__eq,axiom,
    ( bot_bo5496101219168594979_b_c_o
    = ( ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ bot_bo4166481423041325370od_b_c ) ) ) ).

% bot_empty_eq
thf(fact_879_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_880_inf__Int__eq,axiom,
    ! [R: set_li6436108459499378894od_b_c,S: set_li6436108459499378894od_b_c] :
      ( ( inf_in3324677931821484577_b_c_o
        @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ R )
        @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ S ) )
      = ( ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_881_inf__Int__eq,axiom,
    ! [R: set_a,S: set_a] :
      ( ( inf_inf_a_o
        @ ^ [X: a] : ( member_a @ X @ R )
        @ ^ [X: a] : ( member_a @ X @ S ) )
      = ( ^ [X: a] : ( member_a @ X @ ( inf_inf_set_a @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_882_bot__empty__eq2,axiom,
    ( bot_bot_b_c_o
    = ( ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ bot_bo2863795366942399540od_b_c ) ) ) ).

% bot_empty_eq2
thf(fact_883_inf__Int__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( inf_inf_b_c_o
        @ ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ R )
        @ ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ S ) )
      = ( ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ ( inf_in8411426679332095414od_b_c @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_884_insert__mono,axiom,
    ! [C2: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ D )
     => ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ A @ C2 ) @ ( insert6227932334100060350od_b_c @ A @ D ) ) ) ).

% insert_mono
thf(fact_885_subset__insert,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
        = ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_886_subset__insert,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ B ) )
        = ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_887_subset__insertI,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ B @ ( insert6227932334100060350od_b_c @ A @ B ) ) ).

% subset_insertI
thf(fact_888_subset__insertI2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B3 @ B ) ) ) ).

% subset_insertI2
thf(fact_889_insert__Diff__if,axiom,
    ! [X2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ X2 @ B )
       => ( ( minus_5375846548645262261od_b_c @ ( insert6227932334100060350od_b_c @ X2 @ A2 ) @ B )
          = ( minus_5375846548645262261od_b_c @ A2 @ B ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ X2 @ B )
       => ( ( minus_5375846548645262261od_b_c @ ( insert6227932334100060350od_b_c @ X2 @ A2 ) @ B )
          = ( insert6227932334100060350od_b_c @ X2 @ ( minus_5375846548645262261od_b_c @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_890_insert__Diff__if,axiom,
    ! [X2: a,B: set_a,A2: set_a] :
      ( ( ( member_a @ X2 @ B )
       => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
          = ( minus_minus_set_a @ A2 @ B ) ) )
      & ( ~ ( member_a @ X2 @ B )
       => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
          = ( insert_a @ X2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_891_insert__compr,axiom,
    ( insert_a
    = ( ^ [A4: a,B2: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( X = A4 )
              | ( member_a @ X @ B2 ) ) ) ) ) ).

% insert_compr
thf(fact_892_insert__compr,axiom,
    ( insert6227932334100060350od_b_c
    = ( ^ [A4: list_P903359562653991662od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( X = A4 )
              | ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ) ).

% insert_compr
thf(fact_893_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A4 )
              | ( member_nat @ X @ B2 ) ) ) ) ) ).

% insert_compr
thf(fact_894_insert__Collect,axiom,
    ! [A: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( insert6227932334100060350od_b_c @ A @ ( collec2280997390073109977od_b_c @ P ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [U: list_P903359562653991662od_b_c] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_895_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_896_mk__disjoint__insert,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ? [B7: set_li6436108459499378894od_b_c] :
          ( ( A2
            = ( insert6227932334100060350od_b_c @ A @ B7 ) )
          & ~ ( member6330420149250801815od_b_c @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_897_mk__disjoint__insert,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ? [B7: set_a] :
          ( ( A2
            = ( insert_a @ A @ B7 ) )
          & ~ ( member_a @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_898_insert__eq__iff,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ~ ( member6330420149250801815od_b_c @ B3 @ B )
       => ( ( ( insert6227932334100060350od_b_c @ A @ A2 )
            = ( insert6227932334100060350od_b_c @ B3 @ B ) )
          = ( ( ( A = B3 )
             => ( A2 = B ) )
            & ( ( A != B3 )
             => ? [C5: set_li6436108459499378894od_b_c] :
                  ( ( A2
                    = ( insert6227932334100060350od_b_c @ B3 @ C5 ) )
                  & ~ ( member6330420149250801815od_b_c @ B3 @ C5 )
                  & ( B
                    = ( insert6227932334100060350od_b_c @ A @ C5 ) )
                  & ~ ( member6330420149250801815od_b_c @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_899_insert__eq__iff,axiom,
    ! [A: a,A2: set_a,B3: a,B: set_a] :
      ( ~ ( member_a @ A @ A2 )
     => ( ~ ( member_a @ B3 @ B )
       => ( ( ( insert_a @ A @ A2 )
            = ( insert_a @ B3 @ B ) )
          = ( ( ( A = B3 )
             => ( A2 = B ) )
            & ( ( A != B3 )
             => ? [C5: set_a] :
                  ( ( A2
                    = ( insert_a @ B3 @ C5 ) )
                  & ~ ( member_a @ B3 @ C5 )
                  & ( B
                    = ( insert_a @ A @ C5 ) )
                  & ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_900_insert__absorb,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( insert6227932334100060350od_b_c @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_901_insert__absorb,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( insert_a @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_902_insert__ident,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ B )
       => ( ( ( insert6227932334100060350od_b_c @ X2 @ A2 )
            = ( insert6227932334100060350od_b_c @ X2 @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_903_insert__ident,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ~ ( member_a @ X2 @ B )
       => ( ( ( insert_a @ X2 @ A2 )
            = ( insert_a @ X2 @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_904_Set_Oset__insert,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ~ ! [B7: set_li6436108459499378894od_b_c] :
            ( ( A2
              = ( insert6227932334100060350od_b_c @ X2 @ B7 ) )
           => ( member6330420149250801815od_b_c @ X2 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_905_Set_Oset__insert,axiom,
    ! [X2: a,A2: set_a] :
      ( ( member_a @ X2 @ A2 )
     => ~ ! [B7: set_a] :
            ( ( A2
              = ( insert_a @ X2 @ B7 ) )
           => ( member_a @ X2 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_906_insertI2,axiom,
    ! [A: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ B )
     => ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B3 @ B ) ) ) ).

% insertI2
thf(fact_907_insertI2,axiom,
    ! [A: a,B: set_a,B3: a] :
      ( ( member_a @ A @ B )
     => ( member_a @ A @ ( insert_a @ B3 @ B ) ) ) ).

% insertI2
thf(fact_908_insertI1,axiom,
    ! [A: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] : ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ A @ B ) ) ).

% insertI1
thf(fact_909_insertI1,axiom,
    ! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).

% insertI1
thf(fact_910_insertE,axiom,
    ! [A: list_P903359562653991662od_b_c,B3: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B3 @ A2 ) )
     => ( ( A != B3 )
       => ( member6330420149250801815od_b_c @ A @ A2 ) ) ) ).

% insertE
thf(fact_911_insertE,axiom,
    ! [A: a,B3: a,A2: set_a] :
      ( ( member_a @ A @ ( insert_a @ B3 @ A2 ) )
     => ( ( A != B3 )
       => ( member_a @ A @ A2 ) ) ) ).

% insertE
thf(fact_912_subset__singleton__iff,axiom,
    ! [X7: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X7 @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
      = ( ( X7 = bot_bo4166481423041325370od_b_c )
        | ( X7
          = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singleton_iff
thf(fact_913_subset__singletonD,axiom,
    ! [A2: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) )
     => ( ( A2 = bot_bo4166481423041325370od_b_c )
        | ( A2
          = ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singletonD
thf(fact_914_Diff__insert__absorb,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ( minus_5375846548645262261od_b_c @ ( insert6227932334100060350od_b_c @ X2 @ A2 ) @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_915_Diff__insert__absorb,axiom,
    ! [X2: a,A2: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_916_insert__Diff,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( insert6227932334100060350od_b_c @ A @ ( minus_5375846548645262261od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_917_insert__Diff,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_918_subset__Diff__insert,axiom,
    ! [A2: set_a,B: set_a,X2: a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) )
        & ~ ( member_a @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_919_subset__Diff__insert,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( minus_5375846548645262261od_b_c @ B @ ( insert6227932334100060350od_b_c @ X2 @ C2 ) ) )
      = ( ( ord_le282488521294790766od_b_c @ A2 @ ( minus_5375846548645262261od_b_c @ B @ C2 ) )
        & ~ ( member6330420149250801815od_b_c @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_920_subset__insert__iff,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
      = ( ( ( member_a @ X2 @ A2 )
         => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
        & ( ~ ( member_a @ X2 @ A2 )
         => ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_921_subset__insert__iff,axiom,
    ! [A2: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ B ) )
      = ( ( ( member6330420149250801815od_b_c @ X2 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( minus_5375846548645262261od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) ) @ B ) )
        & ( ~ ( member6330420149250801815od_b_c @ X2 @ A2 )
         => ( ord_le282488521294790766od_b_c @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_922_Diff__single__insert,axiom,
    ! [A2: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( minus_5375846548645262261od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) ) @ B )
     => ( ord_le282488521294790766od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_923_observable__io__targets,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [Q2: a] :
              ( ( io_targets_a_b_c @ M @ Io3 @ Q )
             != ( insert_a @ Q2 @ bot_bot_set_a ) ) ) ) ).

% observable_io_targets
thf(fact_924_observable__paths__for__io,axiom,
    ! [M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [P4: list_P6327159017948738492od_c_a] :
              ( ( paths_for_io_a_b_c @ M @ Q @ Io3 )
             != ( insert4789241225314331020od_c_a @ P4 @ bot_bo6236370880139903240od_c_a ) ) ) ) ).

% observable_paths_for_io
thf(fact_925_prefixes__set__Cons__insert,axiom,
    ! [W2: list_b,Xy3: b] :
      ( ( set_list_b2 @ ( prefixes_b @ ( append_b @ W2 @ ( cons_b @ Xy3 @ nil_b ) ) ) )
      = ( insert_list_b @ ( append_b @ W2 @ ( cons_b @ Xy3 @ nil_b ) ) @ ( set_list_b2 @ ( prefixes_b @ W2 ) ) ) ) ).

% prefixes_set_Cons_insert
thf(fact_926_prefixes__set__Cons__insert,axiom,
    ! [W2: list_P903359562653991662od_b_c,Xy3: product_prod_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ W2 @ ( cons_P4529483553340347422od_b_c @ Xy3 @ nil_Product_prod_b_c ) ) ) )
      = ( insert6227932334100060350od_b_c @ ( append2547753245680614915od_b_c @ W2 @ ( cons_P4529483553340347422od_b_c @ Xy3 @ nil_Product_prod_b_c ) ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ W2 ) ) ) ) ).

% prefixes_set_Cons_insert
thf(fact_927_inf__sup__ord_I2_J,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_928_inf__sup__ord_I2_J,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_929_inf__sup__ord_I1_J,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_930_inf__sup__ord_I1_J,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_931_inf__le1,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_932_inf__le1,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_933_inf__le2,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_934_inf__le2,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_935_le__infE,axiom,
    ! [X2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ X2 @ A )
         => ~ ( ord_le282488521294790766od_b_c @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_936_le__infE,axiom,
    ! [X2: nat,A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B3 ) )
     => ~ ( ( ord_less_eq_nat @ X2 @ A )
         => ~ ( ord_less_eq_nat @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_937_le__infI,axiom,
    ! [X2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ A )
     => ( ( ord_le282488521294790766od_b_c @ X2 @ B3 )
       => ( ord_le282488521294790766od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) ) ) ) ).

