%------------------------------------------------------------------------------
% 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_05855_227535__19415676_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1410 ( 502 unt; 132 typ;   0 def)
%            Number of atoms       : 3815 (1121 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11326 ( 448   ~;  59   |; 226   &;8696   @)
%                                         (   0 <=>;1897  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   7 avg)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  :  544 ( 544   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  124 ( 121 usr;  24 con; 0-4 aty)
%            Number of variables   : 3525 ( 239   ^;3188   !;  98   ?;3525   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:27:49.438
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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,
    set_se3924713247505902254od_b_c: $tType ).

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,
    fsm_li6801133765522507155_c_b_c: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    list_l8907847357763382004od_b_c: $tType ).

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__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    list_P903359562653991662od_b_c: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_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_It__Nat__Onat_J,type,
    set_nat: $tType ).

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

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

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

% Explicit typings (121)
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__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_Oacyclic_001tf__a_001tf__b_001tf__c,type,
    acyclic_a_b_c: fsm_a_b_c > $o ).

thf(sy_c_FSM_Oafter_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    after_4052058690717316294_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_FSM_Oafter_001tf__a_001tf__b_001tf__c,type,
    after_a_b_c: fsm_a_b_c > a > list_P903359562653991662od_b_c > a ).

thf(sy_c_FSM_Odistinguishes_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    distin2804555989863659119_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Odistinguishes_001tf__a_001tf__b_001tf__c,type,
    distinguishes_a_b_c: fsm_a_b_c > a > a > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Oinitial_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    initia3567573336347591134_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_FSM_Oinitial_001tf__a_001tf__b_001tf__c,type,
    initial_a_b_c: fsm_a_b_c > a ).

thf(sy_c_FSM_Ois__in__language_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    is_in_7104650932667917939_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Ois__in__language_001tf__a_001tf__b_001tf__c,type,
    is_in_language_a_b_c: fsm_a_b_c > a > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Omaximal__prefix__in__language_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    maxima8146652644187019584_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_FSM_Omaximal__prefix__in__language_001tf__a_001tf__b_001tf__c,type,
    maxima1559550560783484624_a_b_c: fsm_a_b_c > a > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_FSM_Ominimal_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    minima1987601567150520449_c_b_c: fsm_li6801133765522507155_c_b_c > $o ).

thf(sy_c_FSM_Ominimal_001tf__a_001tf__b_001tf__c,type,
    minimal_a_b_c: fsm_a_b_c > $o ).

thf(sy_c_FSM_Ominimally__distinguishes_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    minima9089413714839006869_c_b_c: fsm_li6801133765522507155_c_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Ominimally__distinguishes_001tf__a_001tf__b_001tf__c,type,
    minima243535863231358885_a_b_c: fsm_a_b_c > a > a > list_P903359562653991662od_b_c > $o ).

thf(sy_c_FSM_Oobservable_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    observ6293852833591064631_c_b_c: fsm_li6801133765522507155_c_b_c > $o ).

thf(sy_c_FSM_Oobservable_001tf__a_001tf__b_001tf__c,type,
    observable_a_b_c: fsm_a_b_c > $o ).

thf(sy_c_FSM_Oreachable__states_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    reacha8116992257183400179_c_b_c: fsm_li6801133765522507155_c_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_FSM_Oreachable__states_001tf__a_001tf__b_001tf__c,type,
    reacha1620305530751930115_a_b_c: fsm_a_b_c > set_a ).

thf(sy_c_FSM_Orestrict__to__reachable__states_001tf__a_001tf__b_001tf__c,type,
    restri9132545300209641082_a_b_c: fsm_a_b_c > fsm_a_b_c ).

thf(sy_c_FSM_Ostates_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001tf__b_001tf__c,type,
    states7681702920031268536_c_b_c: fsm_li6801133765522507155_c_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_FSM_Ostates_001tf__a_001tf__b_001tf__c,type,
    states_a_b_c: fsm_a_b_c > set_a ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    finite5583770498833199894od_b_c: set_li6436108459499378894od_b_c > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001tf__a,type,
    finite_card_a: set_a > nat ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    finite3074115686814133143od_b_c: set_li6436108459499378894od_b_c > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    finite1374199133651033463od_b_c: set_se3924713247505902254od_b_c > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
    finite_finite_set_a: set_set_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
    finite_finite_a: set_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    inf_in4978071631833541052od_b_c: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
    inf_inf_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    sup_su3823046536922626210od_b_c: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_001t__Nat__Onat,type,
    lattic9077399895930519397_c_nat: ( list_P903359562653991662od_b_c > nat ) > set_li6436108459499378894od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
    lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    lattic2195833625579453896od_b_c: ( list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ) > ( list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o ) > ( list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
    lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    lattic2204479054038723496od_b_c: ( set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c ) > ( set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > $o ) > ( set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
    lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
    lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    lattic4067905262246253180od_b_c: ( list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
    lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Set__Oset_Itf__a_J,type,
    lattic1258622339881844972_set_a: ( set_a > set_a > set_a ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
    lattic5961991414251573132_set_a: ( a > a > a ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    lattic2734920875441048264od_b_c: ( list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ) > set_li6436108459499378894od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
    lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Set__Oset_Itf__a_J,type,
    lattic2714821017709792056_set_a: ( set_a > set_a > set_a ) > set_set_a > set_a ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
    lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
    lattic1093996805478795353in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    lattic8058834985641542149od_b_c: set_se3924713247505902254od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
    lattic2918178356826803221_set_a: set_set_a > set_a ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    append2547753245680614915od_b_c: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    gen_le7668512674959329659od_b_c: nat > list_P903359562653991662od_b_c > nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    nil_Product_prod_b_c: list_P903359562653991662od_b_c ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    set_li4480668622519654659od_b_c: list_l8907847357763382004od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_List_Oremdups__adj_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    remdup2090892755044397220od_b_c: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    size_s3392097710323735898od_b_c: list_P903359562653991662od_b_c > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_M_Eo_J,type,
    bot_bo5496101219168594979_b_c_o: list_P903359562653991662od_b_c > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
    bot_bot_a_o: a > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    bot_bo4166481423041325370od_b_c: set_li6436108459499378894od_b_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
    bot_bo2794119844231891738od_b_c: set_se3924713247505902254od_b_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    bot_bot_set_set_a: set_set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    ord_le5653067673530651002od_b_c: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
    ord_less_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    ord_le282488521294790766od_b_c: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J,type,
    ord_le6656836712342966862od_b_c: set_se3924713247505902254od_b_c > set_se3924713247505902254od_b_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    collec2280997390073109977od_b_c: ( list_P903359562653991662od_b_c > $o ) > set_li6436108459499378894od_b_c ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Odisjnt_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    disjnt5456880891938978613od_b_c: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > $o ).

thf(sy_c_Set_Odisjnt_001t__Nat__Onat,type,
    disjnt_nat: set_nat > set_nat > $o ).

thf(sy_c_Set_Odisjnt_001tf__a,type,
    disjnt_a: set_a > set_a > $o ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    insert6227932334100060350od_b_c: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
    insert_set_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_Set_Ois__empty_001tf__a,type,
    is_empty_a: set_a > $o ).

thf(sy_c_Sublist_Oprefixes_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    prefix1131979855692807669od_b_c: list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    member6330420149250801815od_b_c: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member6985331446368301687od_b_c: set_li6436108459499378894od_b_c > set_se3924713247505902254od_b_c > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_M,type,
    m: fsm_a_b_c ).

thf(sy_v_S,type,
    s: set_a ).

thf(sy_v_S1____,type,
    s1: set_a ).

thf(sy_v_S2____,type,
    s2: set_a ).

thf(sy_v_Sa____,type,
    sa: set_a ).

thf(sy_v_W____,type,
    w: set_a > set_a > set_li6436108459499378894od_b_c ).

thf(sy_v_k_H____,type,
    k: nat ).

thf(sy_v_k____,type,
    k2: nat ).

thf(sy_v_ka____,type,
    ka: nat ).

thf(sy_v_q1,type,
    q1: a ).

thf(sy_v_q2,type,
    q2: a ).

thf(sy_v_w,type,
    w2: list_P903359562653991662od_b_c ).

thf(sy_v_w_H____,type,
    w3: list_P903359562653991662od_b_c ).

thf(sy_v_wk____,type,
    wk: list_P903359562653991662od_b_c ).

thf(sy_v_wk__suffix____,type,
    wk_suffix: list_P903359562653991662od_b_c ).

% Relevant facts (1274)
thf(fact_0_assms_I2_J,axiom,
    minimal_a_b_c @ m ).

% assms(2)
thf(fact_1__092_060open_062after_AM_Aq1_Aw_H_A_092_060in_062_AS_092_060close_062,axiom,
    member_a @ ( after_a_b_c @ m @ q1 @ w3 ) @ sa ).

% \<open>after M q1 w' \<in> S\<close>
thf(fact_2__092_060open_062after_AM_Aq2_Aw_H_A_092_060in_062_AS_092_060close_062,axiom,
    member_a @ ( after_a_b_c @ m @ q2 @ w3 ) @ sa ).

% \<open>after M q2 w' \<in> S\<close>
thf(fact_3__092_060open_062_Iafter_AM_Aq1_Aw_H_A_092_060in_062_AS1_J_A_092_060noteq_062_A_Iafter_AM_Aq2_Aw_H_A_092_060in_062_AS1_J_092_060close_062,axiom,
    ( ( member_a @ ( after_a_b_c @ m @ q1 @ w3 ) @ s1 )
   != ( member_a @ ( after_a_b_c @ m @ q2 @ w3 ) @ s1 ) ) ).

% \<open>(after M q1 w' \<in> S1) \<noteq> (after M q2 w' \<in> S1)\<close>
thf(fact_4__C_K_C,axiom,
    member6330420149250801815od_b_c @ w3 @ ( w @ sa @ sa ) ).

% "*"
thf(fact_5__092_060open_062w_H_A_092_060noteq_062_Awk_092_060close_062,axiom,
    w3 != wk ).

% \<open>w' \<noteq> wk\<close>
thf(fact_6_assms_I1_J,axiom,
    observable_a_b_c @ m ).

% assms(1)
thf(fact_7_assms_I4_J,axiom,
    member_a @ q2 @ ( states_a_b_c @ m ) ).

% assms(4)
thf(fact_8_assms_I3_J,axiom,
    member_a @ q1 @ ( states_a_b_c @ m ) ).

% assms(3)
thf(fact_9__092_060open_062w_H_A_092_060in_062_ALS_AM_Aq1_092_060close_062,axiom,
    member6330420149250801815od_b_c @ w3 @ ( lS_a_b_c @ m @ q1 ) ).

% \<open>w' \<in> LS M q1\<close>
thf(fact_10__092_060open_062w_H_A_092_060in_062_ALS_AM_Aq2_092_060close_062,axiom,
    member6330420149250801815od_b_c @ w3 @ ( lS_a_b_c @ m @ q2 ) ).

% \<open>w' \<in> LS M q2\<close>
thf(fact_11_assms_I5_J,axiom,
    minima243535863231358885_a_b_c @ m @ q1 @ q2 @ w2 ).

% assms(5)
thf(fact_12__092_060open_062after_AM_Aq1_Awk_A_092_060in_062_AS_092_060close_062,axiom,
    member_a @ ( after_a_b_c @ m @ q1 @ wk ) @ sa ).

% \<open>after M q1 wk \<in> S\<close>
thf(fact_13__092_060open_062after_AM_Aq2_Awk_A_092_060in_062_AS_092_060close_062,axiom,
    member_a @ ( after_a_b_c @ m @ q2 @ wk ) @ sa ).

% \<open>after M q2 wk \<in> S\<close>
thf(fact_14__092_060open_062S2_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
    s2 != bot_bot_set_a ).

% \<open>S2 \<noteq> {}\<close>
thf(fact_15__092_060open_062S1_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
    s1 != bot_bot_set_a ).

% \<open>S1 \<noteq> {}\<close>
thf(fact_16__092_060open_062S_A_061_AS1_A_092_060union_062_AS2_092_060close_062,axiom,
    ( sa
    = ( sup_sup_set_a @ s1 @ s2 ) ) ).

% \<open>S = S1 \<union> S2\<close>
thf(fact_17__092_060open_062w_H_A_092_060noteq_062_Aw_092_060close_062,axiom,
    w3 != w2 ).

% \<open>w' \<noteq> w\<close>
thf(fact_18__092_060open_062minimally__distinguishes_AM_A_Iafter_AM_Aq1_Awk_J_A_Iafter_AM_Aq2_Awk_J_Awk__suffix_092_060close_062,axiom,
    minima243535863231358885_a_b_c @ m @ ( after_a_b_c @ m @ q1 @ wk ) @ ( after_a_b_c @ m @ q2 @ wk ) @ wk_suffix ).

% \<open>minimally_distinguishes M (after M q1 wk) (after M q2 wk) wk_suffix\<close>
thf(fact_19__092_060open_062distinguishes_AM_A_Iafter_AM_Aq1_Awk_J_A_Iafter_AM_Aq2_Awk_J_Awk__suffix_092_060close_062,axiom,
    distinguishes_a_b_c @ m @ ( after_a_b_c @ m @ q1 @ wk ) @ ( after_a_b_c @ m @ q2 @ wk ) @ wk_suffix ).

% \<open>distinguishes M (after M q1 wk) (after M q2 wk) wk_suffix\<close>
thf(fact_20__092_060open_062distinguishes_AM_A_Iafter_AM_Aq1_A_091_093_J_A_Iafter_AM_Aq2_A_091_093_J_Aw_092_060close_062,axiom,
    distinguishes_a_b_c @ m @ ( after_a_b_c @ m @ q1 @ nil_Product_prod_b_c ) @ ( after_a_b_c @ m @ q2 @ nil_Product_prod_b_c ) @ w2 ).

% \<open>distinguishes M (after M q1 []) (after M q2 []) w\<close>
thf(fact_21__092_060open_062wk__suffix_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
    wk_suffix != nil_Product_prod_b_c ).

% \<open>wk_suffix \<noteq> []\<close>
thf(fact_22__092_060open_062wk_A_092_060noteq_062_Aw_092_060close_062,axiom,
    wk != w2 ).

% \<open>wk \<noteq> w\<close>
thf(fact_23__092_060open_062_Iwk__suffix_A_092_060in_062_ALS_AM_A_Iafter_AM_Aq1_Awk_J_J_A_061_A_Iwk__suffix_A_092_060notin_062_ALS_AM_A_Iafter_AM_Aq2_Awk_J_J_092_060close_062,axiom,
    ( ( member6330420149250801815od_b_c @ wk_suffix @ ( lS_a_b_c @ m @ ( after_a_b_c @ m @ q1 @ wk ) ) )
    = ( ~ ( member6330420149250801815od_b_c @ wk_suffix @ ( lS_a_b_c @ m @ ( after_a_b_c @ m @ q2 @ wk ) ) ) ) ) ).

% \<open>(wk_suffix \<in> LS M (after M q1 wk)) = (wk_suffix \<notin> LS M (after M q2 wk))\<close>
thf(fact_24_less_Oprems_I2_J,axiom,
    ord_less_eq_set_a @ sa @ ( states_a_b_c @ m ) ).

% less.prems(2)
thf(fact_25__092_060open_062w_A_061_Awk_A_064_Awk__suffix_092_060close_062,axiom,
    ( w2
    = ( append2547753245680614915od_b_c @ wk @ wk_suffix ) ) ).

% \<open>w = wk @ wk_suffix\<close>
thf(fact_26_minimal_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( minimal_a_b_c @ X )
     => ~ ! [X2: a] :
            ( ( member_a @ X2 @ ( states_a_b_c @ X ) )
           => ! [Xa: a] :
                ( ( member_a @ Xa @ ( states_a_b_c @ X ) )
               => ( ( X2 != Xa )
                 => ( ( lS_a_b_c @ X @ X2 )
                   != ( lS_a_b_c @ X @ Xa ) ) ) ) ) ) ).

% minimal.elims(3)
thf(fact_27_minimal_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( minimal_a_b_c @ X )
     => ! [X3: a] :
          ( ( member_a @ X3 @ ( states_a_b_c @ X ) )
         => ! [Xa2: a] :
              ( ( member_a @ Xa2 @ ( states_a_b_c @ X ) )
             => ( ( X3 != Xa2 )
               => ( ( lS_a_b_c @ X @ X3 )
                 != ( lS_a_b_c @ X @ Xa2 ) ) ) ) ) ) ).

% minimal.elims(2)
thf(fact_28_minimal_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( minimal_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ ( states_a_b_c @ X ) )
             => ! [Y2: a] :
                  ( ( member_a @ Y2 @ ( states_a_b_c @ X ) )
                 => ( ( X4 != Y2 )
                   => ( ( lS_a_b_c @ X @ X4 )
                     != ( lS_a_b_c @ X @ Y2 ) ) ) ) ) ) ) ) ).

% minimal.elims(1)
thf(fact_29_minimal_Osimps,axiom,
    ( minimal_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X4: a] :
          ( ( member_a @ X4 @ ( states_a_b_c @ M ) )
         => ! [Y2: a] :
              ( ( member_a @ Y2 @ ( states_a_b_c @ M ) )
             => ( ( X4 != Y2 )
               => ( ( lS_a_b_c @ M @ X4 )
                 != ( lS_a_b_c @ M @ Y2 ) ) ) ) ) ) ) ).

% minimal.simps
thf(fact_30_after__is__state,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Io: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q ) )
       => ( member6330420149250801815od_b_c @ ( after_4052058690717316294_c_b_c @ M2 @ Q @ Io ) @ ( states7681702920031268536_c_b_c @ M2 ) ) ) ) ).

% after_is_state
thf(fact_31_after__is__state,axiom,
    ! [M2: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q ) )
       => ( member_a @ ( after_a_b_c @ M2 @ Q @ Io ) @ ( states_a_b_c @ M2 ) ) ) ) ).

% after_is_state
thf(fact_32_minimal__alt__def,axiom,
    ( minimal_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q2: a,Q3: a] :
          ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( member_a @ Q3 @ ( states_a_b_c @ M ) )
           => ( ( ( lS_a_b_c @ M @ Q2 )
                = ( lS_a_b_c @ M @ Q3 ) )
             => ( Q2 = Q3 ) ) ) ) ) ) ).

% minimal_alt_def
thf(fact_33_minimally__distinguishes__ex,axiom,
    ! [Q1: a,M2: fsm_a_b_c,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M2 ) )
     => ( ( member_a @ Q22 @ ( states_a_b_c @ M2 ) )
       => ( ( ( lS_a_b_c @ M2 @ Q1 )
           != ( lS_a_b_c @ M2 @ Q22 ) )
         => ~ ! [V: list_P903359562653991662od_b_c] :
                ~ ( minima243535863231358885_a_b_c @ M2 @ Q1 @ Q22 @ V ) ) ) ) ).

% minimally_distinguishes_ex
thf(fact_34_after_Osimps_I1_J,axiom,
    ! [M2: fsm_a_b_c,Q: a] :
      ( ( after_a_b_c @ M2 @ Q @ nil_Product_prod_b_c )
      = Q ) ).

% after.simps(1)
thf(fact_35_assms_I6_J,axiom,
    ord_less_eq_set_a @ s @ ( states_a_b_c @ m ) ).

% assms(6)
thf(fact_36_Un__empty,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( sup_sup_set_a @ A @ B )
        = bot_bot_set_a )
      = ( ( A = bot_bot_set_a )
        & ( B = bot_bot_set_a ) ) ) ).

% Un_empty
thf(fact_37_sup__bot__left,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ X )
      = X ) ).

% sup_bot_left
thf(fact_38_sup__bot__right,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ X @ bot_bot_set_a )
      = X ) ).

% sup_bot_right
thf(fact_39_bot__eq__sup__iff,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ X @ Y ) )
      = ( ( X = bot_bot_set_a )
        & ( Y = bot_bot_set_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_40_sup__eq__bot__iff,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ( sup_sup_set_a @ X @ Y )
        = bot_bot_set_a )
      = ( ( X = bot_bot_set_a )
        & ( Y = bot_bot_set_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_41_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ( sup_sup_set_a @ A2 @ B2 )
        = bot_bot_set_a )
      = ( ( A2 = bot_bot_set_a )
        & ( B2 = bot_bot_set_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_42_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_43_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_a )
        & ( B2 = bot_bot_set_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_44_sup__bot_Oright__neutral,axiom,
    ! [A2: set_a] :
      ( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_45__092_060open_062S1_A_092_060inter_062_AS2_A_061_A_123_125_092_060close_062,axiom,
    ( ( inf_inf_set_a @ s1 @ s2 )
    = bot_bot_set_a ) ).