% le_infI
thf(fact_938_le__infI,axiom,
    ! [X2: nat,A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X2 @ A )
     => ( ( ord_less_eq_nat @ X2 @ B3 )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B3 ) ) ) ) ).

% le_infI
thf(fact_939_inf__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,D2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ D2 )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ ( inf_in4978071631833541052od_b_c @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_940_inf__mono,axiom,
    ! [A: nat,C: nat,B3: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B3 @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_941_le__infI1,axiom,
    ! [A: set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ X2 )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_942_le__infI1,axiom,
    ! [A: nat,X2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_943_le__infI2,axiom,
    ! [B3: set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ X2 )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_944_le__infI2,axiom,
    ! [B3: nat,X2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B3 @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_945_inf_OorderE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( A
        = ( inf_in4978071631833541052od_b_c @ A @ B3 ) ) ) ).

% inf.orderE
thf(fact_946_inf_OorderE,axiom,
    ! [A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( A
        = ( inf_inf_nat @ A @ B3 ) ) ) ).

% inf.orderE
thf(fact_947_inf_OorderI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( A
        = ( inf_in4978071631833541052od_b_c @ A @ B3 ) )
     => ( ord_le282488521294790766od_b_c @ A @ B3 ) ) ).

% inf.orderI
thf(fact_948_inf_OorderI,axiom,
    ! [A: nat,B3: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B3 ) )
     => ( ord_less_eq_nat @ A @ B3 ) ) ).

% inf.orderI
thf(fact_949_inf__unique,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ( ord_le282488521294790766od_b_c @ X3 @ Z )
               => ( ord_le282488521294790766od_b_c @ X3 @ ( F @ Y3 @ Z ) ) ) )
         => ( ( inf_in4978071631833541052od_b_c @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_950_inf__unique,axiom,
    ! [F: nat > nat > nat,X2: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z )
               => ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z ) ) ) )
         => ( ( inf_inf_nat @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_951_le__iff__inf,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [X: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_952_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( inf_inf_nat @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_953_inf_Oabsorb1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B3 )
        = A ) ) ).

% inf.absorb1
thf(fact_954_inf_Oabsorb1,axiom,
    ! [A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( inf_inf_nat @ A @ B3 )
        = A ) ) ).

% inf.absorb1
thf(fact_955_inf_Oabsorb2,axiom,
    ! [B3: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ A )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_956_inf_Oabsorb2,axiom,
    ! [B3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B3 @ A )
     => ( ( inf_inf_nat @ A @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_957_inf__absorb1,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
     => ( ( inf_in4978071631833541052od_b_c @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_958_inf__absorb1,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( inf_inf_nat @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_959_inf__absorb2,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X2 )
     => ( ( inf_in4978071631833541052od_b_c @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_960_inf__absorb2,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( inf_inf_nat @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_961_inf_OboundedE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B3 @ C ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ A @ B3 )
         => ~ ( ord_le282488521294790766od_b_c @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_962_inf_OboundedE,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B3 )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_963_inf_OboundedI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ A @ C )
       => ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_964_inf_OboundedI,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_965_inf__greatest,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z5: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X2 @ Y )
     => ( ( ord_le282488521294790766od_b_c @ X2 @ Z5 )
       => ( ord_le282488521294790766od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ Y @ Z5 ) ) ) ) ).

% inf_greatest
thf(fact_966_inf__greatest,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ X2 @ Z5 )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z5 ) ) ) ) ).

% inf_greatest
thf(fact_967_inf_Oorder__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( A4
          = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_968_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_969_inf_Ocobounded1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ A ) ).

% inf.cobounded1
thf(fact_970_inf_Ocobounded1,axiom,
    ! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ A ) ).

% inf.cobounded1
thf(fact_971_inf_Ocobounded2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_972_inf_Ocobounded2,axiom,
    ! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_973_inf_Oabsorb__iff1,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_974_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_975_inf_Oabsorb__iff2,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_976_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_977_inf_OcoboundedI1,axiom,
    ! [A: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_978_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_979_inf_OcoboundedI2,axiom,
    ! [B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ C )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_980_inf_OcoboundedI2,axiom,
    ! [B3: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B3 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_981_h__obs__Some,axiom,
    ! [M: fsm_a_b_c,Q: a,X2: b,Y: c,Q3: a] :
      ( ( observable_a_b_c @ M )
     => ( ( ( h_obs_a_b_c @ M @ Q @ X2 @ Y )
          = ( some_a @ Q3 ) )
        = ( ( collect_a
            @ ^ [Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y @ Q5 ) ) ) @ ( transitions_a_b_c @ M ) ) )
          = ( insert_a @ Q3 @ bot_bot_set_a ) ) ) ) ).

% h_obs_Some
thf(fact_982_diff__shunt__var,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( minus_5375846548645262261od_b_c @ X2 @ Y )
        = bot_bo4166481423041325370od_b_c )
      = ( ord_le282488521294790766od_b_c @ X2 @ Y ) ) ).

% diff_shunt_var
thf(fact_983_set__as__map__containment,axiom,
    ! [X2: b,Y: c,Zs3: set_Product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Zs3 )
     => ( member_c @ Y
        @ ( case_o8058779280418240651_set_c @ bot_bot_set_c
          @ ^ [Xs2: set_c] : Xs2
          @ ( set_as_map_b_c @ Zs3 @ X2 ) ) ) ) ).

% set_as_map_containment
thf(fact_984_set__as__map__elem,axiom,
    ! [Y: c,Xs: set_Product_prod_b_c,X2: b] :
      ( ( member_c @ Y
        @ ( case_o8058779280418240651_set_c @ bot_bot_set_c
          @ ^ [Xs2: set_c] : Xs2
          @ ( set_as_map_b_c @ Xs @ X2 ) ) )
     => ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ Xs ) ) ).

% set_as_map_elem
thf(fact_985_minus__set__def,axiom,
    ( minus_minus_set_a
    = ( ^ [A3: set_a,B2: set_a] :
          ( collect_a
          @ ( minus_minus_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 )
            @ ^ [X: a] : ( member_a @ X @ B2 ) ) ) ) ) ).

% minus_set_def
thf(fact_986_minus__set__def,axiom,
    ( minus_5375846548645262261od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ( minus_5771323728406603112_b_c_o
            @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A3 )
            @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ B2 ) ) ) ) ) ).

% minus_set_def
thf(fact_987_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A3 )
            @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).

% minus_set_def
thf(fact_988_set__as__map__def,axiom,
    ( set_as_map_b_c
    = ( ^ [S3: set_Product_prod_b_c,X: b] :
          ( if_option_set_c
          @ ? [Z3: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Z3 ) @ S3 )
          @ ( some_set_c
            @ ( collect_c
              @ ^ [Z3: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Z3 ) @ S3 ) ) )
          @ none_set_c ) ) ) ).

% set_as_map_def
thf(fact_989_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_990_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K3: nat,B3: nat] :
      ( ( P @ K3 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_991_nat__le__linear,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
      | ( ord_less_eq_nat @ N @ M3 ) ) ).

% nat_le_linear
thf(fact_992_le__antisym,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( ord_less_eq_nat @ N @ M3 )
       => ( M3 = N ) ) ) ).

% le_antisym
thf(fact_993_eq__imp__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( M3 = N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% eq_imp_le
thf(fact_994_le__trans,axiom,
    ! [I2: nat,J: nat,K3: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ J @ K3 )
       => ( ord_less_eq_nat @ I2 @ K3 ) ) ) ).

% le_trans
thf(fact_995_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_996_eq__diff__iff,axiom,
    ! [K3: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K3 @ M3 )
     => ( ( ord_less_eq_nat @ K3 @ N )
       => ( ( ( minus_minus_nat @ M3 @ K3 )
            = ( minus_minus_nat @ N @ K3 ) )
          = ( M3 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_997_le__diff__iff,axiom,
    ! [K3: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K3 @ M3 )
     => ( ( ord_less_eq_nat @ K3 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
          = ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_998_Nat_Odiff__diff__eq,axiom,
    ! [K3: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K3 @ M3 )
     => ( ( ord_less_eq_nat @ K3 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
          = ( minus_minus_nat @ M3 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_999_diff__le__mono,axiom,
    ! [M3: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1000_diff__le__self,axiom,
    ! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).

% diff_le_self
thf(fact_1001_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1002_diff__le__mono2,axiom,
    ! [M3: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).

% diff_le_mono2
thf(fact_1003_acyclic__paths__up__to__length__code,axiom,
    ( acycli3978232057192500090_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,K4: nat] :
          ( if_set5301400340663253090od_c_a @ ( member_a @ Q6 @ ( states_a_b_c @ M2 ) )
          @ ( acycli8838441729103940241_a_b_c @ nil_Pr1342775757158464060od_c_a @ Q6
            @ ^ [X: a] :
                ( case_o5084979270738236875od_c_a @ bot_bo8402128546026624985od_c_a
                @ ^ [Xs2: set_Pr272002107461315949od_c_a] : Xs2
                @ ( set_as7394328699852255621od_c_a @ ( transitions_a_b_c @ M2 ) @ X ) )
            @ ( insert_a @ Q6 @ bot_bot_set_a )
            @ K4 )
          @ bot_bo6236370880139903240od_c_a ) ) ) ).

% acyclic_paths_up_to_length_code
thf(fact_1004_Succ__def,axiom,
    ( bNF_Gr4730963734287941769od_b_c
    = ( ^ [Kl: set_li336664667694595924od_b_c,Kl2: list_l8907847357763382004od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [K4: list_P903359562653991662od_b_c] : ( member7934871825613078557od_b_c @ ( append6992831555715836169od_b_c @ Kl2 @ ( cons_l342771267123639716od_b_c @ K4 @ nil_li8071162985807626740od_b_c ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1005_Succ__def,axiom,
    ( bNF_Gr6352880689984616693cc_nat
    = ( ^ [Kl: set_list_nat,Kl2: list_nat] :
          ( collect_nat
          @ ^ [K4: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K4 @ nil_nat ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1006_Succ__def,axiom,
    ( bNF_Greatest_Succ_b
    = ( ^ [Kl: set_list_b,Kl2: list_b] :
          ( collect_b
          @ ^ [K4: b] : ( member_list_b @ ( append_b @ Kl2 @ ( cons_b @ K4 @ nil_b ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1007_Succ__def,axiom,
    ( bNF_Gr8135241567188112771od_b_c
    = ( ^ [Kl: set_li6436108459499378894od_b_c,Kl2: list_P903359562653991662od_b_c] :
          ( collec548942224018234067od_b_c
          @ ^ [K4: product_prod_b_c] : ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Kl2 @ ( cons_P4529483553340347422od_b_c @ K4 @ nil_Product_prod_b_c ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1008_SuccI,axiom,
    ! [Kl3: list_l8907847357763382004od_b_c,K3: list_P903359562653991662od_b_c,Kl4: set_li336664667694595924od_b_c] :
      ( ( member7934871825613078557od_b_c @ ( append6992831555715836169od_b_c @ Kl3 @ ( cons_l342771267123639716od_b_c @ K3 @ nil_li8071162985807626740od_b_c ) ) @ Kl4 )
     => ( member6330420149250801815od_b_c @ K3 @ ( bNF_Gr4730963734287941769od_b_c @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1009_SuccI,axiom,
    ! [Kl3: list_a,K3: a,Kl4: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K3 @ nil_a ) ) @ Kl4 )
     => ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1010_SuccI,axiom,
    ! [Kl3: list_b,K3: b,Kl4: set_list_b] :
      ( ( member_list_b @ ( append_b @ Kl3 @ ( cons_b @ K3 @ nil_b ) ) @ Kl4 )
     => ( member_b @ K3 @ ( bNF_Greatest_Succ_b @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1011_SuccI,axiom,
    ! [Kl3: list_P903359562653991662od_b_c,K3: product_prod_b_c,Kl4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Kl3 @ ( cons_P4529483553340347422od_b_c @ K3 @ nil_Product_prod_b_c ) ) @ Kl4 )
     => ( member7862447941013992593od_b_c @ K3 @ ( bNF_Gr8135241567188112771od_b_c @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1012_SuccD,axiom,
    ! [K3: list_P903359562653991662od_b_c,Kl4: set_li336664667694595924od_b_c,Kl3: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ K3 @ ( bNF_Gr4730963734287941769od_b_c @ Kl4 @ Kl3 ) )
     => ( member7934871825613078557od_b_c @ ( append6992831555715836169od_b_c @ Kl3 @ ( cons_l342771267123639716od_b_c @ K3 @ nil_li8071162985807626740od_b_c ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1013_SuccD,axiom,
    ! [K3: a,Kl4: set_list_a,Kl3: list_a] :
      ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) )
     => ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K3 @ nil_a ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1014_SuccD,axiom,
    ! [K3: b,Kl4: set_list_b,Kl3: list_b] :
      ( ( member_b @ K3 @ ( bNF_Greatest_Succ_b @ Kl4 @ Kl3 ) )
     => ( member_list_b @ ( append_b @ Kl3 @ ( cons_b @ K3 @ nil_b ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1015_SuccD,axiom,
    ! [K3: product_prod_b_c,Kl4: set_li6436108459499378894od_b_c,Kl3: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ K3 @ ( bNF_Gr8135241567188112771od_b_c @ Kl4 @ Kl3 ) )
     => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Kl3 @ ( cons_P4529483553340347422od_b_c @ K3 @ nil_Product_prod_b_c ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1016_paths__up__to__length__or__condition__with__witness_H__set,axiom,
    ! [Q: a,M: fsm_a_b_c,P: list_P6327159017948738492od_c_a > option_a,Prev2: list_P6327159017948738492od_c_a,K3: nat] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( paths_8265010931208446681_b_c_a @ ( h_from_a_b_c @ M ) @ P @ Prev2 @ K3 @ Q )
        = ( collec5198149995164206676_c_a_a
          @ ^ [Uu2: produc314987618042074217_c_a_a] :
            ? [P9: list_P6327159017948738492od_c_a,X: a] :
              ( ( Uu2
                = ( produc52780016337147353_c_a_a @ ( append3884164603879732689od_c_a @ Prev2 @ P9 ) @ X ) )
              & ( path_a_b_c @ M @ Q @ P9 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P9 ) @ K3 )
              & ( ( P @ ( append3884164603879732689od_c_a @ Prev2 @ P9 ) )
                = ( some_a @ X ) )
              & ! [P10: list_P6327159017948738492od_c_a,P11: list_P6327159017948738492od_c_a] :
                  ( ( ( P9
                      = ( append3884164603879732689od_c_a @ P10 @ P11 ) )
                    & ( P11 != nil_Pr1342775757158464060od_c_a ) )
                 => ( ( P @ ( append3884164603879732689od_c_a @ Prev2 @ P10 ) )
                    = none_a ) ) ) ) ) ) ).

% paths_up_to_length_or_condition_with_witness'_set
thf(fact_1017_List_Ofinite__set,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] : ( finite3074115686814133143od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) ) ).

% List.finite_set
thf(fact_1018_List_Ofinite__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c] : ( finite3757003021641768849od_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ).

% List.finite_set
thf(fact_1019_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1020_finite__list,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ? [Xs3: list_l8907847357763382004od_b_c] :
          ( ( set_li4480668622519654659od_b_c @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1021_finite__list,axiom,
    ! [A2: set_Product_prod_b_c] :
      ( ( finite3757003021641768849od_b_c @ A2 )
     => ? [Xs3: list_P903359562653991662od_b_c] :
          ( ( set_Product_prod_b_c2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1022_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1023_finite__set__min__param__ex,axiom,
    ! [XS2: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > nat > $o] :
      ( ( finite3074115686814133143od_b_c @ XS2 )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ XS2 )
           => ? [K5: nat] :
              ! [K2: nat] :
                ( ( ord_less_eq_nat @ K5 @ K2 )
               => ( P @ X3 @ K2 ) ) )
       => ? [K: nat] :
          ! [X5: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X5 @ XS2 )
           => ( P @ X5 @ K ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_1024_finite__set__min__param__ex,axiom,
    ! [XS2: set_a,P: a > nat > $o] :
      ( ( finite_finite_a @ XS2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ XS2 )
           => ? [K5: nat] :
              ! [K2: nat] :
                ( ( ord_less_eq_nat @ K5 @ K2 )
               => ( P @ X3 @ K2 ) ) )
       => ? [K: nat] :
          ! [X5: a] :
            ( ( member_a @ X5 @ XS2 )
           => ( P @ X5 @ K ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_1025_finite__set__min__param__ex,axiom,
    ! [XS2: set_nat,P: nat > nat > $o] :
      ( ( finite_finite_nat @ XS2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ XS2 )
           => ? [K5: nat] :
              ! [K2: nat] :
                ( ( ord_less_eq_nat @ K5 @ K2 )
               => ( P @ X3 @ K2 ) ) )
       => ? [K: nat] :
          ! [X5: nat] :
            ( ( member_nat @ X5 @ XS2 )
           => ( P @ X5 @ K ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_1026_fsm__states__finite,axiom,
    ! [M: fsm_a_b_c] : ( finite_finite_a @ ( states_a_b_c @ M ) ) ).

% fsm_states_finite
thf(fact_1027_prefix__free__set__maximal__list__ob,axiom,
    ! [Xs: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ Xs )
     => ( ( member6330420149250801815od_b_c @ X2 @ Xs )
       => ~ ! [X8: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ X2 @ X8 ) @ Xs )
             => ? [Y7: list_P903359562653991662od_b_c] :
                  ( ( Y7 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ X2 @ X8 ) @ Y7 ) @ Xs ) ) ) ) ) ).

% prefix_free_set_maximal_list_ob
thf(fact_1028_finite__set__elem__maximal__extension__ex,axiom,
    ! [Xs: list_P903359562653991662od_b_c,S: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ S )
     => ( ( finite3074115686814133143od_b_c @ S )
       => ? [Ys2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys2 ) @ S )
            & ~ ? [Zs4: list_P903359562653991662od_b_c] :
                  ( ( Zs4 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Xs @ ( append2547753245680614915od_b_c @ Ys2 @ Zs4 ) ) @ S ) ) ) ) ) ).

% finite_set_elem_maximal_extension_ex
thf(fact_1029_finite__snd__helper,axiom,
    ! [Xs: set_Pr2298904015924139405od_b_c,Q: b,P2: c] :
      ( ( finite9181748633584594902od_b_c @ Xs )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [Z3: list_P903359562653991662od_b_c] : ( member2892761686137210070od_b_c @ ( produc1467100693395308445od_b_c @ ( product_Pair_b_c @ Q @ P2 ) @ Z3 ) @ Xs ) ) ) ) ).

% finite_snd_helper
thf(fact_1030_finite__snd__helper,axiom,
    ! [Xs: set_Pr4217935593127992605_c_nat,Q: b,P2: c] :
      ( ( finite6396338376473873022_c_nat @ Xs )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Z3: nat] : ( member4582144196051285374_c_nat @ ( produc573923733243432673_c_nat @ ( product_Pair_b_c @ Q @ P2 ) @ Z3 ) @ Xs ) ) ) ) ).

% finite_snd_helper
thf(fact_1031_finite__lists__length__eq,axiom,
    ! [A2: set_Product_prod_b_c,N: nat] :
      ( ( finite3757003021641768849od_b_c @ A2 )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [Xs2: list_P903359562653991662od_b_c] :
              ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs2 ) @ A2 )
              & ( ( size_s3392097710323735898od_b_c @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1032_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1033_finite__lists__length__eq,axiom,
    ! [A2: set_li6436108459499378894od_b_c,N: nat] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( finite1489335766318066461od_b_c
        @ ( collec686158730324120671od_b_c
          @ ^ [Xs2: list_l8907847357763382004od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ A2 )
              & ( ( size_s4150533072628932448od_b_c @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1034_finite__lists__length__le,axiom,
    ! [A2: set_Product_prod_b_c,N: nat] :
      ( ( finite3757003021641768849od_b_c @ A2 )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [Xs2: list_P903359562653991662od_b_c] :
              ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3392097710323735898od_b_c @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1035_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1036_finite__lists__length__le,axiom,
    ! [A2: set_li6436108459499378894od_b_c,N: nat] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( finite1489335766318066461od_b_c
        @ ( collec686158730324120671od_b_c
          @ ^ [Xs2: list_l8907847357763382004od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s4150533072628932448od_b_c @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1037_prefixes__finite,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( finite3074115686814133143od_b_c
      @ ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
            & ( P @ X ) ) ) ) ).

% prefixes_finite
thf(fact_1038_acyclic__alt__def,axiom,
    ( acyclic_a_b_c
    = ( ^ [M2: fsm_a_b_c] : ( finite3074115686814133143od_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ) ).

% acyclic_alt_def
thf(fact_1039_language__finite,axiom,
    ! [M: fsm_a_b_c,K3: nat] :
      ( finite3074115686814133143od_b_c
      @ ( collec2280997390073109977od_b_c
        @ ^ [Io: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
            & ( ord_less_eq_nat @ ( size_s3392097710323735898od_b_c @ Io ) @ K3 ) ) ) ) ).

% language_finite
thf(fact_1040_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1041_finite__Collect__subsets,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( finite1374199133651033463od_b_c
        @ ( collec1823247286932916153od_b_c
          @ ^ [B2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ B2 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1042_finite__Collect__disjI,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( P @ X )
              | ( Q4 @ X ) ) ) )
      = ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
        & ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1043_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q4 @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q4 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1044_finite__Collect__conjI,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
        | ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) ) )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( P @ X )
              & ( Q4 @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1045_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q4: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q4 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q4 @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1046_finite__Collect__le__nat,axiom,
    ! [K3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K3 ) ) ) ).