% \<open>S1 \<inter> S2 = {}\<close>
thf(fact_46_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A2: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ A2 @ ( collec2280997390073109977od_b_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_50_empty__Collect__eq,axiom,
    ! [P: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P ) )
      = ( ! [X4: a] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_51_Collect__empty__eq,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_52_all__not__in__conv,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ! [X4: list_P903359562653991662od_b_c] :
            ~ ( member6330420149250801815od_b_c @ X4 @ A ) )
      = ( A = bot_bo4166481423041325370od_b_c ) ) ).

% all_not_in_conv
thf(fact_53_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X4: a] :
            ~ ( member_a @ X4 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_54_empty__iff,axiom,
    ! [C: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ C @ bot_bo4166481423041325370od_b_c ) ).

% empty_iff
thf(fact_55_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_56_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_57_subset__antisym,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_58_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ( member_a @ X2 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_59_subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X2 @ A )
         => ( member6330420149250801815od_b_c @ X2 @ B ) )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% subsetI
thf(fact_60_inf__right__idem,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
      = ( inf_inf_set_a @ X @ Y ) ) ).

% inf_right_idem
thf(fact_61_inf_Oright__idem,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_a @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_62_inf__left__idem,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
      = ( inf_inf_set_a @ X @ Y ) ) ).

% inf_left_idem
thf(fact_63_inf_Oleft__idem,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
      = ( inf_inf_set_a @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_64_inf__idem,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ X )
      = X ) ).

% inf_idem
thf(fact_65_inf_Oidem,axiom,
    ! [A2: set_a] :
      ( ( inf_inf_set_a @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_66_sup_Oright__idem,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_67_sup__left__idem,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
      = ( sup_sup_set_a @ X @ Y ) ) ).

% sup_left_idem
thf(fact_68_sup_Oleft__idem,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
      = ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_69_sup__idem,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ X @ X )
      = X ) ).

% sup_idem
thf(fact_70_sup_Oidem,axiom,
    ! [A2: set_a] :
      ( ( sup_sup_set_a @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_71_Int__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A )
        & ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Int_iff
thf(fact_72_Int__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        & ( member_a @ C @ B ) ) ) ).

% Int_iff
thf(fact_73_IntI,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A )
     => ( ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% IntI
thf(fact_74_IntI,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( ( member_a @ C @ B )
       => ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_75_Un__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A )
        | ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Un_iff
thf(fact_76_Un__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        | ( member_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_77_UnCI,axiom,
    ! [C: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ~ ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ A ) )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnCI
thf(fact_78_UnCI,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ A ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnCI
thf(fact_79__092_060open_062S1_A_092_060subseteq_062_AFSM_Ostates_AM_092_060close_062,axiom,
    ord_less_eq_set_a @ s1 @ ( states_a_b_c @ m ) ).

% \<open>S1 \<subseteq> FSM.states M\<close>
thf(fact_80__092_060open_062S2_A_092_060subseteq_062_AFSM_Ostates_AM_092_060close_062,axiom,
    ord_less_eq_set_a @ s2 @ ( states_a_b_c @ m ) ).

% \<open>S2 \<subseteq> FSM.states M\<close>
thf(fact_81__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062wk__suffix_O_A_092_060lbrakk_062w_A_061_Awk_A_064_Awk__suffix_059_Awk__suffix_A_092_060noteq_062_A_091_093_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Wk_suffix: list_P903359562653991662od_b_c] :
        ( ( w2
          = ( append2547753245680614915od_b_c @ wk @ Wk_suffix ) )
       => ( Wk_suffix = nil_Product_prod_b_c ) ) ).

% \<open>\<And>thesis. (\<And>wk_suffix. \<lbrakk>w = wk @ wk_suffix; wk_suffix \<noteq> []\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_82_inf_Obounded__iff,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
      = ( ( ord_less_eq_set_a @ A2 @ B2 )
        & ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_83_inf_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_84_inf_Obounded__iff,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ B2 @ C ) )
      = ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
        & ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_85_le__inf__iff,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
      = ( ( ord_less_eq_set_a @ X @ Y )
        & ( ord_less_eq_set_a @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_86_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_87_le__inf__iff,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z ) )
      = ( ( ord_le282488521294790766od_b_c @ X @ Y )
        & ( ord_le282488521294790766od_b_c @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_88_sup_Obounded__iff,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_a @ B2 @ A2 )
        & ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_89_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_90_sup_Obounded__iff,axiom,
    ! [B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ B2 @ C ) @ A2 )
      = ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
        & ( ord_le282488521294790766od_b_c @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_91_le__sup__iff,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_a @ X @ Z )
        & ( ord_less_eq_set_a @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_92_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_93_le__sup__iff,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ X @ Y ) @ Z )
      = ( ( ord_le282488521294790766od_b_c @ X @ Z )
        & ( ord_le282488521294790766od_b_c @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_94_inf__bot__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% inf_bot_right
thf(fact_95_inf__bot__left,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ X )
      = bot_bot_set_a ) ).

% inf_bot_left
thf(fact_96_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_97_empty__subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ bot_bo4166481423041325370od_b_c @ A ) ).

% empty_subsetI
thf(fact_98_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_99_subset__empty,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ bot_bo4166481423041325370od_b_c )
      = ( A = bot_bo4166481423041325370od_b_c ) ) ).

% subset_empty
thf(fact_100_sup__inf__absorb,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_101_inf__sup__absorb,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_102_Int__subset__iff,axiom,
    ! [C2: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
      = ( ( ord_less_eq_set_a @ C2 @ A )
        & ( ord_less_eq_set_a @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_103_Int__subset__iff,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
      = ( ( ord_le282488521294790766od_b_c @ C2 @ A )
        & ( ord_le282488521294790766od_b_c @ C2 @ B ) ) ) ).

% Int_subset_iff
thf(fact_104_Un__subset__iff,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_a @ A @ C2 )
        & ( ord_less_eq_set_a @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_105_Un__subset__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) @ C2 )
      = ( ( ord_le282488521294790766od_b_c @ A @ C2 )
        & ( ord_le282488521294790766od_b_c @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_106_Un__Int__eq_I1_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_107_Un__Int__eq_I2_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_108_Un__Int__eq_I3_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_109_Un__Int__eq_I4_J,axiom,
    ! [T: set_a,S: set_a] :
      ( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_110_Int__Un__eq_I1_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_111_Int__Un__eq_I2_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_112_Int__Un__eq_I3_J,axiom,
    ! [S: set_a,T: set_a] :
      ( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_113_Int__Un__eq_I4_J,axiom,
    ! [T: set_a,S: set_a] :
      ( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_114_distrib__inf__le,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_115_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_116_distrib__inf__le,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ ( inf_in4978071631833541052od_b_c @ X @ Z ) ) @ ( inf_in4978071631833541052od_b_c @ X @ ( sup_su3823046536922626210od_b_c @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_117_distrib__sup__le,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_118_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_119_distrib__sup__le,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z ) ) @ ( inf_in4978071631833541052od_b_c @ ( sup_su3823046536922626210od_b_c @ X @ Y ) @ ( sup_su3823046536922626210od_b_c @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_120_Un__Int__assoc__eq,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
        = ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
      = ( ord_less_eq_set_a @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_121_Un__Int__assoc__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ( sup_su3823046536922626210od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ C2 )
        = ( inf_in4978071631833541052od_b_c @ A @ ( sup_su3823046536922626210od_b_c @ B @ C2 ) ) )
      = ( ord_le282488521294790766od_b_c @ C2 @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_122_Int__left__commute,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
      = ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).

% Int_left_commute
thf(fact_123_Int__Collect__mono,axiom,
    ! [A: set_a,B: set_a,P: a > $o,Q4: a > $o] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q4 @ X2 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q4 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_124_Int__Collect__mono,axiom,
    ! [A: 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 @ A @ B )
     => ( ! [X2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X2 @ A )
           => ( ( P @ X2 )
             => ( Q4 @ X2 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ ( collec2280997390073109977od_b_c @ P ) ) @ ( inf_in4978071631833541052od_b_c @ B @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_125_Collect__mono__iff,axiom,
    ! [P: a > $o,Q4: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q4 ) )
      = ( ! [X4: a] :
            ( ( P @ X4 )
           => ( Q4 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_126_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 ) )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( P @ X4 )
           => ( Q4 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_127_Int__left__absorb,axiom,
    ! [A: set_a,B: set_a] :
      ( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
      = ( inf_inf_set_a @ A @ B ) ) ).

% Int_left_absorb
thf(fact_128_set__eq__subset,axiom,
    ( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B3 )
          & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_129_set__eq__subset,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] : ( Y3 = Z2 ) )
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A3 @ B3 )
          & ( ord_le282488521294790766od_b_c @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_130_subset__trans,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_131_subset__trans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ C2 )
       => ( ord_le282488521294790766od_b_c @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_132_Int__greatest,axiom,
    ! [C2: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ A )
     => ( ( ord_less_eq_set_a @ C2 @ B )
       => ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_133_Int__greatest,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ A )
     => ( ( ord_le282488521294790766od_b_c @ C2 @ B )
       => ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_134_Collect__mono,axiom,
    ! [P: a > $o,Q4: a > $o] :
      ( ! [X2: a] :
          ( ( P @ X2 )
         => ( Q4 @ X2 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q4 ) ) ) ).

% Collect_mono
thf(fact_135_Collect__mono,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q4: list_P903359562653991662od_b_c > $o] :
      ( ! [X2: list_P903359562653991662od_b_c] :
          ( ( P @ X2 )
         => ( Q4 @ X2 ) )
     => ( ord_le282488521294790766od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q4 ) ) ) ).

% Collect_mono
thf(fact_136_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_137_subset__refl,axiom,
    ! [A: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A @ A ) ).

% subset_refl
thf(fact_138_Int__commute,axiom,
    ( inf_inf_set_a
    = ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).

% Int_commute
thf(fact_139_Int__absorb2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( inf_inf_set_a @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_140_Int__absorb2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_141_Int__absorb1,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( inf_inf_set_a @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_142_Int__absorb1,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ A )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_143_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A3 )
         => ( member_a @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_144_subset__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
        ! [T2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ T2 @ A3 )
         => ( member6330420149250801815od_b_c @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_145_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_146_equalityD2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A = B )
     => ( ord_le282488521294790766od_b_c @ B @ A ) ) ).

% equalityD2
thf(fact_147_equalityD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_148_equalityD1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A = B )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% equalityD1
thf(fact_149_Int__lower2,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_150_Int__lower2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_151_Int__lower1,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_152_Int__lower1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_153_Int__absorb,axiom,
    ! [A: set_a] :
      ( ( inf_inf_set_a @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_154_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( member_a @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_155_subset__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ A3 )
         => ( member6330420149250801815od_b_c @ X4 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_156_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_157_equalityE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A = B )
     => ~ ( ( ord_le282488521294790766od_b_c @ A @ B )
         => ~ ( ord_le282488521294790766od_b_c @ B @ A ) ) ) ).

% equalityE
thf(fact_158_Int__assoc,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
      = ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).

% Int_assoc
thf(fact_159_Int__mono,axiom,
    ! [A: set_a,C2: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_160_Int__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ B @ D )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ ( inf_in4978071631833541052od_b_c @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_161_subsetD,axiom,
    ! [A: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% subsetD
thf(fact_162_subsetD,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( member6330420149250801815od_b_c @ C @ A )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% subsetD
thf(fact_163_in__mono,axiom,
    ! [A: set_a,B: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B ) ) ) ).

% in_mono
thf(fact_164_in__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( member6330420149250801815od_b_c @ X @ A )
       => ( member6330420149250801815od_b_c @ X @ B ) ) ) ).

% in_mono
thf(fact_165_IntD2,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ( member6330420149250801815od_b_c @ C @ B ) ) ).

% IntD2
thf(fact_166_IntD2,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C @ B ) ) ).

% IntD2
thf(fact_167_IntD1,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ( member6330420149250801815od_b_c @ C @ A ) ) ).

% IntD1
thf(fact_168_IntD1,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C @ A ) ) ).

% IntD1
thf(fact_169_IntE,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ~ ( ( member6330420149250801815od_b_c @ C @ A )
         => ~ ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% IntE
thf(fact_170_IntE,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ~ ( ( member_a @ C @ A )
         => ~ ( member_a @ C @ B ) ) ) ).

% IntE
thf(fact_171_inf__left__commute,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
      = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_172_inf_Oleft__commute,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
      = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).

% inf.left_commute
thf(fact_173_inf_OcoboundedI2,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_174_inf_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_175_inf_OcoboundedI2,axiom,
    ! [B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ C )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_176_inf_OcoboundedI1,axiom,
    ! [A2: set_a,C: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_177_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_178_inf_OcoboundedI1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ C )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_179_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_180_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_181_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_182_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( inf_inf_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_183_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_184_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_185_inf_Ocobounded2,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_186_inf_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_187_inf_Ocobounded2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_188_inf_Ocobounded1,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_189_inf_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_190_inf_Ocobounded1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_191_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( A4
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_192_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_193_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_194_inf__greatest,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ X @ Z )
       => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_195_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_196_inf__greatest,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ X @ Z )
       => ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_197_inf_OboundedI,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ A2 @ C )
       => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_198_inf_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_199_inf_OboundedI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ C )
       => ( ord_le282488521294790766od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_200_inf_OboundedE,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_201_inf_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_202_inf_OboundedE,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ B2 @ C ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
         => ~ ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_203_inf__commute,axiom,
    ( inf_inf_set_a
    = ( ^ [X4: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X4 ) ) ) ).

% inf_commute
thf(fact_204_inf__absorb2,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( inf_inf_set_a @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_205_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_206_inf__absorb2,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_207_inf__absorb1,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( inf_inf_set_a @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_208_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_209_inf__absorb1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_210_inf_Ocommute,axiom,
    ( inf_inf_set_a
    = ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).

% inf.commute
thf(fact_211_inf_Oabsorb2,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_212_inf_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_213_inf_Oabsorb2,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_214_inf_Oabsorb1,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_215_inf_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_216_inf_Oabsorb1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( inf_in4978071631833541052od_b_c @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_217_le__iff__inf,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( inf_inf_set_a @ X4 @ Y2 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_218_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y2: nat] :
          ( ( inf_inf_nat @ X4 @ Y2 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_219_le__iff__inf,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ X4 @ Y2 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_220_inf__unique,axiom,
    ! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
      ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ X2 )
     => ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y4 ) @ Y4 )
       => ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ( ord_less_eq_set_a @ X2 @ Z3 )
               => ( ord_less_eq_set_a @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_set_a @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_221_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ X2 )
     => ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ Y4 )
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ( ord_less_eq_nat @ X2 @ Z3 )
               => ( ord_less_eq_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_222_inf__unique,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X2 @ Y4 ) @ X2 )
     => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X2 @ Y4 ) @ Y4 )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c,Z3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ( ord_le282488521294790766od_b_c @ X2 @ Z3 )
               => ( ord_le282488521294790766od_b_c @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_223_inf_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_224_inf_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( inf_inf_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_225_inf_OorderI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( A2
        = ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) )
     => ( ord_le282488521294790766od_b_c @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_226_inf_OorderE,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_227_inf_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_228_inf_OorderE,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( A2
        = ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_229_inf__assoc,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
      = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_230_inf_Oassoc,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
      = ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).

% inf.assoc
thf(fact_231_le__infI2,axiom,
    ! [B2: set_a,X: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ X )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_232_le__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_233_le__infI2,axiom,
    ! [B2: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ X )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_234_le__infI1,axiom,
    ! [A2: set_a,X: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_235_le__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_236_le__infI1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ X )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_237_inf__mono,axiom,
    ! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_238_inf__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_239_inf__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,D2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ C )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ D2 )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) @ ( inf_in4978071631833541052od_b_c @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_240_le__infI,axiom,
    ! [X: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X @ A2 )
     => ( ( ord_less_eq_set_a @ X @ B2 )
       => ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_241_le__infI,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ( ord_less_eq_nat @ X @ B2 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_242_le__infI,axiom,
    ! [X: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ X @ B2 )
       => ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_243_le__infE,axiom,
    ! [X: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_a @ X @ A2 )
         => ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).

% le_infE
thf(fact_244_le__infE,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_nat @ X @ A2 )
         => ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_245_le__infE,axiom,
    ! [X: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ A2 @ B2 ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ X @ A2 )
         => ~ ( ord_le282488521294790766od_b_c @ X @ B2 ) ) ) ).

% le_infE
thf(fact_246_inf__le2,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_247_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_248_inf__le2,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_249_inf__le1,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_250_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_251_inf__le1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_252_inf__sup__ord_I1_J,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_253_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_254_inf__sup__ord_I1_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_255_inf__sup__ord_I2_J,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_256_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_257_inf__sup__ord_I2_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_258_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_a
    = ( ^ [X4: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_259_inf__sup__aci_I2_J,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
      = ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_260_inf__sup__aci_I3_J,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
      = ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_261_inf__sup__aci_I4_J,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
      = ( inf_inf_set_a @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_262_bot__set__def,axiom,
    ( bot_bot_set_a
    = ( collect_a @ bot_bot_a_o ) ) ).

% bot_set_def
thf(fact_263_sup__inf__distrib2,axiom,
    ! [Y: set_a,Z: set_a,X: set_a] :
      ( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_264_sup__inf__distrib1,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_265_inf__sup__distrib2,axiom,
    ! [Y: set_a,Z: set_a,X: set_a] :
      ( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_266_inf__sup__distrib1,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_267_distrib__imp2,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
          ( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
          = ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
     => ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
        = ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_268_distrib__imp1,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
          ( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y4 @ Z3 ) )
          = ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
     => ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
        = ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_269_disjoint__iff__not__equal,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A @ B )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ! [Y2: a] :
                ( ( member_a @ Y2 @ B )
               => ( X4 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_270_Int__empty__right,axiom,
    ! [A: set_a] :
      ( ( inf_inf_set_a @ A @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% Int_empty_right
thf(fact_271_Int__empty__left,axiom,
    ! [B: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ B )
      = bot_bot_set_a ) ).

% Int_empty_left
thf(fact_272_disjoint__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = bot_bo4166481423041325370od_b_c )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ A )
           => ~ ( member6330420149250801815od_b_c @ X4 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_273_disjoint__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A @ B )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ~ ( member_a @ X4 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_274_Int__emptyI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X2 @ A )
         => ~ ( member6330420149250801815od_b_c @ X2 @ B ) )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = bot_bo4166481423041325370od_b_c ) ) ).

% Int_emptyI
thf(fact_275_Int__emptyI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A )
         => ~ ( member_a @ X2 @ B ) )
     => ( ( inf_inf_set_a @ A @ B )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_276_Un__Int__distrib2,axiom,
    ! [B: set_a,C2: set_a,A: set_a] :
      ( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).

% Un_Int_distrib2
thf(fact_277_Int__Un__distrib2,axiom,
    ! [B: set_a,C2: set_a,A: set_a] :
      ( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).

% Int_Un_distrib2
thf(fact_278_Un__Int__distrib,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_279_Int__Un__distrib,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_280_Un__Int__crazy,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
      = ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).

% Un_Int_crazy
thf(fact_281_sup_OcoboundedI2,axiom,
    ! [C: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ C @ B2 )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_282_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_283_sup_OcoboundedI2,axiom,
    ! [C: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C @ B2 )
     => ( ord_le282488521294790766od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_284_sup_OcoboundedI1,axiom,
    ! [C: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C @ A2 )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_285_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_286_sup_OcoboundedI1,axiom,
    ! [C: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C @ A2 )
     => ( ord_le282488521294790766od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_287_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_288_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_289_sup_Oabsorb__iff2,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( sup_su3823046536922626210od_b_c @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_290_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_291_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_292_sup_Oabsorb__iff1,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( ( sup_su3823046536922626210od_b_c @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_293_sup_Ocobounded2,axiom,
    ! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_294_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_295_sup_Ocobounded2,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ B2 @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_296_sup_Ocobounded1,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_297_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_298_sup_Ocobounded1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A2 @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_299_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( A4
          = ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_300_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_301_sup_Oorder__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( A4
          = ( sup_su3823046536922626210od_b_c @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_302_sup_OboundedI,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C @ A2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_303_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_304_sup_OboundedI,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ C @ A2 )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_305_sup_OboundedE,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
         => ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_306_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_307_sup_OboundedE,axiom,
    ! [B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
         => ~ ( ord_le282488521294790766od_b_c @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_308_sup__absorb2,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( sup_sup_set_a @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_309_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_310_sup__absorb2,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( sup_su3823046536922626210od_b_c @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_311_sup__absorb1,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( sup_sup_set_a @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_312_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_313_sup__absorb1,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( sup_su3823046536922626210od_b_c @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_314_sup_Oabsorb2,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_315_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_316_sup_Oabsorb2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( sup_su3823046536922626210od_b_c @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_317_sup_Oabsorb1,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_318_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_319_sup_Oabsorb1,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( sup_su3823046536922626210od_b_c @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_320_sup__unique,axiom,
    ! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
      ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ Y4 @ X2 )
             => ( ( ord_less_eq_set_a @ Z3 @ X2 )
               => ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_set_a @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_321_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X2 )
             => ( ( ord_less_eq_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_322_sup__unique,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X2 @ ( F @ X2 @ Y4 ) )
     => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ Y4 @ ( F @ X2 @ Y4 ) )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c,Z3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ Y4 @ X2 )
             => ( ( ord_le282488521294790766od_b_c @ Z3 @ X2 )
               => ( ord_le282488521294790766od_b_c @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
         => ( ( sup_su3823046536922626210od_b_c @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_323_sup_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_324_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_325_sup_OorderI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( A2
        = ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) )
     => ( ord_le282488521294790766od_b_c @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_326_sup_OorderE,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_327_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_328_sup_OorderE,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( A2
        = ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_329_le__iff__sup,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( sup_sup_set_a @ X4 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_330_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y2: nat] :
          ( ( sup_sup_nat @ X4 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_331_le__iff__sup,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( sup_su3823046536922626210od_b_c @ X4 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_332_sup__least,axiom,
    ! [Y: set_a,X: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( ord_less_eq_set_a @ Z @ X )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_333_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_334_sup__least,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( ord_le282488521294790766od_b_c @ Z @ X )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_335_sup__mono,axiom,
    ! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_336_sup__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_337_sup__mono,axiom,
    ! [A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,D2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ C )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ D2 )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) @ ( sup_su3823046536922626210od_b_c @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_338_sup_Omono,axiom,
    ! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C @ A2 )
     => ( ( ord_less_eq_set_a @ D2 @ B2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_339_sup_Omono,axiom,
    ! [C: nat,A2: nat,D2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D2 @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_340_sup_Omono,axiom,
    ! [C: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,D2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ D2 @ B2 )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ C @ D2 ) @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_341_le__supI2,axiom,
    ! [X: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ X @ B2 )
     => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_342_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_343_le__supI2,axiom,
    ! [X: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ B2 )
     => ( ord_le282488521294790766od_b_c @ X @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_344_le__supI1,axiom,
    ! [X: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X @ A2 )
     => ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_345_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_346_le__supI1,axiom,
    ! [X: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ A2 )
     => ( ord_le282488521294790766od_b_c @ X @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_347_sup__ge2,axiom,
    ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).