% finite_Collect_le_nat
thf(fact_1047_finite__Collect__bounded__ex,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
              ? [Y2: list_P903359562653991662od_b_c] :
                ( ( P @ Y2 )
                & ( Q4 @ X @ Y2 ) ) ) )
        = ( ! [Y2: list_P903359562653991662od_b_c] :
              ( ( P @ Y2 )
             => ( finite3074115686814133143od_b_c
                @ ( collec2280997390073109977od_b_c
                  @ ^ [X: list_P903359562653991662od_b_c] : ( Q4 @ X @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1048_finite__Collect__bounded__ex,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: nat > list_P903359562653991662od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y2: list_P903359562653991662od_b_c] :
                ( ( P @ Y2 )
                & ( Q4 @ X @ Y2 ) ) ) )
        = ( ! [Y2: list_P903359562653991662od_b_c] :
              ( ( P @ Y2 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q4 @ X @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1049_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q4: list_P903359562653991662od_b_c > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [X: list_P903359562653991662od_b_c] :
              ? [Y2: nat] :
                ( ( P @ Y2 )
                & ( Q4 @ X @ Y2 ) ) ) )
        = ( ! [Y2: nat] :
              ( ( P @ Y2 )
             => ( finite3074115686814133143od_b_c
                @ ( collec2280997390073109977od_b_c
                  @ ^ [X: list_P903359562653991662od_b_c] : ( Q4 @ X @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1050_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q4: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y2: nat] :
                ( ( P @ Y2 )
                & ( Q4 @ X @ Y2 ) ) ) )
        = ( ! [Y2: nat] :
              ( ( P @ Y2 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q4 @ X @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1051_pigeonhole__infinite__rel,axiom,
    ! [A2: set_a,B: set_nat,R: a > nat > $o] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ A2 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B )
                  & ( R @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A2 )
                        & ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1052_pigeonhole__infinite__rel,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_nat,R: list_P903359562653991662od_b_c > nat > $o] :
      ( ~ ( finite3074115686814133143od_b_c @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ A2 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B )
                  & ( R @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite3074115686814133143od_b_c
                  @ ( collec2280997390073109977od_b_c
                    @ ^ [A4: list_P903359562653991662od_b_c] :
                        ( ( member6330420149250801815od_b_c @ A4 @ A2 )
                        & ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1053_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B )
                  & ( R @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1054_not__finite__existsD,axiom,
    ! [P: list_P903359562653991662od_b_c > $o] :
      ( ~ ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ? [X_1: list_P903359562653991662od_b_c] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1055_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1056_finite__has__maximal2,axiom,
    ! [A2: set_a,A: a] :
      ( ( finite_finite_a @ A2 )
     => ( ( member_a @ A @ A2 )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A2 )
            & ( ord_less_eq_a @ A @ X3 )
            & ! [Xa3: a] :
                ( ( member_a @ Xa3 @ A2 )
               => ( ( ord_less_eq_a @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1057_finite__has__maximal2,axiom,
    ! [A2: set_se3924713247505902254od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A2 )
     => ( ( member6985331446368301687od_b_c @ A @ A2 )
       => ? [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ A2 )
            & ( ord_le282488521294790766od_b_c @ A @ X3 )
            & ! [Xa3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa3 @ A2 )
               => ( ( ord_le282488521294790766od_b_c @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1058_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1059_finite__has__minimal2,axiom,
    ! [A2: set_a,A: a] :
      ( ( finite_finite_a @ A2 )
     => ( ( member_a @ A @ A2 )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A2 )
            & ( ord_less_eq_a @ X3 @ A )
            & ! [Xa3: a] :
                ( ( member_a @ Xa3 @ A2 )
               => ( ( ord_less_eq_a @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1060_finite__has__minimal2,axiom,
    ! [A2: set_se3924713247505902254od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A2 )
     => ( ( member6985331446368301687od_b_c @ A @ A2 )
       => ? [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ A2 )
            & ( ord_le282488521294790766od_b_c @ X3 @ A )
            & ! [Xa3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa3 @ A2 )
               => ( ( ord_le282488521294790766od_b_c @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1061_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A2 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1062_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_1063_finite__subset,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( finite3074115686814133143od_b_c @ A2 ) ) ) ).

% finite_subset
thf(fact_1064_infinite__super,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T3 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_1065_infinite__super,axiom,
    ! [S: set_li6436108459499378894od_b_c,T3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ S @ T3 )
     => ( ~ ( finite3074115686814133143od_b_c @ S )
       => ~ ( finite3074115686814133143od_b_c @ T3 ) ) ) ).

% infinite_super
thf(fact_1066_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_1067_rev__finite__subset,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ B )
       => ( finite3074115686814133143od_b_c @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1068_finite__image__set2,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) )
       => ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [Uu2: list_P903359562653991662od_b_c] :
              ? [X: list_P903359562653991662od_b_c,Y2: list_P903359562653991662od_b_c] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1069_finite__image__set2,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > nat] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu2: nat] :
              ? [X: list_P903359562653991662od_b_c,Y2: list_P903359562653991662od_b_c] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1070_finite__image__set2,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: nat > $o,F: list_P903359562653991662od_b_c > nat > list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q4 ) )
       => ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [Uu2: list_P903359562653991662od_b_c] :
              ? [X: list_P903359562653991662od_b_c,Y2: nat] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1071_finite__image__set2,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: nat > $o,F: list_P903359562653991662od_b_c > nat > nat] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q4 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu2: nat] :
              ? [X: list_P903359562653991662od_b_c,Y2: nat] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1072_finite__image__set2,axiom,
    ! [P: nat > $o,Q4: list_P903359562653991662od_b_c > $o,F: nat > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) )
       => ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [Uu2: list_P903359562653991662od_b_c] :
              ? [X: nat,Y2: list_P903359562653991662od_b_c] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1073_finite__image__set2,axiom,
    ! [P: nat > $o,Q4: list_P903359562653991662od_b_c > $o,F: nat > list_P903359562653991662od_b_c > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ Q4 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu2: nat] :
              ? [X: nat,Y2: list_P903359562653991662od_b_c] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1074_finite__image__set2,axiom,
    ! [P: nat > $o,Q4: nat > $o,F: nat > nat > list_P903359562653991662od_b_c] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q4 ) )
       => ( finite3074115686814133143od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [Uu2: list_P903359562653991662od_b_c] :
              ? [X: nat,Y2: nat] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1075_finite__image__set2,axiom,
    ! [P: nat > $o,Q4: nat > $o,F: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q4 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu2: nat] :
              ? [X: nat,Y2: nat] :
                ( ( Uu2
                  = ( F @ X @ Y2 ) )
                & ( P @ X )
                & ( Q4 @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1076_finite__image__set,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [X: list_P903359562653991662od_b_c] :
              ( ( Uu2
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_1077_finite__image__set,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,F: list_P903359562653991662od_b_c > nat] :
      ( ( finite3074115686814133143od_b_c @ ( collec2280997390073109977od_b_c @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu2: nat] :
            ? [X: list_P903359562653991662od_b_c] :
              ( ( Uu2
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_1078_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > list_P903359562653991662od_b_c] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite3074115686814133143od_b_c
        @ ( collec2280997390073109977od_b_c
          @ ^ [Uu2: list_P903359562653991662od_b_c] :
            ? [X: nat] :
              ( ( Uu2
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_1079_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu2: nat] :
            ? [X: nat] :
              ( ( Uu2
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_1080_finite__set__of__finite__funs,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,D2: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( finite883499132320816252od_b_c
          @ ( collec7330361759883418810od_b_c
            @ ^ [F3: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
              ! [X: list_P903359562653991662od_b_c] :
                ( ( ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( member6330420149250801815od_b_c @ ( F3 @ X ) @ B ) )
                & ( ~ ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1081_finite__set__of__finite__funs,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_a,D2: a] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ( finite_finite_a @ B )
       => ( finite924386208366894348_b_c_a
          @ ( collec7879005945056494154_b_c_a
            @ ^ [F3: list_P903359562653991662od_b_c > a] :
              ! [X: list_P903359562653991662od_b_c] :
                ( ( ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( member_a @ ( F3 @ X ) @ B ) )
                & ( ~ ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1082_finite__set__of__finite__funs,axiom,
    ! [A2: set_a,B: set_li6436108459499378894od_b_c,D2: list_P903359562653991662od_b_c] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( finite1425346576647452012od_b_c
          @ ( collec8379966313337051818od_b_c
            @ ^ [F3: a > list_P903359562653991662od_b_c] :
              ! [X: a] :
                ( ( ( member_a @ X @ A2 )
                 => ( member6330420149250801815od_b_c @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_a @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1083_finite__set__of__finite__funs,axiom,
    ! [A2: set_a,B: set_a,D2: a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a_a
          @ ( collect_a_a
            @ ^ [F3: a > a] :
              ! [X: a] :
                ( ( ( member_a @ X @ A2 )
                 => ( member_a @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_a @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1084_finite__set__of__finite__funs,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_nat,D2: nat] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( finite2939032633798213464_c_nat
          @ ( collec2331653575325689498_c_nat
            @ ^ [F3: list_P903359562653991662od_b_c > nat] :
              ! [X: list_P903359562653991662od_b_c] :
                ( ( ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( member_nat @ ( F3 @ X ) @ B ) )
                & ( ~ ( member6330420149250801815od_b_c @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1085_finite__set__of__finite__funs,axiom,
    ! [A2: set_a,B: set_nat,D2: nat] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_a_nat
          @ ( collect_a_nat
            @ ^ [F3: a > nat] :
              ! [X: a] :
                ( ( ( member_a @ X @ A2 )
                 => ( member_nat @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_a @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1086_finite__set__of__finite__funs,axiom,
    ! [A2: set_nat,B: set_li6436108459499378894od_b_c,D2: list_P903359562653991662od_b_c] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( finite4158422841392913406od_b_c
          @ ( collec3551043782920389440od_b_c
            @ ^ [F3: nat > list_P903359562653991662od_b_c] :
              ! [X: nat] :
                ( ( ( member_nat @ X @ A2 )
                 => ( member6330420149250801815od_b_c @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_nat @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1087_finite__set__of__finite__funs,axiom,
    ! [A2: set_nat,B: set_a,D2: a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_nat_a
          @ ( collect_nat_a
            @ ^ [F3: nat > a] :
              ! [X: nat] :
                ( ( ( member_nat @ X @ A2 )
                 => ( member_a @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_nat @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1088_finite__set__of__finite__funs,axiom,
    ! [A2: set_nat,B: set_nat,D2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( finite2115694454571419734at_nat
          @ ( collect_nat_nat
            @ ^ [F3: nat > nat] :
              ! [X: nat] :
                ( ( ( member_nat @ X @ A2 )
                 => ( member_nat @ ( F3 @ X ) @ B ) )
                & ( ~ ( member_nat @ X @ A2 )
                 => ( ( F3 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1089_finite__has__minimal,axiom,
    ! [A2: set_se3924713247505902254od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A2 )
     => ( ( A2 != bot_bo2794119844231891738od_b_c )
       => ? [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ A2 )
            & ! [Xa3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa3 @ A2 )
               => ( ( ord_le282488521294790766od_b_c @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1090_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A2 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1091_finite__has__maximal,axiom,
    ! [A2: set_se3924713247505902254od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A2 )
     => ( ( A2 != bot_bo2794119844231891738od_b_c )
       => ? [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ A2 )
            & ! [Xa3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa3 @ A2 )
               => ( ( ord_le282488521294790766od_b_c @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1092_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1093_inter__Set__filter,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B )
        = ( filter4546979855996260255od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ A2 )
          @ B ) ) ) ).

% inter_Set_filter
thf(fact_1094_inter__Set__filter,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( inf_inf_set_a @ A2 @ B )
        = ( filter_a
          @ ^ [X: a] : ( member_a @ X @ A2 )
          @ B ) ) ) ).

% inter_Set_filter
thf(fact_1095_inter__Set__filter,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( inf_inf_set_nat @ A2 @ B )
        = ( filter_nat
          @ ^ [X: nat] : ( member_nat @ X @ A2 )
          @ B ) ) ) ).