% sup_ge2
thf(fact_348_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_349_sup__ge2,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ Y @ ( sup_su3823046536922626210od_b_c @ X @ Y ) ) ).

% sup_ge2
thf(fact_350_sup__ge1,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).

% sup_ge1
thf(fact_351_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_352_sup__ge1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X @ ( sup_su3823046536922626210od_b_c @ X @ Y ) ) ).

% sup_ge1
thf(fact_353_le__supI,axiom,
    ! [A2: set_a,X: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X )
     => ( ( ord_less_eq_set_a @ B2 @ X )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_354_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_355_le__supI,axiom,
    ! [A2: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ X )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ X )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_356_le__supE,axiom,
    ! [A2: set_a,B2: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_a @ A2 @ X )
         => ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).

% le_supE
thf(fact_357_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_358_le__supE,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le282488521294790766od_b_c @ A2 @ X )
         => ~ ( ord_le282488521294790766od_b_c @ B2 @ X ) ) ) ).

% le_supE
thf(fact_359_inf__sup__ord_I3_J,axiom,
    ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_360_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_361_inf__sup__ord_I3_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X @ ( sup_su3823046536922626210od_b_c @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_362_inf__sup__ord_I4_J,axiom,
    ! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_363_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_364_inf__sup__ord_I4_J,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ Y @ ( sup_su3823046536922626210od_b_c @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_365_subset__Un__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( sup_sup_set_a @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_366_subset__Un__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
          ( ( sup_su3823046536922626210od_b_c @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_367_subset__UnE,axiom,
    ! [C2: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
     => ~ ! [A5: set_a] :
            ( ( ord_less_eq_set_a @ A5 @ A )
           => ! [B5: set_a] :
                ( ( ord_less_eq_set_a @ B5 @ B )
               => ( C2
                 != ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_368_subset__UnE,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
     => ~ ! [A5: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ A5 @ A )
           => ! [B5: set_li6436108459499378894od_b_c] :
                ( ( ord_le282488521294790766od_b_c @ B5 @ B )
               => ( C2
                 != ( sup_su3823046536922626210od_b_c @ A5 @ B5 ) ) ) ) ) ).

% subset_UnE
thf(fact_369_Un__absorb2,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( sup_sup_set_a @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_370_Un__absorb2,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ A )
     => ( ( sup_su3823046536922626210od_b_c @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_371_Un__absorb1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( sup_sup_set_a @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_372_Un__absorb1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( sup_su3823046536922626210od_b_c @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_373_Un__upper2,axiom,
    ! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).

% Un_upper2
thf(fact_374_Un__upper2,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ B @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ).

% Un_upper2
thf(fact_375_Un__upper1,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).

% Un_upper1
thf(fact_376_Un__upper1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ).

% Un_upper1
thf(fact_377_Un__least,axiom,
    ! [A: set_a,C2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_378_Un__least,axiom,
    ! [A: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ B @ C2 )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_379_Un__mono,axiom,
    ! [A: set_a,C2: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_380_Un__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ B @ D )
       => ( ord_le282488521294790766od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) @ ( sup_su3823046536922626210od_b_c @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_381_language__prefix,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M2 @ Q ) )
     => ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M2 @ Q ) ) ) ).

% language_prefix
thf(fact_382_ex__in__conv,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ? [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A ) )
      = ( A != bot_bo4166481423041325370od_b_c ) ) ).

% ex_in_conv
thf(fact_383_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X4: a] : ( member_a @ X4 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_384_equals0I,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ! [Y4: list_P903359562653991662od_b_c] :
          ~ ( member6330420149250801815od_b_c @ Y4 @ A )
     => ( A = bot_bo4166481423041325370od_b_c ) ) ).

% equals0I
thf(fact_385_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y4: a] :
          ~ ( member_a @ Y4 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_386_equals0D,axiom,
    ! [A: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( A = bot_bo4166481423041325370od_b_c )
     => ~ ( member6330420149250801815od_b_c @ A2 @ A ) ) ).

% equals0D
thf(fact_387_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_388_emptyE,axiom,
    ! [A2: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) ).

% emptyE
thf(fact_389_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_390_sup__left__commute,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
      = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_391_sup_Oleft__commute,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
      = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_392_sup__commute,axiom,
    ( sup_sup_set_a
    = ( ^ [X4: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X4 ) ) ) ).

% sup_commute
thf(fact_393_sup_Ocommute,axiom,
    ( sup_sup_set_a
    = ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_394_sup__assoc,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
      = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_395_sup_Oassoc,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
      = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_396_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_a
    = ( ^ [X4: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_397_inf__sup__aci_I6_J,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
      = ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_398_inf__sup__aci_I7_J,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
      = ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_399_inf__sup__aci_I8_J,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
      = ( sup_sup_set_a @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_400_observable__after__language__append,axiom,
    ! [M2: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M2 @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_401_observable__after__language__none,axiom,
    ! [M2: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M2 @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_402_observable__after__eq,axiom,
    ! [M2: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( ( after_a_b_c @ M2 @ Q @ Io1 )
          = ( after_a_b_c @ M2 @ Q @ Io2 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M2 @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M2 @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_a_b_c @ M2 @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_403_after__language__iff,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Q: a,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M2 @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_404_after__split,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ Q ) )
       => ( ( after_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q @ Alpha ) @ Gamma )
          = ( after_a_b_c @ M2 @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_405_Un__left__commute,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
      = ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_406_Un__left__absorb,axiom,
    ! [A: set_a,B: set_a] :
      ( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
      = ( sup_sup_set_a @ A @ B ) ) ).

% Un_left_absorb
thf(fact_407_Un__commute,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_408_Un__absorb,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_409_Un__assoc,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
      = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_410_ball__Un,axiom,
    ! [A: set_a,B: set_a,P: a > $o] :
      ( ( ! [X4: a] :
            ( ( member_a @ X4 @ ( sup_sup_set_a @ A @ B ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ( P @ X4 ) )
        & ! [X4: a] :
            ( ( member_a @ X4 @ B )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_411_bex__Un,axiom,
    ! [A: set_a,B: set_a,P: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( sup_sup_set_a @ A @ B ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( P @ X4 ) )
        | ? [X4: a] :
            ( ( member_a @ X4 @ B )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_412_UnI2,axiom,
    ! [C: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ B )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnI2
thf(fact_413_UnI2,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( member_a @ C @ B )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI2
thf(fact_414_UnI1,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnI1
thf(fact_415_UnI1,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI1
thf(fact_416_UnE,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
     => ( ~ ( member6330420149250801815od_b_c @ C @ A )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% UnE
thf(fact_417_UnE,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
     => ( ~ ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% UnE
thf(fact_418_distinguish__prepend,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Q1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Q22: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( distin2804555989863659119_c_b_c @ M2 @ ( after_4052058690717316294_c_b_c @ M2 @ Q1 @ Io ) @ ( after_4052058690717316294_c_b_c @ M2 @ Q22 @ Io ) @ W )
       => ( ( member6330420149250801815od_b_c @ Q1 @ ( states7681702920031268536_c_b_c @ M2 ) )
         => ( ( member6330420149250801815od_b_c @ Q22 @ ( states7681702920031268536_c_b_c @ M2 ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q22 ) )
               => ( distin2804555989863659119_c_b_c @ M2 @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_419_distinguish__prepend,axiom,
    ! [M2: fsm_a_b_c,Q1: a,Io: list_P903359562653991662od_b_c,Q22: a,W: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( distinguishes_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q1 @ Io ) @ ( after_a_b_c @ M2 @ Q22 @ Io ) @ W )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M2 ) )
         => ( ( member_a @ Q22 @ ( states_a_b_c @ M2 ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q22 ) )
               => ( distinguishes_a_b_c @ M2 @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_420_Un__empty__right,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ bot_bot_set_a )
      = A ) ).

% Un_empty_right
thf(fact_421_Un__empty__left,axiom,
    ! [B: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ B )
      = B ) ).

% Un_empty_left
thf(fact_422_minimally__distinguishes__after__append,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Q1: list_P903359562653991662od_b_c,Q22: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( minima1987601567150520449_c_b_c @ M2 )
       => ( ( member6330420149250801815od_b_c @ Q1 @ ( states7681702920031268536_c_b_c @ M2 ) )
         => ( ( member6330420149250801815od_b_c @ Q22 @ ( states7681702920031268536_c_b_c @ M2 ) )
           => ( ( minima9089413714839006869_c_b_c @ M2 @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima9089413714839006869_c_b_c @ M2 @ ( after_4052058690717316294_c_b_c @ M2 @ Q1 @ W ) @ ( after_4052058690717316294_c_b_c @ M2 @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_423_minimally__distinguishes__after__append,axiom,
    ! [M2: fsm_a_b_c,Q1: a,Q22: a,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M2 ) )
         => ( ( member_a @ Q22 @ ( states_a_b_c @ M2 ) )
           => ( ( minima243535863231358885_a_b_c @ M2 @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima243535863231358885_a_b_c @ M2 @ ( after_a_b_c @ M2 @ Q1 @ W ) @ ( after_a_b_c @ M2 @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_424_append__is__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = nil_Product_prod_b_c )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% append_is_Nil_conv
thf(fact_425_Nil__is__append__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( nil_Product_prod_b_c
        = ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% Nil_is_append_conv
thf(fact_426_self__append__conv2,axiom,
    ! [Y: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( Y
        = ( append2547753245680614915od_b_c @ Xs @ Y ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% self_append_conv2
thf(fact_427_append__self__conv2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% append_self_conv2
thf(fact_428_self__append__conv,axiom,
    ! [Y: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Y
        = ( append2547753245680614915od_b_c @ Y @ Ys ) )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% self_append_conv
thf(fact_429_append__self__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% append_self_conv
thf(fact_430_append__Nil2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ Xs @ nil_Product_prod_b_c )
      = Xs ) ).

% append_Nil2
thf(fact_431_append_Oright__neutral,axiom,
    ! [A2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ A2 @ nil_Product_prod_b_c )
      = A2 ) ).

% append.right_neutral
thf(fact_432_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ X @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% boolean_algebra.conj_zero_right
thf(fact_433_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_a] :
      ( ( inf_inf_set_a @ bot_bot_set_a @ X )
      = bot_bot_set_a ) ).

% boolean_algebra.conj_zero_left
thf(fact_434_same__append__eq,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = ( append2547753245680614915od_b_c @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_435_append_Oassoc,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ A2 @ B2 ) @ C )
      = ( append2547753245680614915od_b_c @ A2 @ ( append2547753245680614915od_b_c @ B2 @ C ) ) ) ).

% append.assoc
thf(fact_436_append__assoc,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys ) @ Zs )
      = ( append2547753245680614915od_b_c @ Xs @ ( append2547753245680614915od_b_c @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_437_append__same__eq,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Xs )
        = ( append2547753245680614915od_b_c @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_438_distinguishes__def,axiom,
    ( distinguishes_a_b_c
    = ( ^ [M: fsm_a_b_c,Q12: a,Q23: a,Io3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Io3 @ ( sup_su3823046536922626210od_b_c @ ( lS_a_b_c @ M @ Q12 ) @ ( lS_a_b_c @ M @ Q23 ) ) )
          & ~ ( member6330420149250801815od_b_c @ Io3 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M @ Q12 ) @ ( lS_a_b_c @ M @ Q23 ) ) ) ) ) ) ).

% distinguishes_def
thf(fact_439_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
      ( ( A
        = ( inf_inf_set_a @ K @ A2 ) )
     => ( ( inf_inf_set_a @ A @ B2 )
        = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_440_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
      ( ( B
        = ( inf_inf_set_a @ K @ B2 ) )
     => ( ( inf_inf_set_a @ A2 @ B )
        = ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_441_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
      ( ( A
        = ( sup_sup_set_a @ K @ A2 ) )
     => ( ( sup_sup_set_a @ A @ B2 )
        = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_442_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
      ( ( B
        = ( sup_sup_set_a @ K @ B2 ) )
     => ( ( sup_sup_set_a @ A2 @ B )
        = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_443_append__eq__appendI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xs1: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Us: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2547753245680614915od_b_c @ Xs1 @ Us ) )
       => ( ( append2547753245680614915od_b_c @ Xs @ Ys )
          = ( append2547753245680614915od_b_c @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_444_append__eq__append__conv2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ts: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = ( append2547753245680614915od_b_c @ Zs @ Ts ) )
      = ( ? [Us2: list_P903359562653991662od_b_c] :
            ( ( ( Xs
                = ( append2547753245680614915od_b_c @ Zs @ Us2 ) )
              & ( ( append2547753245680614915od_b_c @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append2547753245680614915od_b_c @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append2547753245680614915od_b_c @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_445_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_a] :
      ( ( sup_sup_set_a @ X @ bot_bot_set_a )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_446_append__Nil,axiom,
    ! [Ys: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_447_append_Oleft__neutral,axiom,
    ! [A2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_448_eq__Nil__appendI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_449_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_450_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_451_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_a,Z: set_a,X: set_a] :
      ( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
      = ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_452_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_a,Z: set_a,X: set_a] :
      ( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
      = ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_453_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_454_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_455_order__refl,axiom,
    ! [X: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X @ X ) ).

% order_refl
thf(fact_456_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_457_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_458_dual__order_Orefl,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_459_bot__empty__eq,axiom,
    ( bot_bo5496101219168594979_b_c_o
    = ( ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ bot_bo4166481423041325370od_b_c ) ) ) ).

% bot_empty_eq
thf(fact_460_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_461_Collect__empty__eq__bot,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( P = bot_bot_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_462_minimally__distinguishes__no__prefix,axiom,
    ! [M2: fsm_a_b_c,U: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c,W3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U @ W ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ V2 @ W ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ( minima243535863231358885_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ U ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ V2 ) @ ( append2547753245680614915od_b_c @ W @ ( append2547753245680614915od_b_c @ W2 @ W3 ) ) )
           => ( ( W2 != nil_Product_prod_b_c )
             => ~ ( distinguishes_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ V2 @ W ) ) @ W3 ) ) ) ) ) ) ).

% minimally_distinguishes_no_prefix
thf(fact_463_fsm__initial,axiom,
    ! [M2: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M2 ) @ ( states_a_b_c @ M2 ) ) ).

% fsm_initial
thf(fact_464_language__contains__empty__sequence,axiom,
    ! [M2: fsm_a_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ).

% language_contains_empty_sequence
thf(fact_465_order__antisym__conv,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_466_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_467_order__antisym__conv,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( ord_le282488521294790766od_b_c @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_468_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_469_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_470_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_471_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_472_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_473_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_474_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_475_ord__le__eq__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_a,C: set_a] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_476_ord__le__eq__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_477_ord__le__eq__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_478_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_479_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_480_ord__eq__le__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_a > set_li6436108459499378894od_b_c,B2: set_a,C: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_481_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_482_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_483_ord__eq__le__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_484_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_li6436108459499378894od_b_c > set_a,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_485_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_li6436108459499378894od_b_c > nat,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_486_ord__eq__le__subst,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_487_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_488_order__eq__refl,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( X = Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_489_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_490_order__eq__refl,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( X = Y )
     => ( ord_le282488521294790766od_b_c @ X @ Y ) ) ).

% order_eq_refl
thf(fact_491_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_492_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_493_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_494_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_495_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_496_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_497_order__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_a,C: set_a] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_498_order__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_499_order__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_500_order__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_501_order__subst1,axiom,
    ! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_502_order__subst1,axiom,
    ! [A2: set_a,F: set_li6436108459499378894od_b_c > set_a,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_503_order__subst1,axiom,
    ! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_504_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_505_order__subst1,axiom,
    ! [A2: nat,F: set_li6436108459499378894od_b_c > nat,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_506_order__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_a > set_li6436108459499378894od_b_c,B2: set_a,C: set_a] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_507_order__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B2: nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_508_order__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le282488521294790766od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_509_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_510_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_511_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] : ( Y3 = Z2 ) )
    = ( ^ [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_512_antisym,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_513_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_514_antisym,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_515_dual__order_Otrans,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C @ B2 )
       => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_516_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_517_dual__order_Otrans,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ C @ B2 )
       => ( ord_le282488521294790766od_b_c @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_518_dual__order_Oantisym,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_519_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_520_dual__order_Oantisym,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_521_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_522_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_523_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] : ( Y3 = Z2 ) )
    = ( ^ [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_524_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
         => ( P @ A6 @ B6 ) )
     => ( ! [A6: nat,B6: nat] :
            ( ( P @ B6 @ A6 )
           => ( P @ A6 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_525_order__trans,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z )
       => ( ord_less_eq_set_a @ X @ Z ) ) ) ).

% order_trans
thf(fact_526_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_527_order__trans,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ Z )
       => ( ord_le282488521294790766od_b_c @ X @ Z ) ) ) ).

% order_trans
thf(fact_528_order_Otrans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_529_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_530_order_Otrans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% order.trans
thf(fact_531_order__antisym,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_532_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_533_order__antisym,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_534_ord__le__eq__trans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_535_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_536_ord__le__eq__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_537_ord__eq__le__trans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_538_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_539_ord__eq__le__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( A2 = B2 )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_540_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y2 )
          & ( ord_less_eq_set_a @ Y2 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_541_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
    = ( ^ [X4: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_542_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] : ( Y3 = Z2 ) )
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ X4 @ Y2 )
          & ( ord_le282488521294790766od_b_c @ Y2 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_543_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_544_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_545_after__language__append__iff,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_546_after__language__subset,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_547_distinguish__prepend__initial,axiom,
    ! [M2: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( distinguishes_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ Io1 @ Io ) ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ Io2 @ Io ) ) @ W )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
           => ( distinguishes_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ Io1 ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ Io2 ) @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ).

% distinguish_prepend_initial
thf(fact_548_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
     => ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_549_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_550_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ bot_bo4166481423041325370od_b_c )
     => ( A2 = bot_bo4166481423041325370od_b_c ) ) ).

% bot.extremum_uniqueI
thf(fact_551_bot_Oextremum__unique,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
      = ( A2 = bot_bot_set_a ) ) ).

% bot.extremum_unique
thf(fact_552_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_553_bot_Oextremum__unique,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ bot_bo4166481423041325370od_b_c )
      = ( A2 = bot_bo4166481423041325370od_b_c ) ) ).

% bot.extremum_unique
thf(fact_554_bot_Oextremum,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).

% bot.extremum
thf(fact_555_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_556_bot_Oextremum,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ bot_bo4166481423041325370od_b_c @ A2 ) ).

% bot.extremum
thf(fact_557_minimally__distinguishes__after__append__initial,axiom,
    ! [M2: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( member6330420149250801815od_b_c @ U @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ V2 @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
           => ( ( minima243535863231358885_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ U ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ V2 ) @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima243535863231358885_a_b_c @ M2 @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ ( append2547753245680614915od_b_c @ V2 @ W ) ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append_initial
thf(fact_558_W__finite,axiom,
    ! [S2: set_a,S3: set_a] : ( finite3074115686814133143od_b_c @ ( w @ S2 @ S3 ) ) ).

% W_finite
thf(fact_559_subset__emptyI,axiom,
    ! [A: set_a] :
      ( ! [X2: a] :
          ~ ( member_a @ X2 @ A )
     => ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).

% subset_emptyI
thf(fact_560_subset__emptyI,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ! [X2: list_P903359562653991662od_b_c] :
          ~ ( member6330420149250801815od_b_c @ X2 @ A )
     => ( ord_le282488521294790766od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ).

% subset_emptyI
thf(fact_561_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( maxima8146652644187019584_c_b_c @ M2 @ Q @ Io ) @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_562_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M2: fsm_a_b_c,Q: a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member_a @ Q @ ( states_a_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( maxima1559550560783484624_a_b_c @ M2 @ Q @ Io ) @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_563__092_060open_062finite_AS_092_060close_062,axiom,
    finite_finite_a @ sa ).

% \<open>finite S\<close>
thf(fact_564_k__def,axiom,
    ( k2
    = ( finite_card_a @ s ) ) ).

% k_def
thf(fact_565_is__in__language__iff,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Q: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Q @ ( states7681702920031268536_c_b_c @ M2 ) )
       => ( ( is_in_7104650932667917939_c_b_c @ M2 @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_566_is__in__language__iff,axiom,
    ! [M2: fsm_a_b_c,Q: a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member_a @ Q @ ( states_a_b_c @ M2 ) )
       => ( ( is_in_language_a_b_c @ M2 @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_567_less_Oprems_I1_J,axiom,
    ( ka
    = ( finite_card_a @ sa ) ) ).

% less.prems(1)
thf(fact_568_fsm__states__finite,axiom,
    ! [M2: fsm_a_b_c] : ( finite_finite_a @ ( states_a_b_c @ M2 ) ) ).

% fsm_states_finite
thf(fact_569__092_060open_062card_AS_A_061_Acard_AS1_A_L_Acard_AS2_092_060close_062,axiom,
    ( ( finite_card_a @ sa )
    = ( plus_plus_nat @ ( finite_card_a @ s1 ) @ ( finite_card_a @ s2 ) ) ) ).

% \<open>card S = card S1 + card S2\<close>
thf(fact_570_finite__Un,axiom,
    ! [F2: set_li6436108459499378894od_b_c,G: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( sup_su3823046536922626210od_b_c @ F2 @ G ) )
      = ( ( finite3074115686814133143od_b_c @ F2 )
        & ( finite3074115686814133143od_b_c @ G ) ) ) ).

% finite_Un
thf(fact_571_finite__Un,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G ) ) ) ).

% finite_Un
thf(fact_572_finite__Un,axiom,
    ! [F2: set_a,G: set_a] :
      ( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
      = ( ( finite_finite_a @ F2 )
        & ( finite_finite_a @ G ) ) ) ).

% finite_Un
thf(fact_573_finite__Int,axiom,
    ! [F2: set_li6436108459499378894od_b_c,G: set_li6436108459499378894od_b_c] :
      ( ( ( finite3074115686814133143od_b_c @ F2 )
        | ( finite3074115686814133143od_b_c @ G ) )
     => ( finite3074115686814133143od_b_c @ ( inf_in4978071631833541052od_b_c @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_574_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_575_finite__Int,axiom,
    ! [F2: set_a,G: set_a] :
      ( ( ( finite_finite_a @ F2 )
        | ( finite_finite_a @ G ) )
     => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_576_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 )
            & ~ ? [Zs2: list_P903359562653991662od_b_c] :
                  ( ( Zs2 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Xs @ ( append2547753245680614915od_b_c @ Ys2 @ Zs2 ) ) @ S ) ) ) ) ) ).

% finite_set_elem_maximal_extension_ex
thf(fact_577_prefix__free__set__maximal__list__ob,axiom,
    ! [Xs: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ Xs )
     => ( ( member6330420149250801815od_b_c @ X @ Xs )
       => ~ ! [X5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ X @ X5 ) @ Xs )
             => ? [Y5: list_P903359562653991662od_b_c] :
                  ( ( Y5 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ X @ X5 ) @ Y5 ) @ Xs ) ) ) ) ) ).

% prefix_free_set_maximal_list_ob
thf(fact_578_card__mono,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).

% card_mono
thf(fact_579_card__mono,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).

% card_mono
thf(fact_580_card__mono,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ).

% card_mono
thf(fact_581_card__Un__le,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ).

% card_Un_le
thf(fact_582_card__Un__Int,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( ( plus_plus_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) )
          = ( plus_plus_nat @ ( finite5583770498833199894od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) @ ( finite5583770498833199894od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_583_card__Un__Int,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
          = ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A @ B ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_584_card__Un__Int,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
          = ( plus_plus_nat @ ( finite_card_a @ ( sup_sup_set_a @ A @ B ) ) @ ( finite_card_a @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_585_finite__set__min__param__ex,axiom,
    ! [XS: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > nat > $o] :
      ( ( finite3074115686814133143od_b_c @ XS )
     => ( ! [X2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X2 @ XS )
           => ? [K2: nat] :
              ! [K3: nat] :
                ( ( ord_less_eq_nat @ K2 @ K3 )
               => ( P @ X2 @ K3 ) ) )
       => ? [K4: nat] :
          ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ XS )
           => ( P @ X3 @ K4 ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_586_finite__set__min__param__ex,axiom,
    ! [XS: set_a,P: a > nat > $o] :
      ( ( finite_finite_a @ XS )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ XS )
           => ? [K2: nat] :
              ! [K3: nat] :
                ( ( ord_less_eq_nat @ K2 @ K3 )
               => ( P @ X2 @ K3 ) ) )
       => ? [K4: nat] :
          ! [X3: a] :
            ( ( member_a @ X3 @ XS )
           => ( P @ X3 @ K4 ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_587_finite__set__min__param__ex,axiom,
    ! [XS: set_nat,P: nat > nat > $o] :
      ( ( finite_finite_nat @ XS )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ XS )
           => ? [K2: nat] :
              ! [K3: nat] :
                ( ( ord_less_eq_nat @ K2 @ K3 )
               => ( P @ X2 @ K3 ) ) )
       => ? [K4: nat] :
          ! [X3: nat] :
            ( ( member_nat @ X3 @ XS )
           => ( P @ X3 @ K4 ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_588_card__Un__disjoint,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( ( ( inf_in4978071631833541052od_b_c @ A @ B )
            = bot_bo4166481423041325370od_b_c )
         => ( ( finite5583770498833199894od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
            = ( plus_plus_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_589_card__Un__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ( inf_inf_set_nat @ A @ B )
            = bot_bot_set_nat )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_590_card__Un__disjoint,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ( ( inf_inf_set_a @ A @ B )
            = bot_bot_set_a )
         => ( ( finite_card_a @ ( sup_sup_set_a @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_591_finite__has__maximal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ( ord_less_eq_set_a @ A2 @ X2 )
            & ! [Xa2: set_a] :
                ( ( member_set_a @ Xa2 @ A )
               => ( ( ord_less_eq_set_a @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_592_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ A2 @ X2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_593_finite__has__maximal2,axiom,
    ! [A: set_se3924713247505902254od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( member6985331446368301687od_b_c @ A2 @ A )
       => ? [X2: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X2 @ A )
            & ( ord_le282488521294790766od_b_c @ A2 @ X2 )
            & ! [Xa2: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa2 @ A )
               => ( ( ord_le282488521294790766od_b_c @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_594_finite__has__minimal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ( ord_less_eq_set_a @ X2 @ A2 )
            & ! [Xa2: set_a] :
                ( ( member_set_a @ Xa2 @ A )
               => ( ( ord_less_eq_set_a @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_595_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ X2 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_596_finite__has__minimal2,axiom,
    ! [A: set_se3924713247505902254od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( member6985331446368301687od_b_c @ A2 @ A )
       => ? [X2: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X2 @ A )
            & ( ord_le282488521294790766od_b_c @ X2 @ A2 )
            & ! [Xa2: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa2 @ A )
               => ( ( ord_le282488521294790766od_b_c @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_597_infinite__imp__nonempty,axiom,
    ! [S: set_li6436108459499378894od_b_c] :
      ( ~ ( finite3074115686814133143od_b_c @ S )
     => ( S != bot_bo4166481423041325370od_b_c ) ) ).

% infinite_imp_nonempty
thf(fact_598_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_599_infinite__imp__nonempty,axiom,
    ! [S: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ( S != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_600_finite_OemptyI,axiom,
    finite3074115686814133143od_b_c @ bot_bo4166481423041325370od_b_c ).

% finite.emptyI
thf(fact_601_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_602_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_603_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_604_finite__subset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ A ) ) ) ).

% finite_subset
thf(fact_605_finite__subset,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( finite3074115686814133143od_b_c @ A ) ) ) ).

% finite_subset
thf(fact_606_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_607_infinite__super,axiom,
    ! [S: set_a,T: set_a] :
      ( ( ord_less_eq_set_a @ S @ T )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ T ) ) ) ).

% infinite_super
thf(fact_608_infinite__super,axiom,
    ! [S: set_li6436108459499378894od_b_c,T: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ S @ T )
     => ( ~ ( finite3074115686814133143od_b_c @ S )
       => ~ ( finite3074115686814133143od_b_c @ T ) ) ) ).

% infinite_super
thf(fact_609_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_610_rev__finite__subset,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( finite_finite_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_611_rev__finite__subset,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( finite3074115686814133143od_b_c @ A ) ) ) ).

% rev_finite_subset
thf(fact_612_infinite__Un,axiom,
    ! [S: set_li6436108459499378894od_b_c,T: set_li6436108459499378894od_b_c] :
      ( ( ~ ( finite3074115686814133143od_b_c @ ( sup_su3823046536922626210od_b_c @ S @ T ) ) )
      = ( ~ ( finite3074115686814133143od_b_c @ S )
        | ~ ( finite3074115686814133143od_b_c @ T ) ) ) ).

% infinite_Un
thf(fact_613_infinite__Un,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_Un
thf(fact_614_infinite__Un,axiom,
    ! [S: set_a,T: set_a] :
      ( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
      = ( ~ ( finite_finite_a @ S )
        | ~ ( finite_finite_a @ T ) ) ) ).

% infinite_Un
thf(fact_615_Un__infinite,axiom,
    ! [S: set_li6436108459499378894od_b_c,T: set_li6436108459499378894od_b_c] :
      ( ~ ( finite3074115686814133143od_b_c @ S )
     => ~ ( finite3074115686814133143od_b_c @ ( sup_su3823046536922626210od_b_c @ S @ T ) ) ) ).

% Un_infinite
thf(fact_616_Un__infinite,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).

% Un_infinite
thf(fact_617_Un__infinite,axiom,
    ! [S: set_a,T: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).

% Un_infinite
thf(fact_618_finite__UnI,axiom,
    ! [F2: set_li6436108459499378894od_b_c,G: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ F2 )
     => ( ( finite3074115686814133143od_b_c @ G )
       => ( finite3074115686814133143od_b_c @ ( sup_su3823046536922626210od_b_c @ F2 @ G ) ) ) ) ).

% finite_UnI
thf(fact_619_finite__UnI,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).

% finite_UnI
thf(fact_620_finite__UnI,axiom,
    ! [F2: set_a,G: set_a] :
      ( ( finite_finite_a @ F2 )
     => ( ( finite_finite_a @ G )
       => ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).

% finite_UnI
thf(fact_621_finite__has__maximal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ! [Xa2: set_a] :
                ( ( member_set_a @ Xa2 @ A )
               => ( ( ord_less_eq_set_a @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_622_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_623_finite__has__maximal,axiom,
    ! [A: set_se3924713247505902254od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ? [X2: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X2 @ A )
            & ! [Xa2: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa2 @ A )
               => ( ( ord_le282488521294790766od_b_c @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_624_finite__has__minimal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ! [Xa2: set_a] :
                ( ( member_set_a @ Xa2 @ A )
               => ( ( ord_less_eq_set_a @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_625_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_626_finite__has__minimal,axiom,
    ! [A: set_se3924713247505902254od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ? [X2: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X2 @ A )
            & ! [Xa2: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa2 @ A )
               => ( ( ord_le282488521294790766od_b_c @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_627_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C2: nat] :
      ( ! [G2: set_nat] :
          ( ( ord_less_eq_set_nat @ G2 @ F2 )
         => ( ( finite_finite_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_628_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_a,C2: nat] :
      ( ! [G2: set_a] :
          ( ( ord_less_eq_set_a @ G2 @ F2 )
         => ( ( finite_finite_a @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C2 ) ) )
     => ( ( finite_finite_a @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_629_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_li6436108459499378894od_b_c,C2: nat] :
      ( ! [G2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ G2 @ F2 )
         => ( ( finite3074115686814133143od_b_c @ G2 )
           => ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ G2 ) @ C2 ) ) )
     => ( ( finite3074115686814133143od_b_c @ F2 )
        & ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_630_infinite__arbitrarily__large,axiom,
    ! [A: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [B7: set_nat] :
          ( ( finite_finite_nat @ B7 )
          & ( ( finite_card_nat @ B7 )
            = N )
          & ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_631_infinite__arbitrarily__large,axiom,
    ! [A: set_a,N: nat] :
      ( ~ ( finite_finite_a @ A )
     => ? [B7: set_a] :
          ( ( finite_finite_a @ B7 )
          & ( ( finite_card_a @ B7 )
            = N )
          & ( ord_less_eq_set_a @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_632_infinite__arbitrarily__large,axiom,
    ! [A: set_li6436108459499378894od_b_c,N: nat] :
      ( ~ ( finite3074115686814133143od_b_c @ A )
     => ? [B7: set_li6436108459499378894od_b_c] :
          ( ( finite3074115686814133143od_b_c @ B7 )
          & ( ( finite5583770498833199894od_b_c @ B7 )
            = N )
          & ( ord_le282488521294790766od_b_c @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_633_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
     => ~ ! [T3: set_nat] :
            ( ( ord_less_eq_set_nat @ T3 @ S )
           => ( ( ( finite_card_nat @ T3 )
                = N )
             => ~ ( finite_finite_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_634_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_a] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
     => ~ ! [T3: set_a] :
            ( ( ord_less_eq_set_a @ T3 @ S )
           => ( ( ( finite_card_a @ T3 )
                = N )
             => ~ ( finite_finite_a @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_635_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ N @ ( finite5583770498833199894od_b_c @ S ) )
     => ~ ! [T3: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ T3 @ S )
           => ( ( ( finite5583770498833199894od_b_c @ T3 )
                = N )
             => ~ ( finite3074115686814133143od_b_c @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_636_exists__subset__between,axiom,
    ! [A: set_nat,N: nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
       => ( ( ord_less_eq_set_nat @ A @ C2 )
         => ( ( finite_finite_nat @ C2 )
           => ? [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ A @ B7 )
                & ( ord_less_eq_set_nat @ B7 @ C2 )
                & ( ( finite_card_nat @ B7 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_637_exists__subset__between,axiom,
    ! [A: set_a,N: nat,C2: set_a] :
      ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
       => ( ( ord_less_eq_set_a @ A @ C2 )
         => ( ( finite_finite_a @ C2 )
           => ? [B7: set_a] :
                ( ( ord_less_eq_set_a @ A @ B7 )
                & ( ord_less_eq_set_a @ B7 @ C2 )
                & ( ( finite_card_a @ B7 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_638_exists__subset__between,axiom,
    ! [A: set_li6436108459499378894od_b_c,N: nat,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite5583770498833199894od_b_c @ C2 ) )
       => ( ( ord_le282488521294790766od_b_c @ A @ C2 )
         => ( ( finite3074115686814133143od_b_c @ C2 )
           => ? [B7: set_li6436108459499378894od_b_c] :
                ( ( ord_le282488521294790766od_b_c @ A @ B7 )
                & ( ord_le282488521294790766od_b_c @ B7 @ C2 )
                & ( ( finite5583770498833199894od_b_c @ B7 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_639_card__subset__eq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_640_card__subset__eq,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( ( finite_card_a @ A )
            = ( finite_card_a @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_641_card__subset__eq,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( ( ( finite5583770498833199894od_b_c @ A )
            = ( finite5583770498833199894od_b_c @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_642_card__seteq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_643_card__seteq,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_644_card__seteq,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ B ) @ ( finite5583770498833199894od_b_c @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_645_Suc,axiom,
    ( ka
    = ( suc @ k ) ) ).

% Suc
thf(fact_646_nat__add__left__cancel__le,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M3 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_647__092_060open_062card_AS2_A_060_Ak_092_060close_062,axiom,
    ord_less_nat @ ( finite_card_a @ s2 ) @ ka ).

% \<open>card S2 < k\<close>
thf(fact_648_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_649_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_650__092_060open_062card_AS1_A_060_Ak_092_060close_062,axiom,
    ord_less_nat @ ( finite_card_a @ s1 ) @ ka ).

% \<open>card S1 < k\<close>
thf(fact_651_Suc__le__mono,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
      = ( ord_less_eq_nat @ N @ M3 ) ) ).

% Suc_le_mono
thf(fact_652_minimal__fixpoint__helper_I2_J,axiom,
    ! [F: nat > nat,P: nat > $o,K: nat,X: nat,X6: nat] :
      ( ( F
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F @ ( suc @ X4 ) ) ) ) )
     => ( ! [X2: nat] :
            ( ( ord_less_eq_nat @ K @ X2 )
           => ( P @ X2 ) )
       => ( ( ord_less_eq_nat @ X @ X6 )
         => ( ( ord_less_nat @ X6 @ ( F @ X ) )
           => ~ ( P @ X6 ) ) ) ) ) ).

% minimal_fixpoint_helper(2)
thf(fact_653_Suc__leD,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% Suc_leD
thf(fact_654_Suc__leI,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).

% Suc_leI
thf(fact_655_le__SucE,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M3 @ N )
       => ( M3
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_656_le__SucI,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_657_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_658_Suc__le__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
      = ( ord_less_nat @ M3 @ N ) ) ).

% Suc_le_eq
thf(fact_659_le__Suc__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M3 @ N )
        | ( M3
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_660_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_661_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_662_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M6 @ N3 )
          & ( M6 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_663_Suc__le__lessD,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
     => ( ord_less_nat @ M3 @ N ) ) ).

% Suc_le_lessD
thf(fact_664_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_665_le__less__Suc__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M3 ) )
        = ( N = M3 ) ) ) ).

% le_less_Suc_eq
thf(fact_666_less__Suc__eq__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_Suc_eq_le
thf(fact_667_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_668_not__less__eq__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).