% inter_Set_filter
thf(fact_1096_finite__subset__induct_H,axiom,
    ! [F4: set_a,A2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F4 )
     => ( ( ord_less_eq_set_a @ F4 @ A2 )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A5: a,F5: set_a] :
                ( ( finite_finite_a @ F5 )
               => ( ( member_a @ A5 @ A2 )
                 => ( ( ord_less_eq_set_a @ F5 @ A2 )
                   => ( ~ ( member_a @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_a @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1097_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F5 @ A2 )
                   => ( ~ ( member_nat @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1098_finite__subset__induct_H,axiom,
    ! [F4: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F4 )
     => ( ( ord_le282488521294790766od_b_c @ F4 @ A2 )
       => ( ( P @ bot_bo4166481423041325370od_b_c )
         => ( ! [A5: list_P903359562653991662od_b_c,F5: set_li6436108459499378894od_b_c] :
                ( ( finite3074115686814133143od_b_c @ F5 )
               => ( ( member6330420149250801815od_b_c @ A5 @ A2 )
                 => ( ( ord_le282488521294790766od_b_c @ F5 @ A2 )
                   => ( ~ ( member6330420149250801815od_b_c @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert6227932334100060350od_b_c @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1099_finite__subset__induct,axiom,
    ! [F4: set_a,A2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F4 )
     => ( ( ord_less_eq_set_a @ F4 @ A2 )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A5: a,F5: set_a] :
                ( ( finite_finite_a @ F5 )
               => ( ( member_a @ A5 @ A2 )
                 => ( ~ ( member_a @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_a @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1100_finite__subset__induct,axiom,
    ! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ~ ( member_nat @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1101_finite__subset__induct,axiom,
    ! [F4: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F4 )
     => ( ( ord_le282488521294790766od_b_c @ F4 @ A2 )
       => ( ( P @ bot_bo4166481423041325370od_b_c )
         => ( ! [A5: list_P903359562653991662od_b_c,F5: set_li6436108459499378894od_b_c] :
                ( ( finite3074115686814133143od_b_c @ F5 )
               => ( ( member6330420149250801815od_b_c @ A5 @ A2 )
                 => ( ~ ( member6330420149250801815od_b_c @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert6227932334100060350od_b_c @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1102_remove__induct,axiom,
    ! [P: set_a > $o,B: set_a] :
      ( ( P @ bot_bot_set_a )
     => ( ( ~ ( finite_finite_a @ B )
         => ( P @ B ) )
       => ( ! [A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ( A7 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A7 @ B )
                 => ( ! [X5: a] :
                        ( ( member_a @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1103_remove__induct,axiom,
    ! [P: set_nat > $o,B: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B )
         => ( P @ B ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1104_remove__induct,axiom,
    ! [P: set_li6436108459499378894od_b_c > $o,B: set_li6436108459499378894od_b_c] :
      ( ( P @ bot_bo4166481423041325370od_b_c )
     => ( ( ~ ( finite3074115686814133143od_b_c @ B )
         => ( P @ B ) )
       => ( ! [A7: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ A7 )
             => ( ( A7 != bot_bo4166481423041325370od_b_c )
               => ( ( ord_le282488521294790766od_b_c @ A7 @ B )
                 => ( ! [X5: list_P903359562653991662od_b_c] :
                        ( ( member6330420149250801815od_b_c @ X5 @ A7 )
                       => ( P @ ( minus_5375846548645262261od_b_c @ A7 @ ( insert6227932334100060350od_b_c @ X5 @ bot_bo4166481423041325370od_b_c ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1105_finite__remove__induct,axiom,
    ! [B: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ B )
     => ( ( P @ bot_bot_set_a )
       => ( ! [A7: set_a] :
              ( ( finite_finite_a @ A7 )
             => ( ( A7 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A7 @ B )
                 => ( ! [X5: a] :
                        ( ( member_a @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1106_finite__remove__induct,axiom,
    ! [B: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1107_finite__remove__induct,axiom,
    ! [B: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( P @ bot_bo4166481423041325370od_b_c )
       => ( ! [A7: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ A7 )
             => ( ( A7 != bot_bo4166481423041325370od_b_c )
               => ( ( ord_le282488521294790766od_b_c @ A7 @ B )
                 => ( ! [X5: list_P903359562653991662od_b_c] :
                        ( ( member6330420149250801815od_b_c @ X5 @ A7 )
                       => ( P @ ( minus_5375846548645262261od_b_c @ A7 @ ( insert6227932334100060350od_b_c @ X5 @ bot_bo4166481423041325370od_b_c ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1108_finite__ranking__induct,axiom,
    ! [S: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o,F: list_P903359562653991662od_b_c > nat] :
      ( ( finite3074115686814133143od_b_c @ S )
     => ( ( P @ bot_bo4166481423041325370od_b_c )
       => ( ! [X3: list_P903359562653991662od_b_c,S4: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ S4 )
             => ( ! [Y5: list_P903359562653991662od_b_c] :
                    ( ( member6330420149250801815od_b_c @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert6227932334100060350od_b_c @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1109_finite__ranking__induct,axiom,
    ! [S: set_a,P: set_a > $o,F: a > nat] :
      ( ( finite_finite_a @ S )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,S4: set_a] :
              ( ( finite_finite_a @ S4 )
             => ( ! [Y5: a] :
                    ( ( member_a @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_a @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1110_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1111_paths__for__io_H__set,axiom,
    ! [Q: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Io3: list_P903359562653991662od_b_c,Prev2: list_P8364599791545511612od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M ) )
     => ( ( paths_4891704155273680592_c_b_c @ ( h_list3189254218192121498_c_b_c @ M ) @ Io3 @ Q @ Prev2 )
        = ( collec1242728128799169191od_b_c
          @ ^ [Uu2: list_P8364599791545511612od_b_c] :
            ? [P9: list_P8364599791545511612od_b_c] :
              ( ( Uu2
                = ( append8574118600258580689od_b_c @ Prev2 @ P9 ) )
              & ( path_l6977963524759401717_c_b_c @ M @ Q @ P9 )
              & ( ( map_Pr4814182253694984182od_b_c
                  @ ^ [T2: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T2 ) ) ) )
                  @ P9 )
                = Io3 ) ) ) ) ) ).

% paths_for_io'_set
thf(fact_1112_paths__for__io_H__set,axiom,
    ! [Q: a,M: fsm_a_b_c,Io3: list_P903359562653991662od_b_c,Prev2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( paths_for_io_a_b_c2 @ ( h_a_b_c @ M ) @ Io3 @ Q @ Prev2 )
        = ( collec6273869032445462695od_c_a
          @ ^ [Uu2: list_P6327159017948738492od_c_a] :
            ? [P9: list_P6327159017948738492od_c_a] :
              ( ( Uu2
                = ( append3884164603879732689od_c_a @ Prev2 @ P9 ) )
              & ( path_a_b_c @ M @ Q @ P9 )
              & ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T2: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
                  @ P9 )
                = Io3 ) ) ) ) ) ).

% paths_for_io'_set
thf(fact_1113_paths__for__io__set__code,axiom,
    ( paths_for_io_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,Io: list_P903359562653991662od_b_c] : ( if_set5301400340663253090od_c_a @ ( member_a @ Q6 @ ( states_a_b_c @ M2 ) ) @ ( paths_for_io_a_b_c2 @ ( h_a_b_c @ M2 ) @ Io @ Q6 @ nil_Pr1342775757158464060od_c_a ) @ bot_bo6236370880139903240od_c_a ) ) ) ).

% paths_for_io_set_code
thf(fact_1114_observable__alt__def__h,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q1: a,X: b,Yq: product_prod_c_a,Yq2: product_prod_c_a] :
          ( ( ( member5074992350434858958od_c_a @ Yq @ ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q1 @ X ) ) )
            & ( member5074992350434858958od_c_a @ Yq2 @ ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q1 @ X ) ) ) )
         => ( ( ( product_fst_c_a @ Yq )
              = ( product_fst_c_a @ Yq2 ) )
           => ( ( product_snd_c_a @ Yq )
              = ( product_snd_c_a @ Yq2 ) ) ) ) ) ) ).

% observable_alt_def_h
thf(fact_1115_finite__less__ub,axiom,
    ! [F: nat > nat,U2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).

% finite_less_ub
thf(fact_1116_image__eqI,axiom,
    ! [B3: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ A2 )
       => ( member6330420149250801815od_b_c @ B3 @ ( image_2850395784482391045od_b_c @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1117_image__eqI,axiom,
    ! [B3: a,F: list_P903359562653991662od_b_c > a,X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ A2 )
       => ( member_a @ B3 @ ( image_3173854338463926933_b_c_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1118_image__eqI,axiom,
    ! [B3: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,X2: a,A2: set_a] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A2 )
       => ( member6330420149250801815od_b_c @ B3 @ ( image_8656872703560917237od_b_c @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1119_image__eqI,axiom,
    ! [B3: a,F: a > a,X2: a,A2: set_a] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A2 )
       => ( member_a @ B3 @ ( image_a_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1120_list_Oset__map,axiom,
    ! [F: product_prod_b_c > b,V2: list_P903359562653991662od_b_c] :
      ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ V2 ) )
      = ( image_7762865866877615260_b_c_b @ F @ ( set_Product_prod_b_c2 @ V2 ) ) ) ).

% list.set_map
thf(fact_1121_list_Oset__map,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,V2: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ V2 ) )
      = ( image_2850395784482391045od_b_c @ F @ ( set_li4480668622519654659od_b_c @ V2 ) ) ) ).

% list.set_map
thf(fact_1122_list_Oset__map,axiom,
    ! [F: product_prod_b_c > list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ V2 ) )
      = ( image_5872333176446027019od_b_c @ F @ ( set_Product_prod_b_c2 @ V2 ) ) ) ).

% list.set_map
thf(fact_1123_list_Oset__map,axiom,
    ! [F: list_P903359562653991662od_b_c > product_prod_b_c,V2: list_l8907847357763382004od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( map_li8582383987200892350od_b_c @ F @ V2 ) )
      = ( image_4511654458200770303od_b_c @ F @ ( set_li4480668622519654659od_b_c @ V2 ) ) ) ).

% list.set_map
thf(fact_1124_list_Oset__map,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,V2: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ V2 ) )
      = ( image_3117045938558108421od_b_c @ F @ ( set_Product_prod_b_c2 @ V2 ) ) ) ).

% list.set_map
thf(fact_1125_finite__UN__I,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c > set_nat] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ! [A5: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ A5 @ A2 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_7944048173531155503et_nat @ B @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1126_finite__UN__I,axiom,
    ! [A2: set_a,B: a > set_nat] :
      ( ( finite_finite_a @ A2 )
     => ( ! [A5: a] :
            ( ( member_a @ A5 @ A2 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1127_finite__UN__I,axiom,
    ! [A2: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1128_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_1129_finite__surj,axiom,
    ! [A2: set_nat,B: set_li6436108459499378894od_b_c,F: nat > list_P903359562653991662od_b_c] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ B @ ( image_2316969277338295127od_b_c @ F @ A2 ) )
       => ( finite3074115686814133143od_b_c @ B ) ) ) ).