% not_less_eq_eq
thf(fact_669_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M7: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M7 ) @ N2 )
             => ( P @ M7 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_670_le__imp__less__Suc,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_671_less__imp__le__nat,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_imp_le_nat
thf(fact_672_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N3: nat] :
          ( ( ord_less_nat @ M6 @ N3 )
          | ( M6 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_673_less__or__eq__imp__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( ord_less_nat @ M3 @ N )
        | ( M3 = N ) )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_674_le__neq__implies__less,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( M3 != N )
       => ( ord_less_nat @ M3 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_675_nat__induct__at__least,axiom,
    ! [M3: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( P @ M3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_676_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_677_transitive__stepwise__le,axiom,
    ! [M3: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ! [X2: nat] : ( R @ X2 @ X2 )
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( ( R @ X2 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X2 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M3 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_678_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_679_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_680_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_681_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_682_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_683_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_684_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_685_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_686_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_687_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_688_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_689_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_690_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_691_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_692_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_693_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_694_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_695_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_696_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_697_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_698_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_699_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A6: nat,B6: nat] :
          ( ( ord_less_nat @ A6 @ B6 )
         => ( P @ A6 @ B6 ) )
     => ( ! [A6: nat] : ( P @ A6 @ A6 )
       => ( ! [A6: nat,B6: nat] :
              ( ( P @ B6 @ A6 )
             => ( P @ A6 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_700_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X7: nat] : ( P2 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N3 )
             => ~ ( P3 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_701_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_702_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_703_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_704_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_705_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X2: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X2 )
             => ( P @ Y6 ) )
         => ( P @ X2 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_706_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_707_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_708_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_709_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_710_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_711_order__le__imp__less__or__eq,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_set_a @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_712_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_713_order__le__imp__less__or__eq,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le5653067673530651002od_b_c @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_714_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_715_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_716_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_717_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le5653067673530651002od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_718_order__less__le__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
      ( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_719_order__less__le__subst1,axiom,
    ! [A2: nat,F: set_a > nat,B2: set_a,C: set_a] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_720_order__less__le__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_a > set_li6436108459499378894od_b_c,B2: set_a,C: set_a] :
      ( ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_721_order__less__le__subst1,axiom,
    ! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
      ( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_722_order__less__le__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_723_order__less__le__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B2: nat,C: nat] :
      ( ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_724_order__less__le__subst1,axiom,
    ! [A2: set_a,F: set_li6436108459499378894od_b_c > set_a,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_725_order__less__le__subst1,axiom,
    ! [A2: nat,F: set_li6436108459499378894od_b_c > nat,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_726_order__less__le__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_727_order__le__less__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_728_order__le__less__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_729_order__le__less__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le5653067673530651002od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: set_a,Y4: set_a] :
              ( ( ord_less_eq_set_a @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_730_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_731_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_732_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le5653067673530651002od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_733_order__le__less__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_a,C: set_a] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_less_set_a @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_734_order__le__less__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_735_order__le__less__subst2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le5653067673530651002od_b_c @ ( F @ B2 ) @ C )
       => ( ! [X2: set_li6436108459499378894od_b_c,Y4: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X2 @ Y4 )
             => ( ord_le282488521294790766od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_736_order__le__less__subst1,axiom,
    ! [A2: set_a,F: nat > set_a,B2: nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_737_order__le__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_738_order__le__less__subst1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,F: nat > set_li6436108459499378894od_b_c,B2: nat,C: nat] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X2: nat,Y4: nat] :
              ( ( ord_less_nat @ X2 @ Y4 )
             => ( ord_le5653067673530651002od_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( ord_le5653067673530651002od_b_c @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_739_order__less__le__trans,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z )
       => ( ord_less_set_a @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_740_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_741_order__less__le__trans,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ Z )
       => ( ord_le5653067673530651002od_b_c @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_742_order__le__less__trans,axiom,
    ! [X: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_set_a @ Y @ Z )
       => ( ord_less_set_a @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_743_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_744_order__le__less__trans,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le5653067673530651002od_b_c @ Y @ Z )
       => ( ord_le5653067673530651002od_b_c @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_745_order__neq__le__trans,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( ord_less_set_a @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_746_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_747_order__neq__le__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( A2 != B2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
       => ( ord_le5653067673530651002od_b_c @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_748_order__le__neq__trans,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_a @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_749_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_750_order__le__neq__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le5653067673530651002od_b_c @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_751_order__less__imp__le,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_set_a @ X @ Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_752_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_753_order__less__imp__le,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ X @ Y )
     => ( ord_le282488521294790766od_b_c @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_754_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_755_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_756_order__less__le,axiom,
    ( ord_less_set_a
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y2 )
          & ( X4 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_757_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y2 )
          & ( X4 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_758_order__less__le,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ X4 @ Y2 )
          & ( X4 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_759_order__le__less,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( ord_less_set_a @ X4 @ Y2 )
          | ( X4 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_760_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y2: nat] :
          ( ( ord_less_nat @ X4 @ Y2 )
          | ( X4 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_761_order__le__less,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( ord_le5653067673530651002od_b_c @ X4 @ Y2 )
          | ( X4 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_762_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_set_a @ B2 @ A2 )
     => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_763_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_764_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ B2 @ A2 )
     => ( ord_le282488521294790766od_b_c @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_765_order_Ostrict__implies__order,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A2 @ B2 )
     => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_766_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_767_order_Ostrict__implies__order,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A2 @ B2 )
     => ( ord_le282488521294790766od_b_c @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_768_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_769_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_770_dual__order_Ostrict__iff__not,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
          & ~ ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_771_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C @ B2 )
       => ( ord_less_set_a @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_772_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_773_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ B2 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ C @ B2 )
       => ( ord_le5653067673530651002od_b_c @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_774_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_set_a @ C @ B2 )
       => ( ord_less_set_a @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_775_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_776_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B2 @ A2 )
     => ( ( ord_le5653067673530651002od_b_c @ C @ B2 )
       => ( ord_le5653067673530651002od_b_c @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_777_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_778_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_779_dual__order_Ostrict__iff__order,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_780_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( ord_less_set_a @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_781_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_782_dual__order_Oorder__iff__strict,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
          ( ( ord_le5653067673530651002od_b_c @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_783_order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_784_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_785_order_Ostrict__iff__not,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
          & ~ ( ord_le282488521294790766od_b_c @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_786_order_Ostrict__trans2,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ord_less_set_a @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_787_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_788_order_Ostrict__trans2,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A2 @ B2 )
     => ( ( ord_le282488521294790766od_b_c @ B2 @ C )
       => ( ord_le5653067673530651002od_b_c @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_789_order_Ostrict__trans1,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_set_a @ B2 @ C )
       => ( ord_less_set_a @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_790_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_791_order_Ostrict__trans1,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B2 )
     => ( ( ord_le5653067673530651002od_b_c @ B2 @ C )
       => ( ord_le5653067673530651002od_b_c @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_792_order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_793_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_794_order_Ostrict__iff__order,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_795_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_set_a @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_796_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_797_order_Oorder__iff__strict,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le5653067673530651002od_b_c @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_798_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_799_less__le__not__le,axiom,
    ( ord_less_set_a
    = ( ^ [X4: set_a,Y2: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y2 )
          & ~ ( ord_less_eq_set_a @ Y2 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_800_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_801_less__le__not__le,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y2: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ X4 @ Y2 )
          & ~ ( ord_le282488521294790766od_b_c @ Y2 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_802_antisym__conv2,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ~ ( ord_less_set_a @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_803_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_804_antisym__conv2,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ~ ( ord_le5653067673530651002od_b_c @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_805_antisym__conv1,axiom,
    ! [X: set_a,Y: set_a] :
      ( ~ ( ord_less_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_806_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_807_antisym__conv1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ~ ( ord_le5653067673530651002od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_808_nless__le,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_809_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_810_nless__le,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B2: set_li6436108459499378894od_b_c] :
      ( ( ~ ( ord_le5653067673530651002od_b_c @ A2 @ B2 ) )
      = ( ~ ( ord_le282488521294790766od_b_c @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_811_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_812_leD,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ~ ( ord_less_set_a @ X @ Y ) ) ).

% leD
thf(fact_813_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_814_leD,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ~ ( ord_le5653067673530651002od_b_c @ X @ Y ) ) ).

% leD
thf(fact_815_bot_Onot__eq__extremum,axiom,
    ! [A2: set_a] :
      ( ( A2 != bot_bot_set_a )
      = ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_816_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_817_bot_Oextremum__strict,axiom,
    ! [A2: set_a] :
      ~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).

% bot.extremum_strict
thf(fact_818_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_819_less__infI1,axiom,
    ! [A2: set_a,X: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A2 @ X )
     => ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_820_less__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_821_less__infI2,axiom,
    ! [B2: set_a,X: set_a,A2: set_a] :
      ( ( ord_less_set_a @ B2 @ X )
     => ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_822_less__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_823_inf_Oabsorb3,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A2 @ B2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_824_inf_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_825_inf_Oabsorb4,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_set_a @ B2 @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_826_inf_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_827_inf_Ostrict__boundedE,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
     => ~ ( ( ord_less_set_a @ A2 @ B2 )
         => ~ ( ord_less_set_a @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_828_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
     => ~ ( ( ord_less_nat @ A2 @ B2 )
         => ~ ( ord_less_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_829_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_a
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( A4
            = ( inf_inf_set_a @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_830_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( A4
            = ( inf_inf_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_831_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_a,C: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A2 @ C )
     => ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_832_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_833_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_set_a @ B2 @ C )
     => ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_834_inf_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_835_less__supI1,axiom,
    ! [X: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_set_a @ X @ A2 )
     => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_836_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_837_less__supI2,axiom,
    ! [X: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_set_a @ X @ B2 )
     => ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_838_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_839_sup_Oabsorb3,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_set_a @ B2 @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_840_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_841_sup_Oabsorb4,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A2 @ B2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_842_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_843_sup_Ostrict__boundedE,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_a @ B2 @ A2 )
         => ~ ( ord_less_set_a @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_844_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_845_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_a
    = ( ^ [B4: set_a,A4: set_a] :
          ( ( A4
            = ( sup_sup_set_a @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_846_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_847_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_set_a @ C @ A2 )
     => ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_848_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_849_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_set_a @ C @ B2 )
     => ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_850_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_851_recursion__renaming__helper,axiom,
    ! [F1: nat > nat,P: nat > $o,F22: nat > nat,K: nat] :
      ( ( F1
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F1 @ ( suc @ X4 ) ) ) ) )
     => ( ( F22
          = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F22 @ ( suc @ X4 ) ) ) ) )
       => ( ! [X2: nat] :
              ( ( ord_less_eq_nat @ K @ X2 )
             => ( P @ X2 ) )
         => ( F1 = F22 ) ) ) ) ).

% recursion_renaming_helper
thf(fact_852_minimal__fixpoint__helper_I1_J,axiom,
    ! [F: nat > nat,P: nat > $o,K: nat,X: nat] :
      ( ( F
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F @ ( suc @ X4 ) ) ) ) )
     => ( ! [X2: nat] :
            ( ( ord_less_eq_nat @ K @ X2 )
           => ( P @ X2 ) )
       => ( P @ ( F @ X ) ) ) ) ).

% minimal_fixpoint_helper(1)
thf(fact_853_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_854_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_855_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_856_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_857_lift__Suc__mono__le,axiom,
    ! [F: nat > set_a,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_858_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_859_lift__Suc__mono__le,axiom,
    ! [F: nat > set_li6436108459499378894od_b_c,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_le282488521294790766od_b_c @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le282488521294790766od_b_c @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_860_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_a,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_861_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_862_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_li6436108459499378894od_b_c,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_le282488521294790766od_b_c @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_le282488521294790766od_b_c @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_863_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M3: nat,K: nat] :
      ( ! [M5: nat,N2: nat] :
          ( ( ord_less_nat @ M5 @ N2 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_864_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_865_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_866_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_867_eq__imp__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( M3 = N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% eq_imp_le
thf(fact_868_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_869_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_870_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_871_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_872_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_873_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_874_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_875_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ~ ! [C3: nat] :
            ( B2
           != ( plus_plus_nat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_876_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_877_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C4: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_878_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_879_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_880_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N3: nat] :
        ? [K5: nat] :
          ( N3
          = ( plus_plus_nat @ M6 @ K5 ) ) ) ) ).

% nat_le_iff_add
thf(fact_881_trans__le__add2,axiom,
    ! [I: nat,J: nat,M3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).

% trans_le_add2
thf(fact_882_trans__le__add1,axiom,
    ! [I: nat,J: nat,M3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).

% trans_le_add1
thf(fact_883_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_884_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_885_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_886_add__leD2,axiom,
    ! [M3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_887_add__leD1,axiom,
    ! [M3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% add_leD1
thf(fact_888_le__add2,axiom,
    ! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).

% le_add2
thf(fact_889_le__add1,axiom,
    ! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).

% le_add1
thf(fact_890_add__leE,axiom,
    ! [M3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M3 @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_891__092_060open_0620_A_060_Acard_AS2_092_060close_062,axiom,
    ord_less_nat @ zero_zero_nat @ ( finite_card_a @ s2 ) ).

% \<open>0 < card S2\<close>
thf(fact_892__092_060open_0620_A_060_Acard_AS1_092_060close_062,axiom,
    ord_less_nat @ zero_zero_nat @ ( finite_card_a @ s1 ) ).

% \<open>0 < card S1\<close>
thf(fact_893__092_060open_062length_Aw_H_A_060_Alength_Awk_092_060close_062,axiom,
    ord_less_nat @ ( size_s3392097710323735898od_b_c @ w3 ) @ ( size_s3392097710323735898od_b_c @ wk ) ).

% \<open>length w' < length wk\<close>
thf(fact_894_card__le__if__inj__on__rel,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,R2: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ! [A6: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ A6 @ A )
           => ? [B8: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: list_P903359562653991662od_b_c,A22: list_P903359562653991662od_b_c,B6: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ A1 @ A )
             => ( ( member6330420149250801815od_b_c @ A22 @ A )
               => ( ( member6330420149250801815od_b_c @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_895_card__le__if__inj__on__rel,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_a,R2: a > list_P903359562653991662od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ! [A6: a] :
            ( ( member_a @ A6 @ A )
           => ? [B8: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: a,A22: a,B6: list_P903359562653991662od_b_c] :
              ( ( member_a @ A1 @ A )
             => ( ( member_a @ A22 @ A )
               => ( ( member6330420149250801815od_b_c @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_896_card__le__if__inj__on__rel,axiom,
    ! [B: set_a,A: set_li6436108459499378894od_b_c,R2: list_P903359562653991662od_b_c > a > $o] :
      ( ( finite_finite_a @ B )
     => ( ! [A6: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ A6 @ A )
           => ? [B8: a] :
                ( ( member_a @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: list_P903359562653991662od_b_c,A22: list_P903359562653991662od_b_c,B6: a] :
              ( ( member6330420149250801815od_b_c @ A1 @ A )
             => ( ( member6330420149250801815od_b_c @ A22 @ A )
               => ( ( member_a @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_897_card__le__if__inj__on__rel,axiom,
    ! [B: set_a,A: set_a,R2: a > a > $o] :
      ( ( finite_finite_a @ B )
     => ( ! [A6: a] :
            ( ( member_a @ A6 @ A )
           => ? [B8: a] :
                ( ( member_a @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: a,A22: a,B6: a] :
              ( ( member_a @ A1 @ A )
             => ( ( member_a @ A22 @ A )
               => ( ( member_a @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_898_card__le__if__inj__on__rel,axiom,
    ! [B: set_nat,A: set_li6436108459499378894od_b_c,R2: list_P903359562653991662od_b_c > nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ! [A6: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ A6 @ A )
           => ? [B8: nat] :
                ( ( member_nat @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: list_P903359562653991662od_b_c,A22: list_P903359562653991662od_b_c,B6: nat] :
              ( ( member6330420149250801815od_b_c @ A1 @ A )
             => ( ( member6330420149250801815od_b_c @ A22 @ A )
               => ( ( member_nat @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_899_card__le__if__inj__on__rel,axiom,
    ! [B: set_nat,A: set_a,R2: a > nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ! [A6: a] :
            ( ( member_a @ A6 @ A )
           => ? [B8: nat] :
                ( ( member_nat @ B8 @ B )
                & ( R2 @ A6 @ B8 ) ) )
       => ( ! [A1: a,A22: a,B6: nat] :
              ( ( member_a @ A1 @ A )
             => ( ( member_a @ A22 @ A )
               => ( ( member_nat @ B6 @ B )
                 => ( ( R2 @ A1 @ B6 )
                   => ( ( R2 @ A22 @ B6 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_900_psubsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_a @ A @ B ) ) ) ).

% psubsetI
thf(fact_901_psubsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( A != B )
       => ( ord_le5653067673530651002od_b_c @ A @ B ) ) ) ).

% psubsetI
thf(fact_902_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_903_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_904_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_905_append__eq__append__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
      ( ( ( ( size_s3392097710323735898od_b_c @ Xs )
          = ( size_s3392097710323735898od_b_c @ Ys ) )
        | ( ( size_s3392097710323735898od_b_c @ Us )
          = ( size_s3392097710323735898od_b_c @ Vs ) ) )
     => ( ( ( append2547753245680614915od_b_c @ Xs @ Us )
          = ( append2547753245680614915od_b_c @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_906_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_907_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_908_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_909_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_910_length__0__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% length_0_conv
thf(fact_911_card_Oinfinite,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ~ ( finite3074115686814133143od_b_c @ A )
     => ( ( finite5583770498833199894od_b_c @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_912_card_Oinfinite,axiom,
    ! [A: set_a] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_card_a @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_913_card_Oinfinite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_card_nat @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_914_card_Oempty,axiom,
    ( ( finite_card_a @ bot_bot_set_a )
    = zero_zero_nat ) ).

% card.empty
thf(fact_915_length__append,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( size_s3392097710323735898od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s3392097710323735898od_b_c @ Xs ) @ ( size_s3392097710323735898od_b_c @ Ys ) ) ) ).

% length_append
thf(fact_916_length__greater__0__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3392097710323735898od_b_c @ Xs ) )
      = ( Xs != nil_Product_prod_b_c ) ) ).

% length_greater_0_conv
thf(fact_917_card__0__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( ( finite5583770498833199894od_b_c @ A )
          = zero_zero_nat )
        = ( A = bot_bo4166481423041325370od_b_c ) ) ) ).

% card_0_eq
thf(fact_918_card__0__eq,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_919_card__0__eq,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( ( finite_card_a @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_a ) ) ) ).

% card_0_eq
thf(fact_920_list_Osize_I3_J,axiom,
    ( ( size_s3392097710323735898od_b_c @ nil_Product_prod_b_c )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_921_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_P903359562653991662od_b_c] :
      ( ( size_s3392097710323735898od_b_c @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_922_neq__if__length__neq,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( size_s3392097710323735898od_b_c @ Xs )
       != ( size_s3392097710323735898od_b_c @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_923_length__induct,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs: list_P903359562653991662od_b_c] :
      ( ! [Xs2: list_P903359562653991662od_b_c] :
          ( ! [Ys3: list_P903359562653991662od_b_c] :
              ( ( ord_less_nat @ ( size_s3392097710323735898od_b_c @ Ys3 ) @ ( size_s3392097710323735898od_b_c @ Xs2 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_924_finite__psubset__induct,axiom,
    ! [A: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ! [A7: set_li6436108459499378894od_b_c] :
            ( ( finite3074115686814133143od_b_c @ A7 )
           => ( ! [B9: set_li6436108459499378894od_b_c] :
                  ( ( ord_le5653067673530651002od_b_c @ B9 @ A7 )
                 => ( P @ B9 ) )
             => ( P @ A7 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_925_finite__psubset__induct,axiom,
    ! [A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ A )
     => ( ! [A7: set_a] :
            ( ( finite_finite_a @ A7 )
           => ( ! [B9: set_a] :
                  ( ( ord_less_set_a @ B9 @ A7 )
                 => ( P @ B9 ) )
             => ( P @ A7 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_926_finite__psubset__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B9: set_nat] :
                  ( ( ord_less_set_nat @ B9 @ A7 )
                 => ( P @ B9 ) )
             => ( P @ A7 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_927_not__psubset__empty,axiom,
    ! [A: set_a] :
      ~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).

% not_psubset_empty
thf(fact_928_maximal__set__cover,axiom,
    ! [X8: set_set_a,S: set_a] :
      ( ( finite_finite_set_a @ X8 )
     => ( ( member_set_a @ S @ X8 )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ X8 )
            & ( ord_less_eq_set_a @ S @ X2 )
            & ! [Xa2: set_a] :
                ( ( member_set_a @ Xa2 @ X8 )
               => ~ ( ord_less_set_a @ X2 @ Xa2 ) ) ) ) ) ).

% maximal_set_cover
thf(fact_929_maximal__set__cover,axiom,
    ! [X8: set_se3924713247505902254od_b_c,S: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ X8 )
     => ( ( member6985331446368301687od_b_c @ S @ X8 )
       => ? [X2: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X2 @ X8 )
            & ( ord_le282488521294790766od_b_c @ S @ X2 )
            & ! [Xa2: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ Xa2 @ X8 )
               => ~ ( ord_le5653067673530651002od_b_c @ X2 @ Xa2 ) ) ) ) ) ).

% maximal_set_cover
thf(fact_930_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_set_a @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_931_subset__iff__psubset__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
          ( ( ord_le5653067673530651002od_b_c @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_932_subset__psubset__trans,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_set_a @ B @ C2 )
       => ( ord_less_set_a @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_933_subset__psubset__trans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( ord_le5653067673530651002od_b_c @ B @ C2 )
       => ( ord_le5653067673530651002od_b_c @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_934_subset__not__subset__eq,axiom,
    ( ord_less_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B3 )
          & ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_935_subset__not__subset__eq,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A3 @ B3 )
          & ~ ( ord_le282488521294790766od_b_c @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_936_psubset__subset__trans,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_set_a @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_937_psubset__subset__trans,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ C2 )
       => ( ord_le5653067673530651002od_b_c @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_938_psubset__imp__subset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_set_a @ A @ B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_939_psubset__imp__subset,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A @ B )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_940_psubset__eq,axiom,
    ( ord_less_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_941_psubset__eq,axiom,
    ( ord_le5653067673530651002od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_942_psubsetE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_set_a @ A @ B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ( ord_less_eq_set_a @ B @ A ) ) ) ).

% psubsetE
thf(fact_943_psubsetE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A @ B )
     => ~ ( ( ord_le282488521294790766od_b_c @ A @ B )
         => ( ord_le282488521294790766od_b_c @ B @ A ) ) ) ).

% psubsetE
thf(fact_944_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_945_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_946_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_947_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_948_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_949_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_950_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_951_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_952_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_953_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_954_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_955_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_956_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_957_finite__maxlen,axiom,
    ! [M2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ M2 )
     => ? [N2: nat] :
        ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ M2 )
         => ( ord_less_nat @ ( size_s3392097710323735898od_b_c @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_958_min__length__elem,axiom,
    ! [Xs: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ Xs )
     => ( ( Xs != bot_bo4166481423041325370od_b_c )
       => ? [X2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X2 @ Xs )
            & ~ ? [Xa2: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Xa2 @ Xs )
                  & ( ord_less_nat @ ( size_s3392097710323735898od_b_c @ Xa2 ) @ ( size_s3392097710323735898od_b_c @ X2 ) ) ) ) ) ) ).