% finite_surj
thf(fact_1130_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 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A2 )
            & ( finite_finite_nat @ C6 )
            & ( B
              = ( image_nat_nat @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1131_finite__subset__image,axiom,
    ! [B: set_nat,F: list_P903359562653991662od_b_c > nat,A2: set_li6436108459499378894od_b_c] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ ( image_4545125072327328889_c_nat @ F @ A2 ) )
       => ? [C6: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ C6 @ A2 )
            & ( finite3074115686814133143od_b_c @ C6 )
            & ( B
              = ( image_4545125072327328889_c_nat @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1132_finite__subset__image,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: nat > list_P903359562653991662od_b_c,A2: set_nat] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ ( image_2316969277338295127od_b_c @ F @ A2 ) )
       => ? [C6: set_nat] :
            ( ( ord_less_eq_set_nat @ C6 @ A2 )
            & ( finite_finite_nat @ C6 )
            & ( B
              = ( image_2316969277338295127od_b_c @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1133_finite__subset__image,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
       => ? [C6: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ C6 @ A2 )
            & ( finite3074115686814133143od_b_c @ C6 )
            & ( B
              = ( image_2850395784482391045od_b_c @ F @ C6 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1134_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A2 )
            & ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1135_ex__finite__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A2: set_li6436108459499378894od_b_c,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_4545125072327328889_c_nat @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_li6436108459499378894od_b_c] :
            ( ( finite3074115686814133143od_b_c @ B2 )
            & ( ord_le282488521294790766od_b_c @ B2 @ A2 )
            & ( P @ ( image_4545125072327328889_c_nat @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1136_ex__finite__subset__image,axiom,
    ! [F: nat > list_P903359562653991662od_b_c,A2: set_nat,P: set_li6436108459499378894od_b_c > $o] :
      ( ( ? [B2: set_li6436108459499378894od_b_c] :
            ( ( finite3074115686814133143od_b_c @ B2 )
            & ( ord_le282488521294790766od_b_c @ B2 @ ( image_2316969277338295127od_b_c @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A2 )
            & ( P @ ( image_2316969277338295127od_b_c @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1137_ex__finite__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( ? [B2: set_li6436108459499378894od_b_c] :
            ( ( finite3074115686814133143od_b_c @ B2 )
            & ( ord_le282488521294790766od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_li6436108459499378894od_b_c] :
            ( ( finite3074115686814133143od_b_c @ B2 )
            & ( ord_le282488521294790766od_b_c @ B2 @ A2 )
            & ( P @ ( image_2850395784482391045od_b_c @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1138_all__finite__subset__image,axiom,
    ! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A2 ) )
           => ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1139_all__finite__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A2: set_li6436108459499378894od_b_c,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_4545125072327328889_c_nat @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ( finite3074115686814133143od_b_c @ B2 )
              & ( ord_le282488521294790766od_b_c @ B2 @ A2 ) )
           => ( P @ ( image_4545125072327328889_c_nat @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1140_all__finite__subset__image,axiom,
    ! [F: nat > list_P903359562653991662od_b_c,A2: set_nat,P: set_li6436108459499378894od_b_c > $o] :
      ( ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ( finite3074115686814133143od_b_c @ B2 )
              & ( ord_le282488521294790766od_b_c @ B2 @ ( image_2316969277338295127od_b_c @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A2 ) )
           => ( P @ ( image_2316969277338295127od_b_c @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1141_all__finite__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ( finite3074115686814133143od_b_c @ B2 )
              & ( ord_le282488521294790766od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A2 ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ( finite3074115686814133143od_b_c @ B2 )
              & ( ord_le282488521294790766od_b_c @ B2 @ A2 ) )
           => ( P @ ( image_2850395784482391045od_b_c @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1142_pigeonhole__infinite,axiom,
    ! [A2: set_a,F: a > nat] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finite_finite_nat @ ( image_a_nat @ F @ A2 ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A2 )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A4: a] :
                      ( ( member_a @ A4 @ A2 )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1143_pigeonhole__infinite,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > nat] :
      ( ~ ( finite3074115686814133143od_b_c @ A2 )
     => ( ( finite_finite_nat @ ( image_4545125072327328889_c_nat @ F @ A2 ) )
       => ? [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
            & ~ ( finite3074115686814133143od_b_c
                @ ( collec2280997390073109977od_b_c
                  @ ^ [A4: list_P903359562653991662od_b_c] :
                      ( ( member6330420149250801815od_b_c @ A4 @ A2 )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1144_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A4: nat] :
                      ( ( member_nat @ A4 @ A2 )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1145_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu2: list_P903359562653991662od_b_c] :
          ? [X: list_P903359562653991662od_b_c] :
            ( ( Uu2
              = ( F @ X ) )
            & ( member6330420149250801815od_b_c @ X @ A2 ) ) )
      = ( image_2850395784482391045od_b_c @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1146_Setcompr__eq__image,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A2: set_a] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu2: list_P903359562653991662od_b_c] :
          ? [X: a] :
            ( ( Uu2
              = ( F @ X ) )
            & ( member_a @ X @ A2 ) ) )
      = ( image_8656872703560917237od_b_c @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1147_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A2: set_li6436108459499378894od_b_c] :
      ( ( collect_nat
        @ ^ [Uu2: nat] :
          ? [X: list_P903359562653991662od_b_c] :
            ( ( Uu2
              = ( F @ X ) )
            & ( member6330420149250801815od_b_c @ X @ A2 ) ) )
      = ( image_4545125072327328889_c_nat @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1148_Setcompr__eq__image,axiom,
    ! [F: a > nat,A2: set_a] :
      ( ( collect_nat
        @ ^ [Uu2: nat] :
          ? [X: a] :
            ( ( Uu2
              = ( F @ X ) )
            & ( member_a @ X @ A2 ) ) )
      = ( image_a_nat @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1149_setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu2: list_P903359562653991662od_b_c] :
          ? [X: list_P903359562653991662od_b_c] :
            ( ( Uu2
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_2850395784482391045od_b_c @ F @ ( collec2280997390073109977od_b_c @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1150_setcompr__eq__image,axiom,
    ! [F: nat > list_P903359562653991662od_b_c,P: nat > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu2: list_P903359562653991662od_b_c] :
          ? [X: nat] :
            ( ( Uu2
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_2316969277338295127od_b_c @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1151_setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,P: list_P903359562653991662od_b_c > $o] :
      ( ( collect_nat
        @ ^ [Uu2: nat] :
          ? [X: list_P903359562653991662od_b_c] :
            ( ( Uu2
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_4545125072327328889_c_nat @ F @ ( collec2280997390073109977od_b_c @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1152_setcompr__eq__image,axiom,
    ! [F: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu2: nat] :
          ? [X: nat] :
            ( ( Uu2
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1153_imageE,axiom,
    ! [B3: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ A2 ) ) ) ).

% imageE
thf(fact_1154_imageE,axiom,
    ! [B3: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,A2: set_a] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( image_8656872703560917237od_b_c @ F @ A2 ) )
     => ~ ! [X3: a] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A2 ) ) ) ).

% imageE
thf(fact_1155_imageE,axiom,
    ! [B3: a,F: list_P903359562653991662od_b_c > a,A2: set_li6436108459499378894od_b_c] :
      ( ( member_a @ B3 @ ( image_3173854338463926933_b_c_a @ F @ A2 ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ A2 ) ) ) ).

% imageE
thf(fact_1156_imageE,axiom,
    ! [B3: a,F: a > a,A2: set_a] :
      ( ( member_a @ B3 @ ( image_a_a @ F @ A2 ) )
     => ~ ! [X3: a] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A2 ) ) ) ).

% imageE
thf(fact_1157_Compr__image__eq,axiom,
    ! [F: a > a,A2: set_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_a_a @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1158_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > a,A2: set_li6436108459499378894od_b_c,P: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_3173854338463926933_b_c_a @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_3173854338463926933_b_c_a @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1159_Compr__image__eq,axiom,
    ! [F: nat > a,A2: set_nat,P: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_nat_a @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_a @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1160_Compr__image__eq,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A2: set_a,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( image_8656872703560917237od_b_c @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_8656872703560917237od_b_c @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1161_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_2850395784482391045od_b_c @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1162_Compr__image__eq,axiom,
    ! [F: nat > list_P903359562653991662od_b_c,A2: set_nat,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ ( image_2316969277338295127od_b_c @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_2316969277338295127od_b_c @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1163_Compr__image__eq,axiom,
    ! [F: a > nat,A2: set_a,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_a_nat @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_a_nat @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1164_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A2: set_li6436108459499378894od_b_c,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_4545125072327328889_c_nat @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_4545125072327328889_c_nat @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1165_Compr__image__eq,axiom,
    ! [F: nat > nat,A2: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1166_image__set,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( image_2850395784482391045od_b_c @ F @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_1167_image__set,axiom,
    ! [F: list_P903359562653991662od_b_c > product_prod_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( image_4511654458200770303od_b_c @ F @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( set_Product_prod_b_c2 @ ( map_li8582383987200892350od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_1168_image__set,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ F @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_1169_image__set,axiom,
    ! [F: product_prod_b_c > list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_5872333176446027019od_b_c @ F @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( set_li4480668622519654659od_b_c @ ( map_Pr719690668591373258od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_1170_image__set,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_3117045938558108421od_b_c @ F @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( set_Product_prod_b_c2 @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_1171_imageI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( image_2850395784482391045od_b_c @ F @ A2 ) ) ) ).

% imageI
thf(fact_1172_imageI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( member_a @ ( F @ X2 ) @ ( image_3173854338463926933_b_c_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_1173_imageI,axiom,
    ! [X2: a,A2: set_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ A2 )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( image_8656872703560917237od_b_c @ F @ A2 ) ) ) ).

% imageI
thf(fact_1174_imageI,axiom,
    ! [X2: a,A2: set_a,F: a > a] :
      ( ( member_a @ X2 @ A2 )
     => ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_1175_rev__image__eqI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member6330420149250801815od_b_c @ B3 @ ( image_2850395784482391045od_b_c @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1176_rev__image__eqI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: a,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ A2 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_a @ B3 @ ( image_3173854338463926933_b_c_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1177_rev__image__eqI,axiom,
    ! [X2: a,A2: set_a,B3: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ A2 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member6330420149250801815od_b_c @ B3 @ ( image_8656872703560917237od_b_c @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1178_rev__image__eqI,axiom,
    ! [X2: a,A2: set_a,B3: a,F: a > a] :
      ( ( member_a @ X2 @ A2 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_a @ B3 @ ( image_a_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1179_subset__image__iff,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
      = ( ? [AA: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ AA @ A2 )
            & ( B
              = ( image_2850395784482391045od_b_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1180_subset__imageE,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
     => ~ ! [C6: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ C6 @ A2 )
           => ( B
             != ( image_2850395784482391045od_b_c @ F @ C6 ) ) ) ) ).

% subset_imageE
thf(fact_1181_image__subsetI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a,B: set_a] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A2 )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_3173854338463926933_b_c_a @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_1182_image__subsetI,axiom,
    ! [A2: set_a,F: a > a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_1183_image__subsetI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A2 )
         => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B ) )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_1184_image__subsetI,axiom,
    ! [A2: set_a,F: a > list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B ) )
     => ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ A2 ) @ B ) ) ).

% image_subsetI
thf(fact_1185_image__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A2 ) @ ( image_2850395784482391045od_b_c @ F @ B ) ) ) ).

% image_mono
thf(fact_1186_all__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A2 ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
           => ( P @ ( image_2850395784482391045od_b_c @ F @ B2 ) ) ) ) ) ).