% min_length_elem
thf(fact_959_max__length__elem,axiom,
    ! [Xs: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ Xs )
     => ( ( Xs != bot_bo4166481423041325370od_b_c )
       => ? [X2: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X2 @ Xs )
            & ~ ? [Xa2: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Xa2 @ Xs )
                  & ( ord_less_nat @ ( size_s3392097710323735898od_b_c @ X2 ) @ ( size_s3392097710323735898od_b_c @ Xa2 ) ) ) ) ) ) ).

% max_length_elem
thf(fact_960_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K4: nat] :
            ( ( ord_less_eq_nat @ K4 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K4 )
               => ~ ( P @ I3 ) )
            & ( P @ K4 ) ) ) ) ).

% ex_least_nat_le
thf(fact_961_psubset__card__mono,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le5653067673530651002od_b_c @ A @ B )
       => ( ord_less_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ).

% psubset_card_mono
thf(fact_962_psubset__card__mono,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_set_a @ A @ B )
       => ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).

% psubset_card_mono
thf(fact_963_psubset__card__mono,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_set_nat @ A @ B )
       => ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).

% psubset_card_mono
thf(fact_964_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_965_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_966_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_967_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_968_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_969_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_970_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_nat @ X4 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_971_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_nat @ X2 @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_972_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M2: nat] :
      ( ( P @ X )
     => ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( ord_less_eq_nat @ X2 @ M2 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X3: nat] :
                    ( ( P @ X3 )
                   => ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_973_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_974_finite__subset__mapping__limit,axiom,
    ! [F: nat > set_nat] :
      ( ( finite_finite_nat @ ( F @ zero_zero_nat ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ord_less_eq_set_nat @ ( F @ J2 ) @ ( F @ I2 ) ) )
       => ~ ! [K4: nat] :
              ~ ! [K6: nat] :
                  ( ( ord_less_eq_nat @ K4 @ K6 )
                 => ( ( F @ K6 )
                    = ( F @ K4 ) ) ) ) ) ).

% finite_subset_mapping_limit
thf(fact_975_finite__subset__mapping__limit,axiom,
    ! [F: nat > set_a] :
      ( ( finite_finite_a @ ( F @ zero_zero_nat ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ord_less_eq_set_a @ ( F @ J2 ) @ ( F @ I2 ) ) )
       => ~ ! [K4: nat] :
              ~ ! [K6: nat] :
                  ( ( ord_less_eq_nat @ K4 @ K6 )
                 => ( ( F @ K6 )
                    = ( F @ K4 ) ) ) ) ) ).

% finite_subset_mapping_limit
thf(fact_976_finite__subset__mapping__limit,axiom,
    ! [F: nat > set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( F @ zero_zero_nat ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ord_le282488521294790766od_b_c @ ( F @ J2 ) @ ( F @ I2 ) ) )
       => ~ ! [K4: nat] :
              ~ ! [K6: nat] :
                  ( ( ord_less_eq_nat @ K4 @ K6 )
                 => ( ( F @ K6 )
                    = ( F @ K4 ) ) ) ) ) ).

% finite_subset_mapping_limit
thf(fact_977_card__eq__0__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ( finite5583770498833199894od_b_c @ A )
        = zero_zero_nat )
      = ( ( A = bot_bo4166481423041325370od_b_c )
        | ~ ( finite3074115686814133143od_b_c @ A ) ) ) ).

% card_eq_0_iff
thf(fact_978_card__eq__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ( finite_card_nat @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A ) ) ) ).

% card_eq_0_iff
thf(fact_979_card__eq__0__iff,axiom,
    ! [A: set_a] :
      ( ( ( finite_card_a @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_a )
        | ~ ( finite_finite_a @ A ) ) ) ).

% card_eq_0_iff
thf(fact_980_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K4: nat] :
            ( ( ord_less_nat @ K4 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K4 )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K4 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_981_card__ge__0__finite,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite5583770498833199894od_b_c @ A ) )
     => ( finite3074115686814133143od_b_c @ A ) ) ).

% card_ge_0_finite
thf(fact_982_card__ge__0__finite,axiom,
    ! [A: set_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
     => ( finite_finite_a @ A ) ) ).

% card_ge_0_finite
thf(fact_983_card__ge__0__finite,axiom,
    ! [A: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
     => ( finite_finite_nat @ A ) ) ).

% card_ge_0_finite
thf(fact_984_minimally__distinguishes__def,axiom,
    ( minima243535863231358885_a_b_c
    = ( ^ [M: fsm_a_b_c,Q12: a,Q23: a,Io3: list_P903359562653991662od_b_c] :
          ( ( distinguishes_a_b_c @ M @ Q12 @ Q23 @ Io3 )
          & ! [Io4: list_P903359562653991662od_b_c] :
              ( ( distinguishes_a_b_c @ M @ Q12 @ Q23 @ Io4 )
             => ( ord_less_eq_nat @ ( size_s3392097710323735898od_b_c @ Io3 ) @ ( size_s3392097710323735898od_b_c @ Io4 ) ) ) ) ) ) ).

% minimally_distinguishes_def
thf(fact_985_card__psubset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
         => ( ord_less_set_nat @ A @ B ) ) ) ) ).

% card_psubset
thf(fact_986_card__psubset,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) )
         => ( ord_less_set_a @ A @ B ) ) ) ) ).

% card_psubset
thf(fact_987_card__psubset,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( ( ord_less_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) )
         => ( ord_le5653067673530651002od_b_c @ A @ B ) ) ) ) ).

% card_psubset
thf(fact_988_card__gt__0__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite5583770498833199894od_b_c @ A ) )
      = ( ( A != bot_bo4166481423041325370od_b_c )
        & ( finite3074115686814133143od_b_c @ A ) ) ) ).

% card_gt_0_iff
thf(fact_989_card__gt__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
      = ( ( A != bot_bot_set_nat )
        & ( finite_finite_nat @ A ) ) ) ).

% card_gt_0_iff
thf(fact_990_card__gt__0__iff,axiom,
    ! [A: set_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
      = ( ( A != bot_bot_set_a )
        & ( finite_finite_a @ A ) ) ) ).

% card_gt_0_iff
thf(fact_991_card__le__Suc0__iff__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( ord_less_eq_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ A )
             => ! [Y2: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Y2 @ A )
                 => ( X4 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_992_card__le__Suc0__iff__eq,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ A )
             => ! [Y2: a] :
                  ( ( member_a @ Y2 @ A )
                 => ( X4 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_993_card__le__Suc0__iff__eq,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A )
             => ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A )
                 => ( X4 = Y2 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_994_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X2 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_995_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ X8 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X8 )
                & ( ord_less_nat @ X2 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_996_arg__min__if__finite_I2_J,axiom,
    ! [S: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > nat] :
      ( ( finite3074115686814133143od_b_c @ S )
     => ( ( S != bot_bo4166481423041325370od_b_c )
       => ~ ? [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ S )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic9077399895930519397_c_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_997_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( ( member_nat @ X3 @ S )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_998_arg__min__if__finite_I2_J,axiom,
    ! [S: set_a,F: a > nat] :
      ( ( finite_finite_a @ S )
     => ( ( S != bot_bot_set_a )
       => ~ ? [X3: a] :
              ( ( member_a @ X3 @ S )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_999_arg__min__least,axiom,
    ! [S: set_li6436108459499378894od_b_c,Y: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > nat] :
      ( ( finite3074115686814133143od_b_c @ S )
     => ( ( S != bot_bo4166481423041325370od_b_c )
       => ( ( member6330420149250801815od_b_c @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic9077399895930519397_c_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1000_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1001_arg__min__least,axiom,
    ! [S: set_a,Y: a,F: a > nat] :
      ( ( finite_finite_a @ S )
     => ( ( S != bot_bot_set_a )
       => ( ( member_a @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1002_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M3: nat] :
      ( ! [K4: nat] :
          ( ( ord_less_nat @ N @ K4 )
         => ( P @ K4 ) )
     => ( ! [K4: nat] :
            ( ( ord_less_eq_nat @ K4 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K4 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K4 ) ) )
       => ( P @ M3 ) ) ) ).

% nat_descend_induct
thf(fact_1003_psubsetD,axiom,
    ! [A: set_a,B: set_a,C: a] :
      ( ( ord_less_set_a @ A @ B )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% psubsetD
thf(fact_1004_psubsetD,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( ord_le5653067673530651002od_b_c @ A @ B )
     => ( ( member6330420149250801815od_b_c @ C @ A )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% psubsetD
thf(fact_1005_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) ) @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) )
             => ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
               != ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_1006_complete__interval,axiom,
    ! [A2: nat,B2: nat,P: nat > $o] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B2 )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A2 @ C3 )
              & ( ord_less_eq_nat @ C3 @ B2 )
              & ! [X3: nat] :
                  ( ( ( ord_less_eq_nat @ A2 @ X3 )
                    & ( ord_less_nat @ X3 @ C3 ) )
                 => ( P @ X3 ) )
              & ! [D3: nat] :
                  ( ! [X2: nat] :
                      ( ( ( ord_less_eq_nat @ A2 @ X2 )
                        & ( ord_less_nat @ X2 @ D3 ) )
                     => ( P @ X2 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1007_verit__comp__simplify1_I3_J,axiom,
    ! [B10: nat,A8: nat] :
      ( ( ~ ( ord_less_eq_nat @ B10 @ A8 ) )
      = ( ord_less_nat @ A8 @ B10 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1008_reachable__state__is__state,axiom,
    ! [Q: a,M2: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( member_a @ Q @ ( states_a_b_c @ M2 ) ) ) ).

% reachable_state_is_state
thf(fact_1009_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_nat @ A2 @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_1010_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1011_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1012_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1013_after__reachable,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Io: list_P903359562653991662od_b_c,Q: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Q @ ( reacha8116992257183400179_c_b_c @ M2 ) )
         => ( member6330420149250801815od_b_c @ ( after_4052058690717316294_c_b_c @ M2 @ Q @ Io ) @ ( reacha8116992257183400179_c_b_c @ M2 ) ) ) ) ) ).

% after_reachable
thf(fact_1014_after__reachable,axiom,
    ! [M2: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q ) )
       => ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M2 ) )
         => ( member_a @ ( after_a_b_c @ M2 @ Q @ Io ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ) ) ).

% after_reachable
thf(fact_1015_after__reachable__initial,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_lis2930931384350476499_c_b_c @ M2 @ ( initia3567573336347591134_c_b_c @ M2 ) ) )
       => ( member6330420149250801815od_b_c @ ( after_4052058690717316294_c_b_c @ M2 @ ( initia3567573336347591134_c_b_c @ M2 ) @ Io ) @ ( reacha8116992257183400179_c_b_c @ M2 ) ) ) ) ).

% after_reachable_initial
thf(fact_1016_after__reachable__initial,axiom,
    ! [M2: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
       => ( member_a @ ( after_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ Io ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ) ).

% after_reachable_initial
thf(fact_1017_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
                = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_1018_minf_I8_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ~ ( ord_less_eq_nat @ T4 @ X3 ) ) ).

% minf(8)
thf(fact_1019_minf_I6_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ord_less_eq_nat @ X3 @ T4 ) ) ).

% minf(6)
thf(fact_1020_pinf_I8_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ord_less_eq_nat @ T4 @ X3 ) ) ).

% pinf(8)
thf(fact_1021_pinf_I6_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ~ ( ord_less_eq_nat @ X3 @ T4 ) ) ).

% pinf(6)
thf(fact_1022_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1023_gen__length__def,axiom,
    ( gen_le7668512674959329659od_b_c
    = ( ^ [N3: nat,Xs3: list_P903359562653991662od_b_c] : ( plus_plus_nat @ N3 @ ( size_s3392097710323735898od_b_c @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1024_length__code,axiom,
    ( size_s3392097710323735898od_b_c
    = ( gen_le7668512674959329659od_b_c @ zero_zero_nat ) ) ).

% length_code
thf(fact_1025_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le7668512674959329659od_b_c @ N @ nil_Product_prod_b_c )
      = N ) ).

% gen_length_code(1)
thf(fact_1026_remdups__adj__length__ge1,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s3392097710323735898od_b_c @ ( remdup2090892755044397220od_b_c @ Xs ) ) ) ) ).

% remdups_adj_length_ge1
thf(fact_1027_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( states_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) )
      = ( reacha1620305530751930115_a_b_c @ M2 ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_1028_Set_Ois__empty__def,axiom,
    ( is_empty_a
    = ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).

% Set.is_empty_def
thf(fact_1029_remdups__adj__Nil__iff,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( ( remdup2090892755044397220od_b_c @ Xs )
        = nil_Product_prod_b_c )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% remdups_adj_Nil_iff
thf(fact_1030_remdups__adj_Osimps_I1_J,axiom,
    ( ( remdup2090892755044397220od_b_c @ nil_Product_prod_b_c )
    = nil_Product_prod_b_c ) ).

% remdups_adj.simps(1)
thf(fact_1031_restrict__to__reachable__states__observable,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( observable_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_1032_remdups__adj__length,axiom,
    ! [Xs: list_P903359562653991662od_b_c] : ( ord_less_eq_nat @ ( size_s3392097710323735898od_b_c @ ( remdup2090892755044397220od_b_c @ Xs ) ) @ ( size_s3392097710323735898od_b_c @ Xs ) ) ).

% remdups_adj_length
thf(fact_1033_restrict__to__reachable__states__minimal,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M2 )
     => ( minimal_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) ) ) ).

% restrict_to_reachable_states_minimal
thf(fact_1034_restrict__to__reachable__states__language,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( lS_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) @ ( initial_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) ) )
      = ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ).

% restrict_to_reachable_states_language
thf(fact_1035_restrict__to__reachable__states__reachable__states,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( reacha1620305530751930115_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) )
      = ( states_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_1036_acyclic__alt__def,axiom,
    ( acyclic_a_b_c
    = ( ^ [M: fsm_a_b_c] : ( finite3074115686814133143od_b_c @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ) ).

% acyclic_alt_def
thf(fact_1037_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1038_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1039_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic2204479054038723496od_b_c @ inf_in4978071631833541052od_b_c @ ord_le282488521294790766od_b_c @ ord_le5653067673530651002od_b_c ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_1040_card__Un__disjnt,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( finite3074115686814133143od_b_c @ B )
       => ( ( disjnt5456880891938978613od_b_c @ A @ B )
         => ( ( finite5583770498833199894od_b_c @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
            = ( plus_plus_nat @ ( finite5583770498833199894od_b_c @ A ) @ ( finite5583770498833199894od_b_c @ B ) ) ) ) ) ) ).

% card_Un_disjnt
thf(fact_1041_card__Un__disjnt,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( disjnt_nat @ A @ B )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ) ).

% card_Un_disjnt
thf(fact_1042_card__Un__disjnt,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ( disjnt_a @ A @ B )
         => ( ( finite_card_a @ ( sup_sup_set_a @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ) ).

% card_Un_disjnt
thf(fact_1043_disjnt__self__iff__empty,axiom,
    ! [S: set_a] :
      ( ( disjnt_a @ S @ S )
      = ( S = bot_bot_set_a ) ) ).

% disjnt_self_iff_empty
thf(fact_1044_disjnt__Un1,axiom,
    ! [A: set_a,B: set_a,C2: set_a] :
      ( ( disjnt_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
      = ( ( disjnt_a @ A @ C2 )
        & ( disjnt_a @ B @ C2 ) ) ) ).

% disjnt_Un1
thf(fact_1045_disjnt__Un2,axiom,
    ! [C2: set_a,A: set_a,B: set_a] :
      ( ( disjnt_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
      = ( ( disjnt_a @ C2 @ A )
        & ( disjnt_a @ C2 @ B ) ) ) ).

% disjnt_Un2
thf(fact_1046_disjnt__empty2,axiom,
    ! [A: set_a] : ( disjnt_a @ A @ bot_bot_set_a ) ).

% disjnt_empty2
thf(fact_1047_disjnt__empty1,axiom,
    ! [A: set_a] : ( disjnt_a @ bot_bot_set_a @ A ) ).

% disjnt_empty1
thf(fact_1048_disjnt__iff,axiom,
    ( disjnt_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [X4: a] :
          ~ ( ( member_a @ X4 @ A3 )
            & ( member_a @ X4 @ B3 ) ) ) ) ).

% disjnt_iff
thf(fact_1049_disjnt__iff,axiom,
    ( disjnt5456880891938978613od_b_c
    = ( ^ [A3: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ~ ( ( member6330420149250801815od_b_c @ X4 @ A3 )
            & ( member6330420149250801815od_b_c @ X4 @ B3 ) ) ) ) ).

% disjnt_iff
thf(fact_1050_disjnt__subset1,axiom,
    ! [X8: set_a,Y7: set_a,Z4: set_a] :
      ( ( disjnt_a @ X8 @ Y7 )
     => ( ( ord_less_eq_set_a @ Z4 @ X8 )
       => ( disjnt_a @ Z4 @ Y7 ) ) ) ).

% disjnt_subset1
thf(fact_1051_disjnt__subset1,axiom,
    ! [X8: set_li6436108459499378894od_b_c,Y7: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] :
      ( ( disjnt5456880891938978613od_b_c @ X8 @ Y7 )
     => ( ( ord_le282488521294790766od_b_c @ Z4 @ X8 )
       => ( disjnt5456880891938978613od_b_c @ Z4 @ Y7 ) ) ) ).

% disjnt_subset1
thf(fact_1052_disjnt__subset2,axiom,
    ! [X8: set_a,Y7: set_a,Z4: set_a] :
      ( ( disjnt_a @ X8 @ Y7 )
     => ( ( ord_less_eq_set_a @ Z4 @ Y7 )
       => ( disjnt_a @ X8 @ Z4 ) ) ) ).

% disjnt_subset2
thf(fact_1053_disjnt__subset2,axiom,
    ! [X8: set_li6436108459499378894od_b_c,Y7: set_li6436108459499378894od_b_c,Z4: set_li6436108459499378894od_b_c] :
      ( ( disjnt5456880891938978613od_b_c @ X8 @ Y7 )
     => ( ( ord_le282488521294790766od_b_c @ Z4 @ Y7 )
       => ( disjnt5456880891938978613od_b_c @ X8 @ Z4 ) ) ) ).

% disjnt_subset2
thf(fact_1054_disjnt__def,axiom,
    ( disjnt_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ( inf_inf_set_a @ A3 @ B3 )
          = bot_bot_set_a ) ) ) ).

% disjnt_def
thf(fact_1055_LS__from__LS__acyclic,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
        = ( lS_acyclic_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ).

% LS_from_LS_acyclic
thf(fact_1056_semilattice__order__set_Osubset__imp,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,B: set_nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( A != bot_bot_set_nat )
         => ( ( finite_finite_nat @ B )
           => ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_1057_semilattice__order__set_Osubset__imp,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,B: set_a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( A != bot_bot_set_a )
         => ( ( finite_finite_a @ B )
           => ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_1058_semilattice__order__set_Osubset__imp,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Less_eq: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Less: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( lattic2195833625579453896od_b_c @ F @ Less_eq @ Less )
     => ( ( ord_le282488521294790766od_b_c @ A @ B )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ( finite3074115686814133143od_b_c @ B )
           => ( Less_eq @ ( lattic2734920875441048264od_b_c @ F @ B ) @ ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_1059_Sup__fin_Ounion,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ( B != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
              = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1060_Sup__fin_Ounion,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ( ( finite_finite_set_a @ B )
         => ( ( B != bot_bot_set_set_a )
           => ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A @ B ) )
              = ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1061_inf__Sup__absorb,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
          = A2 ) ) ) ).

% inf_Sup_absorb
thf(fact_1062_inf__Sup__absorb,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) )
          = A2 ) ) ) ).

% inf_Sup_absorb
thf(fact_1063_Sup__fin__def,axiom,
    ( lattic2918178356826803221_set_a
    = ( lattic2714821017709792056_set_a @ sup_sup_set_a ) ) ).

% Sup_fin_def
thf(fact_1064_Sup__fin_OcoboundedI,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ( ord_less_eq_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_1065_Sup__fin_OcoboundedI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_1066_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se3924713247505902254od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( member6985331446368301687od_b_c @ A2 @ A )
       => ( ord_le282488521294790766od_b_c @ A2 @ ( lattic8058834985641542149od_b_c @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_1067_Sup__fin_Oin__idem,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
          = ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_1068_Sup__fin_Oin__idem,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ X @ A )
       => ( ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) )
          = ( lattic2918178356826803221_set_a @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_1069_semilattice__order__set_OcoboundedI,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Less_eq: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Less: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,A: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( lattic2195833625579453896od_b_c @ F @ Less_eq @ Less )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( member6330420149250801815od_b_c @ A2 @ A )
         => ( Less_eq @ ( lattic2734920875441048264od_b_c @ F @ A ) @ A2 ) ) ) ) ).