% all_subset_image
thf(fact_1187_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1188_dom__map__entry,axiom,
    ! [K3: b,F: c > c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_entry_b_c @ K3 @ F @ Xs ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% dom_map_entry
thf(fact_1189_dom__map__default,axiom,
    ! [K3: b,V2: c,F: c > c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_default_b_c @ K3 @ V2 @ F @ Xs ) ) )
      = ( insert_b @ K3 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% dom_map_default
thf(fact_1190_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M5: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_eq_nat @ X @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1191_UN__I,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( member6330420149250801815od_b_c @ B3 @ ( B @ A ) )
       => ( member6330420149250801815od_b_c @ B3 @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ B @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_1192_UN__I,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B3: a,B: list_P903359562653991662od_b_c > set_a] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ( member_a @ B3 @ ( B @ A ) )
       => ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_6303648431024068597_set_a @ B @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_1193_UN__I,axiom,
    ! [A: a,A2: set_a,B3: list_P903359562653991662od_b_c,B: a > set_li6436108459499378894od_b_c] :
      ( ( member_a @ A @ A2 )
     => ( ( member6330420149250801815od_b_c @ B3 @ ( B @ A ) )
       => ( member6330420149250801815od_b_c @ B3 @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ B @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_1194_UN__I,axiom,
    ! [A: a,A2: set_a,B3: a,B: a > set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( member_a @ B3 @ ( B @ A ) )
       => ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_1195_set__concat,axiom,
    ! [Xs: list_l5832389826979350138od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( concat8136016674533020335od_b_c @ Xs ) )
      = ( comple1250193257546725051od_b_c @ ( image_5994471740994765151od_b_c @ set_li4480668622519654659od_b_c @ ( set_li2649297040355570313od_b_c @ Xs ) ) ) ) ).

% set_concat
thf(fact_1196_set__concat,axiom,
    ! [Xs: list_l8907847357763382004od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( concat3284542956341965481od_b_c @ Xs ) )
      = ( comple7928098250179550773od_b_c @ ( image_2896857402404799071od_b_c @ set_Product_prod_b_c2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_concat
thf(fact_1197_set__Cons__sing__Nil,axiom,
    ! [A2: set_b] :
      ( ( set_Cons_b @ A2 @ ( insert_list_b @ nil_b @ bot_bot_set_list_b ) )
      = ( image_b_list_b
        @ ^ [X: b] : ( cons_b @ X @ nil_b )
        @ A2 ) ) ).

% set_Cons_sing_Nil
thf(fact_1198_set__Cons__sing__Nil,axiom,
    ! [A2: set_Product_prod_b_c] :
      ( ( set_Co7706362885935805726od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) )
      = ( image_5872333176446027019od_b_c
        @ ^ [X: product_prod_b_c] : ( cons_P4529483553340347422od_b_c @ X @ nil_Product_prod_b_c )
        @ A2 ) ) ).

% set_Cons_sing_Nil
thf(fact_1199_None__notin__image__Some,axiom,
    ! [A2: set_a] :
      ~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A2 ) ) ).

% None_notin_image_Some
thf(fact_1200_in__image__insert__iff,axiom,
    ! [B: set_se3924713247505902254od_b_c,X2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ! [C6: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ C6 @ B )
         => ~ ( member6330420149250801815od_b_c @ X2 @ C6 ) )
     => ( ( member6985331446368301687od_b_c @ A2 @ ( image_8181729759120431109od_b_c @ ( insert6227932334100060350od_b_c @ X2 ) @ B ) )
        = ( ( member6330420149250801815od_b_c @ X2 @ A2 )
          & ( member6985331446368301687od_b_c @ ( minus_5375846548645262261od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_1201_in__image__insert__iff,axiom,
    ! [B: set_set_a,X2: a,A2: set_a] :
      ( ! [C6: set_a] :
          ( ( member_set_a @ C6 @ B )
         => ~ ( member_a @ X2 @ C6 ) )
     => ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B ) )
        = ( ( member_a @ X2 @ A2 )
          & ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_1202_Sup__eqI,axiom,
    ! [A2: set_se3924713247505902254od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ! [Y3: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ Y3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ Y3 @ X2 ) )
     => ( ! [Y3: set_li6436108459499378894od_b_c] :
            ( ! [Z6: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Z6 @ A2 )
               => ( ord_le282488521294790766od_b_c @ Z6 @ Y3 ) )
           => ( ord_le282488521294790766od_b_c @ X2 @ Y3 ) )
       => ( ( comple1250193257546725051od_b_c @ A2 )
          = X2 ) ) ) ).

% Sup_eqI
thf(fact_1203_Sup__mono,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] :
      ( ! [A5: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ A5 @ A2 )
         => ? [X5: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ X5 @ B )
              & ( ord_le282488521294790766od_b_c @ A5 @ X5 ) ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Sup_mono
thf(fact_1204_Sup__least,axiom,
    ! [A2: set_se3924713247505902254od_b_c,Z5: set_li6436108459499378894od_b_c] :
      ( ! [X3: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ X3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ X3 @ Z5 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ Z5 ) ) ).

% Sup_least
thf(fact_1205_Sup__upper,axiom,
    ! [X2: set_li6436108459499378894od_b_c,A2: set_se3924713247505902254od_b_c] :
      ( ( member6985331446368301687od_b_c @ X2 @ A2 )
     => ( ord_le282488521294790766od_b_c @ X2 @ ( comple1250193257546725051od_b_c @ A2 ) ) ) ).

% Sup_upper
thf(fact_1206_Sup__le__iff,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ B3 )
      = ( ! [X: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X @ A2 )
           => ( ord_le282488521294790766od_b_c @ X @ B3 ) ) ) ) ).

% Sup_le_iff
thf(fact_1207_Sup__upper2,axiom,
    ! [U2: set_li6436108459499378894od_b_c,A2: set_se3924713247505902254od_b_c,V2: set_li6436108459499378894od_b_c] :
      ( ( member6985331446368301687od_b_c @ U2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ V2 @ U2 )
       => ( ord_le282488521294790766od_b_c @ V2 @ ( comple1250193257546725051od_b_c @ A2 ) ) ) ) ).

% Sup_upper2
thf(fact_1208_Union__subsetI,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] :
      ( ! [X3: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ X3 @ A2 )
         => ? [Y5: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ Y5 @ B )
              & ( ord_le282488521294790766od_b_c @ X3 @ Y5 ) ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Union_subsetI
thf(fact_1209_Union__upper,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: set_se3924713247505902254od_b_c] :
      ( ( member6985331446368301687od_b_c @ B @ A2 )
     => ( ord_le282488521294790766od_b_c @ B @ ( comple1250193257546725051od_b_c @ A2 ) ) ) ).

% Union_upper
thf(fact_1210_Union__least,axiom,
    ! [A2: set_se3924713247505902254od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ! [X9: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ X9 @ A2 )
         => ( ord_le282488521294790766od_b_c @ X9 @ C2 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ C2 ) ) ).

% Union_least
thf(fact_1211_Union__mono,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ A2 @ B )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Union_mono
thf(fact_1212_UN__E,axiom,
    ! [B3: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ B @ A2 ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ~ ( member6330420149250801815od_b_c @ B3 @ ( B @ X3 ) ) ) ) ).

% UN_E
thf(fact_1213_UN__E,axiom,
    ! [B3: list_P903359562653991662od_b_c,B: a > set_li6436108459499378894od_b_c,A2: set_a] :
      ( ( member6330420149250801815od_b_c @ B3 @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ B @ A2 ) ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ~ ( member6330420149250801815od_b_c @ B3 @ ( B @ X3 ) ) ) ) ).

% UN_E
thf(fact_1214_UN__E,axiom,
    ! [B3: a,B: list_P903359562653991662od_b_c > set_a,A2: set_li6436108459499378894od_b_c] :
      ( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_6303648431024068597_set_a @ B @ A2 ) ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ~ ( member_a @ B3 @ ( B @ X3 ) ) ) ) ).

% UN_E
thf(fact_1215_UN__E,axiom,
    ! [B3: a,B: a > set_a,A2: set_a] :
      ( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A2 ) ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ~ ( member_a @ B3 @ ( B @ X3 ) ) ) ) ).

% UN_E
thf(fact_1216_SUP__eq,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,G: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ! [I3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ I3 @ A2 )
         => ? [X5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X5 @ B )
              & ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ ( G @ X5 ) ) ) )
     => ( ! [J2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ J2 @ B )
           => ? [X5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X5 @ A2 )
                & ( ord_le282488521294790766od_b_c @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) )
          = ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ G @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1217_SUP__eq,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_a,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,G: a > set_li6436108459499378894od_b_c] :
      ( ! [I3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ I3 @ A2 )
         => ? [X5: a] :
              ( ( member_a @ X5 @ B )
              & ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ ( G @ X5 ) ) ) )
     => ( ! [J2: a] :
            ( ( member_a @ J2 @ B )
           => ? [X5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X5 @ A2 )
                & ( ord_le282488521294790766od_b_c @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) )
          = ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ G @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1218_SUP__eq,axiom,
    ! [A2: set_a,B: set_li6436108459499378894od_b_c,F: a > set_li6436108459499378894od_b_c,G: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ A2 )
         => ? [X5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X5 @ B )
              & ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ ( G @ X5 ) ) ) )
     => ( ! [J2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ J2 @ B )
           => ? [X5: a] :
                ( ( member_a @ X5 @ A2 )
                & ( ord_le282488521294790766od_b_c @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) )
          = ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ G @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1219_SUP__eq,axiom,
    ! [A2: set_a,B: set_a,F: a > set_li6436108459499378894od_b_c,G: a > set_li6436108459499378894od_b_c] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ A2 )
         => ? [X5: a] :
              ( ( member_a @ X5 @ B )
              & ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ ( G @ X5 ) ) ) )
     => ( ! [J2: a] :
            ( ( member_a @ J2 @ B )
           => ? [X5: a] :
                ( ( member_a @ X5 @ A2 )
                & ( ord_le282488521294790766od_b_c @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) )
          = ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ G @ B ) ) ) ) ) ).

% SUP_eq
thf(fact_1220_less__eq__Sup,axiom,
    ! [A2: set_se3924713247505902254od_b_c,U2: set_li6436108459499378894od_b_c] :
      ( ! [V: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ V @ A2 )
         => ( ord_le282488521294790766od_b_c @ U2 @ V ) )
     => ( ( A2 != bot_bo2794119844231891738od_b_c )
       => ( ord_le282488521294790766od_b_c @ U2 @ ( comple1250193257546725051od_b_c @ A2 ) ) ) ) ).

% less_eq_Sup
thf(fact_1221_Sup__subset__mono,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ A2 @ B )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Sup_subset_mono
thf(fact_1222_Union__Int__subset,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] : ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( inf_in5524221907227607964od_b_c @ A2 @ B ) ) @ ( inf_in4978071631833541052od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Union_Int_subset
thf(fact_1223_SUP__upper2,axiom,
    ! [I2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,U2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ I2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ U2 @ ( F @ I2 ) )
       => ( ord_le282488521294790766od_b_c @ U2 @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) ) ) ) ).

% SUP_upper2
thf(fact_1224_SUP__upper2,axiom,
    ! [I2: a,A2: set_a,U2: set_li6436108459499378894od_b_c,F: a > set_li6436108459499378894od_b_c] :
      ( ( member_a @ I2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ U2 @ ( F @ I2 ) )
       => ( ord_le282488521294790766od_b_c @ U2 @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) ) ) ) ).

% SUP_upper2
thf(fact_1225_SUP__upper,axiom,
    ! [I2: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ I2 @ A2 )
     => ( ord_le282488521294790766od_b_c @ ( F @ I2 ) @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) ) ) ).

% SUP_upper
thf(fact_1226_SUP__upper,axiom,
    ! [I2: a,A2: set_a,F: a > set_li6436108459499378894od_b_c] :
      ( ( member_a @ I2 @ A2 )
     => ( ord_le282488521294790766od_b_c @ ( F @ I2 ) @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) ) ) ).

% SUP_upper
thf(fact_1227_SUP__least,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,U2: set_li6436108459499378894od_b_c] :
      ( ! [I3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ I3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ U2 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) @ U2 ) ) ).

% SUP_least
thf(fact_1228_SUP__least,axiom,
    ! [A2: set_a,F: a > set_li6436108459499378894od_b_c,U2: set_li6436108459499378894od_b_c] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ U2 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) @ U2 ) ) ).

% SUP_least
thf(fact_1229_SUP__eqI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ! [I3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ I3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ X2 ) )
     => ( ! [Y3: set_li6436108459499378894od_b_c] :
            ( ! [I4: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ I4 @ A2 )
               => ( ord_le282488521294790766od_b_c @ ( F @ I4 ) @ Y3 ) )
           => ( ord_le282488521294790766od_b_c @ X2 @ Y3 ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_1230_SUP__eqI,axiom,
    ! [A2: set_a,F: a > set_li6436108459499378894od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( F @ I3 ) @ X2 ) )
     => ( ! [Y3: set_li6436108459499378894od_b_c] :
            ( ! [I4: a] :
                ( ( member_a @ I4 @ A2 )
               => ( ord_le282488521294790766od_b_c @ ( F @ I4 ) @ Y3 ) )
           => ( ord_le282488521294790766od_b_c @ X2 @ Y3 ) )
       => ( ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_1231_UN__upper,axiom,
    ! [A: list_P903359562653991662od_b_c,A2: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A2 )
     => ( ord_le282488521294790766od_b_c @ ( B @ A ) @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ B @ A2 ) ) ) ) ).

% UN_upper
thf(fact_1232_UN__upper,axiom,
    ! [A: a,A2: set_a,B: a > set_li6436108459499378894od_b_c] :
      ( ( member_a @ A @ A2 )
     => ( ord_le282488521294790766od_b_c @ ( B @ A ) @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ B @ A2 ) ) ) ) ).

% UN_upper
thf(fact_1233_UN__least,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( B @ X3 ) @ C2 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ B @ A2 ) ) @ C2 ) ) ).

% UN_least
thf(fact_1234_UN__least,axiom,
    ! [A2: set_a,B: a > set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ( ord_le282488521294790766od_b_c @ ( B @ X3 ) @ C2 ) )
     => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ B @ A2 ) ) @ C2 ) ) ).

% UN_least
thf(fact_1235_UN__mono,axiom,
    ! [A2: set_a,B: set_a,F: a > set_li6436108459499378894od_b_c,G: a > set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ G @ B ) ) ) ) ) ).

% UN_mono
thf(fact_1236_UN__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,G: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ G @ B ) ) ) ) ) ).

% UN_mono
thf(fact_1237_SUP__eq__iff,axiom,
    ! [I: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( I != bot_bo4166481423041325370od_b_c )
     => ( ! [I3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ I3 @ I )
           => ( ord_le282488521294790766od_b_c @ C @ ( F @ I3 ) ) )
       => ( ( ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ I ) )
            = C )
          = ( ! [X: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X @ I )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1238_SUP__eq__iff,axiom,
    ! [I: set_a,C: set_li6436108459499378894od_b_c,F: a > set_li6436108459499378894od_b_c] :
      ( ( I != bot_bot_set_a )
     => ( ! [I3: a] :
            ( ( member_a @ I3 @ I )
           => ( ord_le282488521294790766od_b_c @ C @ ( F @ I3 ) ) )
       => ( ( ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ I ) )
            = C )
          = ( ! [X: a] :
                ( ( member_a @ X @ I )
               => ( ( F @ X )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_1239_Sup__inter__less__eq,axiom,
    ! [A2: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] : ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( inf_in5524221907227607964od_b_c @ A2 @ B ) ) @ ( inf_in4978071631833541052od_b_c @ ( comple1250193257546725051od_b_c @ A2 ) @ ( comple1250193257546725051od_b_c @ B ) ) ) ).

% Sup_inter_less_eq
thf(fact_1240_SUP__subset__mono,axiom,
    ! [A2: set_a,B: set_a,F: a > set_li6436108459499378894od_b_c,G: a > set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ G @ B ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1241_SUP__subset__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,G: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ G @ B ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1242_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_1243_finite__subset__Union,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B8: set_se3924713247505902254od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ ( comple1250193257546725051od_b_c @ B8 ) )
       => ~ ! [F6: set_se3924713247505902254od_b_c] :
              ( ( finite1374199133651033463od_b_c @ F6 )
             => ( ( ord_le6656836712342966862od_b_c @ F6 @ B8 )
               => ~ ( ord_le282488521294790766od_b_c @ A2 @ ( comple1250193257546725051od_b_c @ F6 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1244_SUP__Sup__eq,axiom,
    ! [S: set_se3924713247505902254od_b_c] :
      ( ( comple6218144523844345314_b_c_o
        @ ( image_2661030049660093720_b_c_o
          @ ^ [I5: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ I5 )
          @ S ) )
      = ( ^ [X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ ( comple1250193257546725051od_b_c @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1245_SUP__Sup__eq,axiom,
    ! [S: set_set_a] :
      ( ( complete_Sup_Sup_a_o
        @ ( image_set_a_a_o
          @ ^ [I5: set_a,X: a] : ( member_a @ X @ I5 )
          @ S ) )
      = ( ^ [X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1246_Sup__set__def,axiom,
    ( comple2307003609928055243_set_a
    = ( ^ [A3: set_set_a] :
          ( collect_a
          @ ^ [X: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X ) @ A3 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1247_Sup__set__def,axiom,
    ( comple1250193257546725051od_b_c
    = ( ^ [A3: set_se3924713247505902254od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X: list_P903359562653991662od_b_c] : ( complete_Sup_Sup_o @ ( image_1238484426057152719_b_c_o @ ( member6330420149250801815od_b_c @ X ) @ A3 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1248_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A3: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A3 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1249_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_a_o
    = ( ^ [S5: set_a_o,X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1250_Sup__SUP__eq,axiom,
    ( comple6218144523844345314_b_c_o
    = ( ^ [S5: set_li313786449249895621_b_c_o,X: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X @ ( comple1250193257546725051od_b_c @ ( image_7656793960495356344od_b_c @ collec2280997390073109977od_b_c @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1251_Sup__SUP__eq,axiom,
    ( comple8317665133742190828_nat_o
    = ( ^ [S5: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1252_SUP__Sup__eq2,axiom,
    ! [S: set_se3344102371033692456od_b_c] :
      ( ( comple1411054825784354266_b_c_o
        @ ( image_6949117070849169066_b_c_o
          @ ^ [I5: set_Product_prod_b_c,X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ I5 )
          @ S ) )
      = ( ^ [X: b,Y2: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ ( comple7928098250179550773od_b_c @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1253_cSup__eq__maximum,axiom,
    ! [Z5: set_li6436108459499378894od_b_c,X7: set_se3924713247505902254od_b_c] :
      ( ( member6985331446368301687od_b_c @ Z5 @ X7 )
     => ( ! [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ X7 )
           => ( ord_le282488521294790766od_b_c @ X3 @ Z5 ) )
       => ( ( comple1250193257546725051od_b_c @ X7 )
          = Z5 ) ) ) ).

% cSup_eq_maximum
thf(fact_1254_cSup__eq__maximum,axiom,
    ! [Z5: nat,X7: set_nat] :
      ( ( member_nat @ Z5 @ X7 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ Z5 ) )
       => ( ( complete_Sup_Sup_nat @ X7 )
          = Z5 ) ) ) ).

% cSup_eq_maximum
thf(fact_1255_cSup__least,axiom,
    ! [X7: set_se3924713247505902254od_b_c,Z5: set_li6436108459499378894od_b_c] :
      ( ( X7 != bot_bo2794119844231891738od_b_c )
     => ( ! [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ X7 )
           => ( ord_le282488521294790766od_b_c @ X3 @ Z5 ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ X7 ) @ Z5 ) ) ) ).

% cSup_least
thf(fact_1256_cSup__least,axiom,
    ! [X7: set_nat,Z5: nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ Z5 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X7 ) @ Z5 ) ) ) ).

% cSup_least
thf(fact_1257_cSup__eq__non__empty,axiom,
    ! [X7: set_se3924713247505902254od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( X7 != bot_bo2794119844231891738od_b_c )
     => ( ! [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ X7 )
           => ( ord_le282488521294790766od_b_c @ X3 @ A ) )
       => ( ! [Y3: set_li6436108459499378894od_b_c] :
              ( ! [X5: set_li6436108459499378894od_b_c] :
                  ( ( member6985331446368301687od_b_c @ X5 @ X7 )
                 => ( ord_le282488521294790766od_b_c @ X5 @ Y3 ) )
             => ( ord_le282488521294790766od_b_c @ A @ Y3 ) )
         => ( ( comple1250193257546725051od_b_c @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1258_cSup__eq__non__empty,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ord_less_eq_nat @ X3 @ A ) )
       => ( ! [Y3: nat] :
              ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ X7 )
                 => ( ord_less_eq_nat @ X5 @ Y3 ) )
             => ( ord_less_eq_nat @ A @ Y3 ) )
         => ( ( complete_Sup_Sup_nat @ X7 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1259_le__cSup__finite,axiom,
    ! [X7: set_se3924713247505902254od_b_c,X2: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ X7 )
     => ( ( member6985331446368301687od_b_c @ X2 @ X7 )
       => ( ord_le282488521294790766od_b_c @ X2 @ ( comple1250193257546725051od_b_c @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_1260_le__cSup__finite,axiom,
    ! [X7: set_nat,X2: nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( member_nat @ X2 @ X7 )
       => ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X7 ) ) ) ) ).

% le_cSup_finite
thf(fact_1261_cSUP__least,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,M: set_li6436108459499378894od_b_c] :
      ( ( A2 != bot_bo4166481423041325370od_b_c )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ M ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ F @ A2 ) ) @ M ) ) ) ).

% cSUP_least
thf(fact_1262_cSUP__least,axiom,
    ! [A2: set_a,F: a > set_li6436108459499378894od_b_c,M: set_li6436108459499378894od_b_c] :
      ( ( A2 != bot_bot_set_a )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ X3 ) @ M ) )
       => ( ord_le282488521294790766od_b_c @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ F @ A2 ) ) @ M ) ) ) ).

% cSUP_least
thf(fact_1263_cSUP__least,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > nat,M: nat] :
      ( ( A2 != bot_bo4166481423041325370od_b_c )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_4545125072327328889_c_nat @ F @ A2 ) ) @ M ) ) ) ).

% cSUP_least
thf(fact_1264_cSUP__least,axiom,
    ! [A2: set_a,F: a > nat,M: nat] :
      ( ( A2 != bot_bot_set_a )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A2 ) ) @ M ) ) ) ).

% cSUP_least

% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_Itf__c_J_J_T,axiom,
    ! [X2: option_set_c,Y: option_set_c] :
      ( ( if_option_set_c @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_Itf__c_J_J_T,axiom,
    ! [X2: option_set_c,Y: option_set_c] :
      ( ( if_option_set_c @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ( if_set5301400340663253090od_c_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ( if_set5301400340663253090od_c_a @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member6330420149250801815od_b_c @ io @ ( set_li4480668622519654659od_b_c @ ( langua5143504920213240017_a_b_c @ m @ q2 @ xsa ) ) ).

%------------------------------------------------------------------------------