% semilattice_order_set.coboundedI
thf(fact_1070_semilattice__order__set_OcoboundedI,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,A2: a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( finite_finite_a @ A )
       => ( ( member_a @ A2 @ A )
         => ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ A2 ) ) ) ) ).

% semilattice_order_set.coboundedI
thf(fact_1071_semilattice__order__set_OcoboundedI,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,A2: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A )
       => ( ( member_nat @ A2 @ A )
         => ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ A ) @ A2 ) ) ) ) ).

% semilattice_order_set.coboundedI
thf(fact_1072_Sup__fin_Obounded__iff,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
          = ( ! [X4: set_a] :
                ( ( member_set_a @ X4 @ A )
               => ( ord_less_eq_set_a @ X4 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1073_Sup__fin_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A )
               => ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1074_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se3924713247505902254od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ( ( ord_le282488521294790766od_b_c @ ( lattic8058834985641542149od_b_c @ A ) @ X )
          = ( ! [X4: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ X4 @ A )
               => ( ord_le282488521294790766od_b_c @ X4 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_1075_Sup__fin_OboundedI,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ( ! [A6: set_a] :
              ( ( member_set_a @ A6 @ A )
             => ( ord_less_eq_set_a @ A6 @ X ) )
         => ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1076_Sup__fin_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A6: nat] :
              ( ( member_nat @ A6 @ A )
             => ( ord_less_eq_nat @ A6 @ X ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1077_Sup__fin_OboundedI,axiom,
    ! [A: set_se3924713247505902254od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ( ! [A6: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ A6 @ A )
             => ( ord_le282488521294790766od_b_c @ A6 @ X ) )
         => ( ord_le282488521294790766od_b_c @ ( lattic8058834985641542149od_b_c @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_1078_Sup__fin_OboundedE,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X )
         => ! [A9: set_a] :
              ( ( member_set_a @ A9 @ A )
             => ( ord_less_eq_set_a @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1079_Sup__fin_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
         => ! [A9: nat] :
              ( ( member_nat @ A9 @ A )
             => ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1080_Sup__fin_OboundedE,axiom,
    ! [A: set_se3924713247505902254od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( finite1374199133651033463od_b_c @ A )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ( ( ord_le282488521294790766od_b_c @ ( lattic8058834985641542149od_b_c @ A ) @ X )
         => ! [A9: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ A9 @ A )
             => ( ord_le282488521294790766od_b_c @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_1081_semilattice__order__set_OboundedE,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Less_eq: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Less: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic2195833625579453896od_b_c @ F @ Less_eq @ Less )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ( Less_eq @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) )
           => ! [A9: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ A9 @ A )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_1082_semilattice__order__set_OboundedE,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
           => ! [A9: nat] :
                ( ( member_nat @ A9 @ A )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_1083_semilattice__order__set_OboundedE,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
           => ! [A9: a] :
                ( ( member_a @ A9 @ A )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_1084_semilattice__order__set_OboundedI,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Less_eq: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Less: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic2195833625579453896od_b_c @ F @ Less_eq @ Less )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ! [A6: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ A6 @ A )
               => ( Less_eq @ X @ A6 ) )
           => ( Less_eq @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_1085_semilattice__order__set_OboundedI,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ! [A6: nat] :
                ( ( member_nat @ A6 @ A )
               => ( Less_eq @ X @ A6 ) )
           => ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_1086_semilattice__order__set_OboundedI,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ! [A6: a] :
                ( ( member_a @ A6 @ A )
               => ( Less_eq @ X @ A6 ) )
           => ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_1087_semilattice__order__set_Obounded__iff,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Less_eq: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Less: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic2195833625579453896od_b_c @ F @ Less_eq @ Less )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ( Less_eq @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) )
            = ( ! [X4: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ X4 @ A )
                 => ( Less_eq @ X @ X4 ) ) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
thf(fact_1088_semilattice__order__set_Obounded__iff,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
            = ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ A )
                 => ( Less_eq @ X @ X4 ) ) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
thf(fact_1089_semilattice__order__set_Obounded__iff,axiom,
    ! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
      ( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
            = ( ! [X4: a] :
                  ( ( member_a @ X4 @ A )
                 => ( Less_eq @ X @ X4 ) ) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
thf(fact_1090_Sup__fin_Osubset__imp,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( A != bot_bot_set_set_a )
       => ( ( finite_finite_set_a @ B )
         => ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1091_Sup__fin_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1092_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ A @ B )
     => ( ( A != bot_bo2794119844231891738od_b_c )
       => ( ( finite1374199133651033463od_b_c @ B )
         => ( ord_le282488521294790766od_b_c @ ( lattic8058834985641542149od_b_c @ A ) @ ( lattic8058834985641542149od_b_c @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_1093_Sup__fin_Osubset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
            = ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1094_Sup__fin_Osubset,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( B != bot_bot_set_set_a )
       => ( ( ord_le3724670747650509150_set_a @ B @ A )
         => ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A ) )
            = ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1095_semilattice__set_Ounion,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ( finite3074115686814133143od_b_c @ B )
           => ( ( B != bot_bo4166481423041325370od_b_c )
             => ( ( lattic2734920875441048264od_b_c @ F @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
                = ( F @ ( lattic2734920875441048264od_b_c @ F @ A ) @ ( lattic2734920875441048264od_b_c @ F @ B ) ) ) ) ) ) ) ) ).

% semilattice_set.union
thf(fact_1096_semilattice__set_Ounion,axiom,
    ! [F: nat > nat > nat,A: set_nat,B: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( finite_finite_nat @ B )
           => ( ( B != bot_bot_set_nat )
             => ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
                = ( F @ ( lattic7742739596368939638_F_nat @ F @ A ) @ ( lattic7742739596368939638_F_nat @ F @ B ) ) ) ) ) ) ) ) ).

% semilattice_set.union
thf(fact_1097_semilattice__set_Ounion,axiom,
    ! [F: a > a > a,A: set_a,B: set_a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ( finite_finite_a @ B )
           => ( ( B != bot_bot_set_a )
             => ( ( lattic5116578512385870296ce_F_a @ F @ ( sup_sup_set_a @ A @ B ) )
                = ( F @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ ( lattic5116578512385870296ce_F_a @ F @ B ) ) ) ) ) ) ) ) ).

% semilattice_set.union
thf(fact_1098_semilattice__set_Osubset,axiom,
    ! [F: nat > nat > nat,A: set_nat,B: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ( B != bot_bot_set_nat )
         => ( ( ord_less_eq_set_nat @ B @ A )
           => ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A ) )
              = ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_1099_semilattice__set_Osubset,axiom,
    ! [F: a > a > a,A: set_a,B: set_a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ( B != bot_bot_set_a )
         => ( ( ord_less_eq_set_a @ B @ A )
           => ( ( F @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
              = ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_1100_semilattice__set_Osubset,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( B != bot_bo4166481423041325370od_b_c )
         => ( ( ord_le282488521294790766od_b_c @ B @ A )
           => ( ( F @ ( lattic2734920875441048264od_b_c @ F @ B ) @ ( lattic2734920875441048264od_b_c @ F @ A ) )
              = ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_1101_Sup__fin_Oinsert,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
          = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1102_Sup__fin_Oinsert,axiom,
    ! [A: set_set_a,X: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X @ A ) )
          = ( sup_sup_set_a @ X @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1103_insertCI,axiom,
    ! [A2: a,B: set_a,B2: a] :
      ( ( ~ ( member_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1104_insertCI,axiom,
    ! [A2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c] :
      ( ( ~ ( member6330420149250801815od_b_c @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1105_insert__iff,axiom,
    ! [A2: a,B2: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1106_insert__iff,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member6330420149250801815od_b_c @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1107_finite__insert,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ A ) )
      = ( finite3074115686814133143od_b_c @ A ) ) ).

% finite_insert
thf(fact_1108_finite__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
      = ( finite_finite_a @ A ) ) ).

% finite_insert
thf(fact_1109_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_1110_singletonI,axiom,
    ! [A2: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) ) ).

% singletonI
thf(fact_1111_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_1112_insert__subset,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
      = ( ( member_a @ X @ B )
        & ( ord_less_eq_set_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_1113_insert__subset,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ X @ A ) @ B )
      = ( ( member6330420149250801815od_b_c @ X @ B )
        & ( ord_le282488521294790766od_b_c @ A @ B ) ) ) ).

% insert_subset
thf(fact_1114_Int__insert__left__if0,axiom,
    ! [A2: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A2 @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ B ) @ C2 )
        = ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1115_Int__insert__left__if0,axiom,
    ! [A2: a,C2: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
        = ( inf_inf_set_a @ B @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1116_Int__insert__left__if1,axiom,
    ! [A2: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ B ) @ C2 )
        = ( insert6227932334100060350od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1117_Int__insert__left__if1,axiom,
    ! [A2: a,C2: set_a,B: set_a] :
      ( ( member_a @ A2 @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1118_insert__inter__insert,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
      = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_1119_Int__insert__right__if0,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ( inf_in4978071631833541052od_b_c @ A @ ( insert6227932334100060350od_b_c @ A2 @ B ) )
        = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1120_Int__insert__right__if0,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
        = ( inf_inf_set_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1121_Int__insert__right__if1,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ( inf_in4978071631833541052od_b_c @ A @ ( insert6227932334100060350od_b_c @ A2 @ B ) )
        = ( insert6227932334100060350od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1122_Int__insert__right__if1,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1123_Un__insert__left,axiom,
    ! [A2: a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
      = ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1124_Un__insert__right,axiom,
    ! [A: set_a,A2: a,B: set_a] :
      ( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
      = ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_1125_disjnt__insert2,axiom,
    ! [Y7: set_a,A2: a,X8: set_a] :
      ( ( disjnt_a @ Y7 @ ( insert_a @ A2 @ X8 ) )
      = ( ~ ( member_a @ A2 @ Y7 )
        & ( disjnt_a @ Y7 @ X8 ) ) ) ).

% disjnt_insert2
thf(fact_1126_disjnt__insert2,axiom,
    ! [Y7: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c,X8: set_li6436108459499378894od_b_c] :
      ( ( disjnt5456880891938978613od_b_c @ Y7 @ ( insert6227932334100060350od_b_c @ A2 @ X8 ) )
      = ( ~ ( member6330420149250801815od_b_c @ A2 @ Y7 )
        & ( disjnt5456880891938978613od_b_c @ Y7 @ X8 ) ) ) ).

% disjnt_insert2
thf(fact_1127_disjnt__insert1,axiom,
    ! [A2: a,X8: set_a,Y7: set_a] :
      ( ( disjnt_a @ ( insert_a @ A2 @ X8 ) @ Y7 )
      = ( ~ ( member_a @ A2 @ Y7 )
        & ( disjnt_a @ X8 @ Y7 ) ) ) ).

% disjnt_insert1
thf(fact_1128_disjnt__insert1,axiom,
    ! [A2: list_P903359562653991662od_b_c,X8: set_li6436108459499378894od_b_c,Y7: set_li6436108459499378894od_b_c] :
      ( ( disjnt5456880891938978613od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ X8 ) @ Y7 )
      = ( ~ ( member6330420149250801815od_b_c @ A2 @ Y7 )
        & ( disjnt5456880891938978613od_b_c @ X8 @ Y7 ) ) ) ).

% disjnt_insert1
thf(fact_1129_singleton__insert__inj__eq,axiom,
    ! [B2: a,A2: a,A: set_a] :
      ( ( ( insert_a @ B2 @ bot_bot_set_a )
        = ( insert_a @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1130_singleton__insert__inj__eq,axiom,
    ! [B2: list_P903359562653991662od_b_c,A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ B2 @ bot_bo4166481423041325370od_b_c )
        = ( insert6227932334100060350od_b_c @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le282488521294790766od_b_c @ A @ ( insert6227932334100060350od_b_c @ B2 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1131_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B2: a] :
      ( ( ( insert_a @ A2 @ A )
        = ( insert_a @ B2 @ bot_bot_set_a ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1132_singleton__insert__inj__eq_H,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ A2 @ A )
        = ( insert6227932334100060350od_b_c @ B2 @ bot_bo4166481423041325370od_b_c ) )
      = ( ( A2 = B2 )
        & ( ord_le282488521294790766od_b_c @ A @ ( insert6227932334100060350od_b_c @ B2 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1133_disjoint__insert_I2_J,axiom,
    ! [A: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ A @ ( insert6227932334100060350od_b_c @ B2 @ B ) ) )
      = ( ~ ( member6330420149250801815od_b_c @ B2 @ A )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1134_disjoint__insert_I2_J,axiom,
    ! [A: set_a,B2: a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
      = ( ~ ( member_a @ B2 @ A )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1135_disjoint__insert_I1_J,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ B @ ( insert6227932334100060350od_b_c @ A2 @ A ) )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A2 @ B )
        & ( ( inf_in4978071631833541052od_b_c @ B @ A )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% disjoint_insert(1)
thf(fact_1136_disjoint__insert_I1_J,axiom,
    ! [B: set_a,A2: a,A: set_a] :
      ( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ B @ A )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1137_insert__disjoint_I2_J,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ A ) @ B ) )
      = ( ~ ( member6330420149250801815od_b_c @ A2 @ B )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1138_insert__disjoint_I2_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
      = ( ~ ( member_a @ A2 @ B )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1139_insert__disjoint_I1_J,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ A ) @ B )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A2 @ B )
        & ( ( inf_in4978071631833541052od_b_c @ A @ B )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% insert_disjoint(1)
thf(fact_1140_insert__disjoint_I1_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ A @ B )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1141_card__insert__disjoint,axiom,
    ! [A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ~ ( member6330420149250801815od_b_c @ X @ A )
       => ( ( finite5583770498833199894od_b_c @ ( insert6227932334100060350od_b_c @ X @ A ) )
          = ( suc @ ( finite5583770498833199894od_b_c @ A ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_1142_card__insert__disjoint,axiom,
    ! [A: set_a,X: a] :
      ( ( finite_finite_a @ A )
     => ( ~ ( member_a @ X @ A )
       => ( ( finite_card_a @ ( insert_a @ X @ A ) )
          = ( suc @ ( finite_card_a @ A ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_1143_card__insert__disjoint,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ X @ A )
       => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
          = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_1144_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ~ ( member6330420149250801815od_b_c @ X @ A )
         => ( ( A != bot_bo4166481423041325370od_b_c )
           => ( ( lattic2734920875441048264od_b_c @ F @ ( insert6227932334100060350od_b_c @ X @ A ) )
              = ( F @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_1145_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: nat > nat > nat,A: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ~ ( member_nat @ X @ A )
         => ( ( A != bot_bot_set_nat )
           => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
              = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_1146_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: a > a > a,A: set_a,X: a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ~ ( member_a @ X @ A )
         => ( ( A != bot_bot_set_a )
           => ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
              = ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_1147_semilattice__set_Oinsert,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ( lattic2734920875441048264od_b_c @ F @ ( insert6227932334100060350od_b_c @ X @ A ) )
            = ( F @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ) ).

% semilattice_set.insert
thf(fact_1148_semilattice__set_Oinsert,axiom,
    ! [F: nat > nat > nat,A: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
            = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).

% semilattice_set.insert
thf(fact_1149_semilattice__set_Oinsert,axiom,
    ! [F: a > a > a,A: set_a,X: a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
            = ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).

% semilattice_set.insert
thf(fact_1150_semilattice__set_Oclosed,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( A != bot_bo4166481423041325370od_b_c )
         => ( ! [X2: list_P903359562653991662od_b_c,Y4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ ( F @ X2 @ Y4 ) @ ( insert6227932334100060350od_b_c @ X2 @ ( insert6227932334100060350od_b_c @ Y4 @ bot_bo4166481423041325370od_b_c ) ) )
           => ( member6330420149250801815od_b_c @ ( lattic2734920875441048264od_b_c @ F @ A ) @ A ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_1151_semilattice__set_Oclosed,axiom,
    ! [F: nat > nat > nat,A: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ! [X2: nat,Y4: nat] : ( member_nat @ ( F @ X2 @ Y4 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
           => ( member_nat @ ( lattic7742739596368939638_F_nat @ F @ A ) @ A ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_1152_semilattice__set_Oclosed,axiom,
    ! [F: a > a > a,A: set_a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ( A != bot_bot_set_a )
         => ( ! [X2: a,Y4: a] : ( member_a @ ( F @ X2 @ Y4 ) @ ( insert_a @ X2 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
           => ( member_a @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ A ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_1153_semilattice__set_Osingleton,axiom,
    ! [F: a > a > a,X: a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ bot_bot_set_a ) )
        = X ) ) ).

% semilattice_set.singleton
thf(fact_1154_singleton__inject,axiom,
    ! [A2: a,B2: a] :
      ( ( ( insert_a @ A2 @ bot_bot_set_a )
        = ( insert_a @ B2 @ bot_bot_set_a ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1155_insert__not__empty,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a @ A2 @ A )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_1156_doubleton__eq__iff,axiom,
    ! [A2: a,B2: a,C: a,D2: a] :
      ( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
        = ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1157_singleton__iff,axiom,
    ! [B2: list_P903359562653991662od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B2 @ ( insert6227932334100060350od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1158_singleton__iff,axiom,
    ! [B2: a,A2: a] :
      ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1159_singletonD,axiom,
    ! [B2: list_P903359562653991662od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B2 @ ( insert6227932334100060350od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1160_singletonD,axiom,
    ! [B2: a,A2: a] :
      ( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1161_insertE,axiom,
    ! [A2: a,B2: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_1162_insertE,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member6330420149250801815od_b_c @ A2 @ A ) ) ) ).

% insertE
thf(fact_1163_insertI1,axiom,
    ! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).

% insertI1
thf(fact_1164_insertI1,axiom,
    ! [A2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] : ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ A2 @ B ) ) ).

% insertI1
thf(fact_1165_insertI2,axiom,
    ! [A2: a,B: set_a,B2: a] :
      ( ( member_a @ A2 @ B )
     => ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1166_insertI2,axiom,
    ! [A2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ B )
     => ( member6330420149250801815od_b_c @ A2 @ ( insert6227932334100060350od_b_c @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1167_Set_Oset__insert,axiom,
    ! [X: a,A: set_a] :
      ( ( member_a @ X @ A )
     => ~ ! [B7: set_a] :
            ( ( A
              = ( insert_a @ X @ B7 ) )
           => ( member_a @ X @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1168_Set_Oset__insert,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ A )
     => ~ ! [B7: set_li6436108459499378894od_b_c] :
            ( ( A
              = ( insert6227932334100060350od_b_c @ X @ B7 ) )
           => ( member6330420149250801815od_b_c @ X @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1169_insert__ident,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ~ ( member_a @ X @ B )
       => ( ( ( insert_a @ X @ A )
            = ( insert_a @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1170_insert__ident,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ A )
     => ( ~ ( member6330420149250801815od_b_c @ X @ B )
       => ( ( ( insert6227932334100060350od_b_c @ X @ A )
            = ( insert6227932334100060350od_b_c @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1171_insert__absorb,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1172_insert__absorb,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ( insert6227932334100060350od_b_c @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1173_insert__eq__iff,axiom,
    ! [A2: a,A: set_a,B2: a,B: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ~ ( member_a @ B2 @ B )
       => ( ( ( insert_a @ A2 @ A )
            = ( insert_a @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C5: set_a] :
                  ( ( A
                    = ( insert_a @ B2 @ C5 ) )
                  & ~ ( member_a @ B2 @ C5 )
                  & ( B
                    = ( insert_a @ A2 @ C5 ) )
                  & ~ ( member_a @ A2 @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1174_insert__eq__iff,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ~ ( member6330420149250801815od_b_c @ B2 @ B )
       => ( ( ( insert6227932334100060350od_b_c @ A2 @ A )
            = ( insert6227932334100060350od_b_c @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C5: set_li6436108459499378894od_b_c] :
                  ( ( A
                    = ( insert6227932334100060350od_b_c @ B2 @ C5 ) )
                  & ~ ( member6330420149250801815od_b_c @ B2 @ C5 )
                  & ( B
                    = ( insert6227932334100060350od_b_c @ A2 @ C5 ) )
                  & ~ ( member6330420149250801815od_b_c @ A2 @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1175_mk__disjoint__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ? [B7: set_a] :
          ( ( A
            = ( insert_a @ A2 @ B7 ) )
          & ~ ( member_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1176_mk__disjoint__insert,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ A )
     => ? [B7: set_li6436108459499378894od_b_c] :
          ( ( A
            = ( insert6227932334100060350od_b_c @ A2 @ B7 ) )
          & ~ ( member6330420149250801815od_b_c @ A2 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1177_Int__insert__left,axiom,
    ! [A2: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A2 @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ B ) @ C2 )
          = ( insert6227932334100060350od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A2 @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ B ) @ C2 )
          = ( inf_in4978071631833541052od_b_c @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1178_Int__insert__left,axiom,
    ! [A2: a,C2: set_a,B: set_a] :
      ( ( ( member_a @ A2 @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
          = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
      & ( ~ ( member_a @ A2 @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
          = ( inf_inf_set_a @ B @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1179_Int__insert__right,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A2 @ A )
       => ( ( inf_in4978071631833541052od_b_c @ A @ ( insert6227932334100060350od_b_c @ A2 @ B ) )
          = ( insert6227932334100060350od_b_c @ A2 @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A2 @ A )
       => ( ( inf_in4978071631833541052od_b_c @ A @ ( insert6227932334100060350od_b_c @ A2 @ B ) )
          = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1180_Int__insert__right,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
          = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
      & ( ~ ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_insert_right
thf(fact_1181_insert__subsetI,axiom,
    ! [X: a,A: set_a,X8: set_a] :
      ( ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ X8 @ A )
       => ( ord_less_eq_set_a @ ( insert_a @ X @ X8 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_1182_insert__subsetI,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,X8: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ A )
     => ( ( ord_le282488521294790766od_b_c @ X8 @ A )
       => ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ X @ X8 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_1183_insert__mono,axiom,
    ! [C2: set_a,D: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ C2 @ D )
     => ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1184_insert__mono,axiom,
    ! [C2: set_li6436108459499378894od_b_c,D: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ D )
     => ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ C2 ) @ ( insert6227932334100060350od_b_c @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1185_subset__insert,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% subset_insert
thf(fact_1186_subset__insert,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ A )
     => ( ( ord_le282488521294790766od_b_c @ A @ ( insert6227932334100060350od_b_c @ X @ B ) )
        = ( ord_le282488521294790766od_b_c @ A @ B ) ) ) ).

% subset_insert
thf(fact_1187_subset__insertI,axiom,
    ! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1188_subset__insertI,axiom,
    ! [B: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ B @ ( insert6227932334100060350od_b_c @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1189_subset__insertI2,axiom,
    ! [A: set_a,B: set_a,B2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1190_subset__insertI2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ord_le282488521294790766od_b_c @ A @ ( insert6227932334100060350od_b_c @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1191_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic1258622339881844972_set_a @ inf_inf_set_a ).

% Inf_fin.semilattice_set_axioms
thf(fact_1192_Sup__fin_Osemilattice__set__axioms,axiom,
    lattic1258622339881844972_set_a @ sup_sup_set_a ).

% Sup_fin.semilattice_set_axioms
thf(fact_1193_disjnt__insert,axiom,
    ! [X: a,N6: set_a,M2: set_a] :
      ( ~ ( member_a @ X @ N6 )
     => ( ( disjnt_a @ M2 @ N6 )
       => ( disjnt_a @ ( insert_a @ X @ M2 ) @ N6 ) ) ) ).

% disjnt_insert
thf(fact_1194_disjnt__insert,axiom,
    ! [X: list_P903359562653991662od_b_c,N6: set_li6436108459499378894od_b_c,M2: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ N6 )
     => ( ( disjnt5456880891938978613od_b_c @ M2 @ N6 )
       => ( disjnt5456880891938978613od_b_c @ ( insert6227932334100060350od_b_c @ X @ M2 ) @ N6 ) ) ) ).

% disjnt_insert
thf(fact_1195_finite_OinsertI,axiom,
    ! [A: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( finite3074115686814133143od_b_c @ ( insert6227932334100060350od_b_c @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1196_finite_OinsertI,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1197_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1198_infinite__finite__induct,axiom,
    ! [P: set_li6436108459499378894od_b_c > $o,A: set_li6436108459499378894od_b_c] :
      ( ! [A7: set_li6436108459499378894od_b_c] :
          ( ~ ( finite3074115686814133143od_b_c @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo4166481423041325370od_b_c )
       => ( ! [X2: list_P903359562653991662od_b_c,F3: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ F3 )
             => ( ~ ( member6330420149250801815od_b_c @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert6227932334100060350od_b_c @ X2 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1199_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1200_infinite__finite__induct,axiom,
    ! [P: set_a > $o,A: set_a] :
      ( ! [A7: set_a] :
          ( ~ ( finite_finite_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X2: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1201_finite__ne__induct,axiom,
    ! [F2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F2 )
     => ( ( F2 != bot_bo4166481423041325370od_b_c )
       => ( ! [X2: list_P903359562653991662od_b_c] : ( P @ ( insert6227932334100060350od_b_c @ X2 @ bot_bo4166481423041325370od_b_c ) )
         => ( ! [X2: list_P903359562653991662od_b_c,F3: set_li6436108459499378894od_b_c] :
                ( ( finite3074115686814133143od_b_c @ F3 )
               => ( ( F3 != bot_bo4166481423041325370od_b_c )
                 => ( ~ ( member6330420149250801815od_b_c @ X2 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert6227932334100060350od_b_c @ X2 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1202_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
         => ( ! [X2: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X2 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1203_finite__ne__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( F2 != bot_bot_set_a )
       => ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
         => ( ! [X2: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( F3 != bot_bot_set_a )
                 => ( ~ ( member_a @ X2 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1204_finite__induct,axiom,
    ! [F2: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F2 )
     => ( ( P @ bot_bo4166481423041325370od_b_c )
       => ( ! [X2: list_P903359562653991662od_b_c,F3: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ F3 )
             => ( ~ ( member6330420149250801815od_b_c @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert6227932334100060350od_b_c @ X2 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1205_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1206_finite__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X2: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X2 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1207_finite_Osimps,axiom,
    ( finite3074115686814133143od_b_c
    = ( ^ [A4: set_li6436108459499378894od_b_c] :
          ( ( A4 = bot_bo4166481423041325370od_b_c )
          | ? [A3: set_li6436108459499378894od_b_c,B4: list_P903359562653991662od_b_c] :
              ( ( A4
                = ( insert6227932334100060350od_b_c @ B4 @ A3 ) )
              & ( finite3074115686814133143od_b_c @ A3 ) ) ) ) ) ).

% finite.simps
thf(fact_1208_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
          ( ( A4 = bot_bot_set_nat )
          | ? [A3: set_nat,B4: nat] :
              ( ( A4
                = ( insert_nat @ B4 @ A3 ) )
              & ( finite_finite_nat @ A3 ) ) ) ) ) ).

% finite.simps
thf(fact_1209_finite_Osimps,axiom,
    ( finite_finite_a
    = ( ^ [A4: set_a] :
          ( ( A4 = bot_bot_set_a )
          | ? [A3: set_a,B4: a] :
              ( ( A4
                = ( insert_a @ B4 @ A3 ) )
              & ( finite_finite_a @ A3 ) ) ) ) ) ).

% finite.simps
thf(fact_1210_finite_Ocases,axiom,
    ! [A2: set_li6436108459499378894od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A2 )
     => ( ( A2 != bot_bo4166481423041325370od_b_c )
       => ~ ! [A7: set_li6436108459499378894od_b_c] :
              ( ? [A6: list_P903359562653991662od_b_c] :
                  ( A2
                  = ( insert6227932334100060350od_b_c @ A6 @ A7 ) )
             => ~ ( finite3074115686814133143od_b_c @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1211_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A6: nat] :
                  ( A2
                  = ( insert_nat @ A6 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1212_finite_Ocases,axiom,
    ! [A2: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( A2 != bot_bot_set_a )
       => ~ ! [A7: set_a] :
              ( ? [A6: a] :
                  ( A2
                  = ( insert_a @ A6 @ A7 ) )
             => ~ ( finite_finite_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1213_subset__singletonD,axiom,
    ! [A: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
     => ( ( A = bot_bot_set_a )
        | ( A
          = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_1214_subset__singletonD,axiom,
    ! [A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( insert6227932334100060350od_b_c @ X @ bot_bo4166481423041325370od_b_c ) )
     => ( ( A = bot_bo4166481423041325370od_b_c )
        | ( A
          = ( insert6227932334100060350od_b_c @ X @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singletonD
thf(fact_1215_subset__singleton__iff,axiom,
    ! [X8: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ X8 @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( ( X8 = bot_bot_set_a )
        | ( X8
          = ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_1216_subset__singleton__iff,axiom,
    ! [X8: set_li6436108459499378894od_b_c,A2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X8 @ ( insert6227932334100060350od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) )
      = ( ( X8 = bot_bo4166481423041325370od_b_c )
        | ( X8
          = ( insert6227932334100060350od_b_c @ A2 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singleton_iff
thf(fact_1217_singleton__Un__iff,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ( ( insert_a @ X @ bot_bot_set_a )
        = ( sup_sup_set_a @ A @ B ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1218_Un__singleton__iff,axiom,
    ! [A: set_a,B: set_a,X: a] :
      ( ( ( sup_sup_set_a @ A @ B )
        = ( insert_a @ X @ bot_bot_set_a ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1219_insert__is__Un,axiom,
    ( insert_a
    = ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).

% insert_is_Un
thf(fact_1220_card__insert__le,axiom,
    ! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X @ A ) ) ) ).

% card_insert_le
thf(fact_1221_semilattice__set_Oin__idem,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( lattic4067905262246253180od_b_c @ F )
     => ( ( finite3074115686814133143od_b_c @ A )
       => ( ( member6330420149250801815od_b_c @ X @ A )
         => ( ( F @ X @ ( lattic2734920875441048264od_b_c @ F @ A ) )
            = ( lattic2734920875441048264od_b_c @ F @ A ) ) ) ) ) ).

% semilattice_set.in_idem
thf(fact_1222_semilattice__set_Oin__idem,axiom,
    ! [F: a > a > a,A: set_a,X: a] :
      ( ( lattic5961991414251573132_set_a @ F )
     => ( ( finite_finite_a @ A )
       => ( ( member_a @ X @ A )
         => ( ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
            = ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ).

% semilattice_set.in_idem
thf(fact_1223_semilattice__set_Oin__idem,axiom,
    ! [F: nat > nat > nat,A: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A )
       => ( ( member_nat @ X @ A )
         => ( ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
            = ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ).

% semilattice_set.in_idem
thf(fact_1224_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 )
       => ( ! [X2: list_P903359562653991662od_b_c,S4: set_li6436108459499378894od_b_c] :
              ( ( finite3074115686814133143od_b_c @ S4 )
             => ( ! [Y6: list_P903359562653991662od_b_c] :
                    ( ( member6330420149250801815od_b_c @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert6227932334100060350od_b_c @ X2 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1225_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X2 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1226_finite__ranking__induct,axiom,
    ! [S: set_a,P: set_a > $o,F: a > nat] :
      ( ( finite_finite_a @ S )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X2: a,S4: set_a] :
              ( ( finite_finite_a @ S4 )
             => ( ! [Y6: a] :
                    ( ( member_a @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_a @ X2 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1227_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B6: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                   => ( ord_less_nat @ B6 @ X3 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B6 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1228_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B6: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X3: nat] :
                    ( ( member_nat @ X3 @ A7 )
                   => ( ord_less_nat @ X3 @ B6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B6 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1229_finite__subset__induct,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A6: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A6 @ A )
                 => ( ~ ( member_nat @ A6 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1230_finite__subset__induct,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A6: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A6 @ A )
                 => ( ~ ( member_a @ A6 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1231_finite__subset__induct,axiom,
    ! [F2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F2 )
     => ( ( ord_le282488521294790766od_b_c @ F2 @ A )
       => ( ( P @ bot_bo4166481423041325370od_b_c )
         => ( ! [A6: list_P903359562653991662od_b_c,F3: set_li6436108459499378894od_b_c] :
                ( ( finite3074115686814133143od_b_c @ F3 )
               => ( ( member6330420149250801815od_b_c @ A6 @ A )
                 => ( ~ ( member6330420149250801815od_b_c @ A6 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert6227932334100060350od_b_c @ A6 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1232_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A6: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A6 @ A )
                 => ( ( ord_less_eq_set_nat @ F3 @ A )
                   => ( ~ ( member_nat @ A6 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1233_finite__subset__induct_H,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A6: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A6 @ A )
                 => ( ( ord_less_eq_set_a @ F3 @ A )
                   => ( ~ ( member_a @ A6 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1234_finite__subset__induct_H,axiom,
    ! [F2: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,P: set_li6436108459499378894od_b_c > $o] :
      ( ( finite3074115686814133143od_b_c @ F2 )
     => ( ( ord_le282488521294790766od_b_c @ F2 @ A )
       => ( ( P @ bot_bo4166481423041325370od_b_c )
         => ( ! [A6: list_P903359562653991662od_b_c,F3: set_li6436108459499378894od_b_c] :
                ( ( finite3074115686814133143od_b_c @ F3 )
               => ( ( member6330420149250801815od_b_c @ A6 @ A )
                 => ( ( ord_le282488521294790766od_b_c @ F3 @ A )
                   => ( ~ ( member6330420149250801815od_b_c @ A6 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert6227932334100060350od_b_c @ A6 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1235_card__Suc__eq__finite,axiom,
    ! [A: set_li6436108459499378894od_b_c,K: nat] :
      ( ( ( finite5583770498833199894od_b_c @ A )
        = ( suc @ K ) )
      = ( ? [B4: list_P903359562653991662od_b_c,B3: set_li6436108459499378894od_b_c] :
            ( ( A
              = ( insert6227932334100060350od_b_c @ B4 @ B3 ) )
            & ~ ( member6330420149250801815od_b_c @ B4 @ B3 )
            & ( ( finite5583770498833199894od_b_c @ B3 )
              = K )
            & ( finite3074115686814133143od_b_c @ B3 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_1236_card__Suc__eq__finite,axiom,
    ! [A: set_a,K: nat] :
      ( ( ( finite_card_a @ A )
        = ( suc @ K ) )
      = ( ? [B4: a,B3: set_a] :
            ( ( A
              = ( insert_a @ B4 @ B3 ) )
            & ~ ( member_a @ B4 @ B3 )
            & ( ( finite_card_a @ B3 )
              = K )
            & ( finite_finite_a @ B3 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_1237_card__Suc__eq__finite,axiom,
    ! [A: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A )
        = ( suc @ K ) )
      = ( ? [B4: nat,B3: set_nat] :
            ( ( A
              = ( insert_nat @ B4 @ B3 ) )
            & ~ ( member_nat @ B4 @ B3 )
            & ( ( finite_card_nat @ B3 )
              = K )
            & ( finite_finite_nat @ B3 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_1238_card__insert__if,axiom,
    ! [A: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( finite3074115686814133143od_b_c @ A )
     => ( ( ( member6330420149250801815od_b_c @ X @ A )
         => ( ( finite5583770498833199894od_b_c @ ( insert6227932334100060350od_b_c @ X @ A ) )
            = ( finite5583770498833199894od_b_c @ A ) ) )
        & ( ~ ( member6330420149250801815od_b_c @ X @ A )
         => ( ( finite5583770498833199894od_b_c @ ( insert6227932334100060350od_b_c @ X @ A ) )
            = ( suc @ ( finite5583770498833199894od_b_c @ A ) ) ) ) ) ) ).

% card_insert_if
thf(fact_1239_card__insert__if,axiom,
    ! [A: set_a,X: a] :
      ( ( finite_finite_a @ A )
     => ( ( ( member_a @ X @ A )
         => ( ( finite_card_a @ ( insert_a @ X @ A ) )
            = ( finite_card_a @ A ) ) )
        & ( ~ ( member_a @ X @ A )
         => ( ( finite_card_a @ ( insert_a @ X @ A ) )
            = ( suc @ ( finite_card_a @ A ) ) ) ) ) ) ).

% card_insert_if
thf(fact_1240_card__insert__if,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( member_nat @ X @ A )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
            = ( finite_card_nat @ A ) ) )
        & ( ~ ( member_nat @ X @ A )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
            = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).

% card_insert_if
thf(fact_1241_card__1__singleton__iff,axiom,
    ! [A: set_a] :
      ( ( ( finite_card_a @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: a] :
            ( A
            = ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).

% card_1_singleton_iff
thf(fact_1242_card__eq__SucD,axiom,
    ! [A: set_li6436108459499378894od_b_c,K: nat] :
      ( ( ( finite5583770498833199894od_b_c @ A )
        = ( suc @ K ) )
     => ? [B6: list_P903359562653991662od_b_c,B7: set_li6436108459499378894od_b_c] :
          ( ( A
            = ( insert6227932334100060350od_b_c @ B6 @ B7 ) )
          & ~ ( member6330420149250801815od_b_c @ B6 @ B7 )
          & ( ( finite5583770498833199894od_b_c @ B7 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bo4166481423041325370od_b_c ) ) ) ) ).

% card_eq_SucD
thf(fact_1243_card__eq__SucD,axiom,
    ! [A: set_a,K: nat] :
      ( ( ( finite_card_a @ A )
        = ( suc @ K ) )
     => ? [B6: a,B7: set_a] :
          ( ( A
            = ( insert_a @ B6 @ B7 ) )
          & ~ ( member_a @ B6 @ B7 )
          & ( ( finite_card_a @ B7 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_a ) ) ) ) ).

% card_eq_SucD
thf(fact_1244_card__Suc__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c,K: nat] :
      ( ( ( finite5583770498833199894od_b_c @ A )
        = ( suc @ K ) )
      = ( ? [B4: list_P903359562653991662od_b_c,B3: set_li6436108459499378894od_b_c] :
            ( ( A
              = ( insert6227932334100060350od_b_c @ B4 @ B3 ) )
            & ~ ( member6330420149250801815od_b_c @ B4 @ B3 )
            & ( ( finite5583770498833199894od_b_c @ B3 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B3 = bot_bo4166481423041325370od_b_c ) ) ) ) ) ).

% card_Suc_eq
thf(fact_1245_card__Suc__eq,axiom,
    ! [A: set_a,K: nat] :
      ( ( ( finite_card_a @ A )
        = ( suc @ K ) )
      = ( ? [B4: a,B3: set_a] :
            ( ( A
              = ( insert_a @ B4 @ B3 ) )
            & ~ ( member_a @ B4 @ B3 )
            & ( ( finite_card_a @ B3 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B3 = bot_bot_set_a ) ) ) ) ) ).

% card_Suc_eq
thf(fact_1246_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1247_diff__is__0__eq_H,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( minus_minus_nat @ M3 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1248_diff__is__0__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( minus_minus_nat @ M3 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M3 @ N ) ) ).

% diff_is_0_eq
thf(fact_1249_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1250_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1251_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1252_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1253_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1254_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_1255_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1256_diff__le__self,axiom,
    ! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).

% diff_le_self
thf(fact_1257_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_1258_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M3 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1259_le__diff__iff,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1260_eq__diff__iff,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M3 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M3 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1261_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1262_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1263_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1264_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1265_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1266_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1267_less__diff__iff,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M3 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1268_Suc__diff__le,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_eq_nat @ N @ M3 )
     => ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
        = ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1269_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1270_monotone__function__with__limit__witness__helper,axiom,
    ! [F: nat > nat,K: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( ( ( F @ I2 )
                = ( F @ J2 ) )
             => ! [M5: nat] :
                  ( ( ord_less_eq_nat @ J2 @ M5 )
                 => ( ( F @ I2 )
                    = ( F @ M5 ) ) ) ) )
       => ( ! [I2: nat] : ( ord_less_eq_nat @ ( F @ I2 ) @ K )
         => ~ ! [X2: nat] :
                ( ( ( F @ ( suc @ X2 ) )
                  = ( F @ X2 ) )
               => ~ ( ord_less_eq_nat @ X2 @ ( minus_minus_nat @ K @ ( F @ zero_zero_nat ) ) ) ) ) ) ) ).

% monotone_function_with_limit_witness_helper
thf(fact_1271__092_060open_062_092_060And_062S_H_H_AS_H_O_AW_AS_H_AS_H_H_A_092_060subseteq_062_Aset_A_Iprefixes_Aw_J_092_060close_062,axiom,
    ! [S2: set_a,S3: set_a] : ( ord_le282488521294790766od_b_c @ ( w @ S2 @ S3 ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ w2 ) ) ) ).

% \<open>\<And>S'' S'. W S' S'' \<subseteq> set (prefixes w)\<close>
thf(fact_1272__092_060open_062w_H_A_092_060in_062_Aset_A_Iprefixes_Aw_J_092_060close_062,axiom,
    member6330420149250801815od_b_c @ w3 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ w2 ) ) ).

% \<open>w' \<in> set (prefixes w)\<close>
thf(fact_1273__092_060open_062wk_A_092_060in_062_Aset_A_Iprefixes_Aw_J_092_060close_062,axiom,
    member6330420149250801815od_b_c @ wk @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ w2 ) ) ).

% \<open>wk \<in> set (prefixes w)\<close>

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( ( member_a @ ( after_a_b_c @ m @ q1 @ w3 ) @ s1 )
      & ( member_a @ ( after_a_b_c @ m @ q2 @ w3 ) @ s2 ) )
    | ( ( member_a @ ( after_a_b_c @ m @ q1 @ w3 ) @ s2 )
      & ( member_a @ ( after_a_b_c @ m @ q2 @ w3 ) @ s1 ) ) ) ).

%------------------------------------------------------------------------------