%------------------------------------------------------------------------------
% 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/0051_State_Cover/prob_00666_031695__20308764_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1464 ( 680 unt; 192 typ;   0 def)
%            Number of atoms       : 3261 (1437 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 9674 ( 403   ~;  66   |; 265   &;7892   @)
%                                         (   0 <=>;1048  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   5 avg)
%            Number of types       :   17 (  16 usr)
%            Number of type conns  :  733 ( 733   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  179 ( 176 usr;  19 con; 0-6 aty)
%            Number of variables   : 3317 ( 390   ^;2826   !; 101   ?;3317   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:29:02.737
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    option7233216073105783496od_c_a: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    set_op7949082993927878370od_c_a: $tType ).

thf(ty_n_t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    option503927706846959746od_c_a: $tType ).

thf(ty_n_t__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    list_o7717683381156371842od_c_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    set_li1159382662694783132od_c_a: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    list_P6327159017948738492od_c_a: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    set_option_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    set_option_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__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

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

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

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

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

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

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

% Explicit typings (176)
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_Oacyclic__paths__up__to__length_001tf__a_001tf__b_001tf__c,type,
    acycli3978232057192500090_a_b_c: fsm_a_b_c > a > nat > set_li1159382662694783132od_c_a ).

thf(sy_c_FSM_Oadd__state_001tf__a_001tf__b_001tf__c,type,
    add_state_a_b_c: fsm_a_b_c > a > fsm_a_b_c ).

thf(sy_c_FSM_Ocompleted__path_001tf__a_001tf__b_001tf__c,type,
    completed_path_a_b_c: fsm_a_b_c > a > list_P6327159017948738492od_c_a > $o ).

thf(sy_c_FSM_Odeadlock__state_001tf__a_001tf__b_001tf__c,type,
    deadlock_state_a_b_c: fsm_a_b_c > a > $o ).

thf(sy_c_FSM_Ofilter__states_001tf__a_001tf__b_001tf__c,type,
    filter_states_a_b_c: fsm_a_b_c > ( a > $o ) > fsm_a_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_Omaximal__acyclic__paths_001tf__a_001tf__b_001tf__c,type,
    maxima334016647137088875_a_b_c: fsm_a_b_c > set_li1159382662694783132od_c_a ).

thf(sy_c_FSM_Opath_001tf__a_001tf__b_001tf__c,type,
    path_a_b_c: fsm_a_b_c > a > list_P6327159017948738492od_c_a > $o ).

thf(sy_c_FSM_Opaths__up__to__length_001tf__a_001tf__b_001tf__c,type,
    paths_5655032219139660648_a_b_c: fsm_a_b_c > a > nat > set_li1159382662694783132od_c_a ).

thf(sy_c_FSM_Oreachable_001tf__a_001tf__b_001tf__c,type,
    reachable_a_b_c: fsm_a_b_c > a > $o ).

thf(sy_c_FSM_Oreachable__k_001tf__a_001tf__b_001tf__c,type,
    reachable_k_a_b_c: fsm_a_b_c > a > nat > set_a ).

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_Osize_001tf__a_001tf__b_001tf__c,type,
    size_a_b_c: fsm_a_b_c > nat ).

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_FSM_Otarget_001tf__a_001tf__b_001tf__c,type,
    target_a_b_c: a > list_P6327159017948738492od_c_a > a ).

thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up230880734466166339od_c_a: ( list_P6327159017948738492od_c_a > option503927706846959746od_c_a ) > list_P6327159017948738492od_c_a > option503927706846959746od_c_a > list_P6327159017948738492od_c_a > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up4729603247160217221tion_a: ( list_P6327159017948738492od_c_a > option_a ) > list_P6327159017948738492od_c_a > option_a > list_P6327159017948738492od_c_a > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up4606601786420396787od_c_a: ( nat > option503927706846959746od_c_a ) > nat > option503927706846959746od_c_a > nat > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_Itf__a_J,type,
    fun_upd_nat_option_a: ( nat > option_a ) > nat > option_a > nat > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up3067012645132738557od_c_a: ( option503927706846959746od_c_a > option503927706846959746od_c_a ) > option503927706846959746od_c_a > option503927706846959746od_c_a > option503927706846959746od_c_a > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up4228323085938402111tion_a: ( option503927706846959746od_c_a > option_a ) > option503927706846959746od_c_a > option_a > option503927706846959746od_c_a > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up4491647651108784803od_c_a: ( option_nat > option503927706846959746od_c_a ) > option_nat > option503927706846959746od_c_a > option_nat > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up1391842941490748133tion_a: ( option_nat > option_a ) > option_nat > option_a > option_nat > option_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up2061198598404480699od_c_a: ( option_a > option503927706846959746od_c_a ) > option_a > option503927706846959746od_c_a > option_a > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
    fun_up1079276522633388797tion_a: ( option_a > option_a ) > option_a > option_a > option_a > option_a ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_It__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    fun_up5326097583011407431od_c_a: ( a > option7233216073105783496od_c_a ) > a > option7233216073105783496od_c_a > a > option7233216073105783496od_c_a ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    fun_up7010226539926187649od_c_a: ( a > option503927706846959746od_c_a ) > a > option503927706846959746od_c_a > a > option503927706846959746od_c_a ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_Itf__a_J,type,
    fun_upd_a_option_a: ( a > option_a ) > a > option_a > a > option_a ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    inj_on3621648716004224823od_c_a: ( list_P6327159017948738492od_c_a > option503927706846959746od_c_a ) > set_li1159382662694783132od_c_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
    inj_on_a_nat: ( a > nat ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    inj_on8536872568654550389od_c_a: ( a > option503927706846959746od_c_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
    inj_on_a_set_a: ( a > set_a ) > set_a > $o ).

thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
    minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    minus_4060711634664891779od_c_a: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    minus_1221259694836641097od_c_a: set_op7949082993927878370od_c_a > set_op7949082993927878370od_c_a > set_op7949082993927878370od_c_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    minus_5999362281193037231on_nat: set_option_nat > set_option_nat > set_option_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    minus_1574173051537231627tion_a: set_option_a > set_option_a > set_option_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
    uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_M_Eo_J,type,
    uminus_uminus_a_o: ( a > $o ) > a > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
    uminus_uminus_set_a: set_a > set_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
    uniq_nat: ( nat > $o ) > $o ).

thf(sy_c_HOL_OUniq_001tf__a,type,
    uniq_a: ( a > $o ) > $o ).

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

thf(sy_c_If_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    if_opt5277955309491978056od_c_a: $o > option503927706846959746od_c_a > option503927706846959746od_c_a > option503927706846959746od_c_a ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_Itf__a_J,type,
    semila2496817875450240012_set_a: ( set_a > set_a > set_a ) > set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
    sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
    sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).

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__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    sup_su500200128730103920od_c_a: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a ).

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__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    sup_su7734108422889136310od_c_a: set_op7949082993927878370od_c_a > set_op7949082993927878370od_c_a > set_op7949082993927878370od_c_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    sup_su7692501479792361346on_nat: set_option_nat > set_option_nat > set_option_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    sup_sup_set_option_a: set_option_a > set_option_a > set_option_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_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic8739620818006775868at_nat: ( nat > nat ) > ( nat > $o ) > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001tf__a_001t__Nat__Onat,type,
    lattic1189635703294652468_a_nat: ( a > nat ) > ( a > $o ) > a ).

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_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_List_Ogen__length_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    gen_le1648446388249104713od_c_a: nat > list_P6327159017948738492od_c_a > nat ).

thf(sy_c_List_Olist_ONil_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    nil_op474191768008379010od_c_a: list_o7717683381156371842od_c_a ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    nil_Pr1342775757158464060od_c_a: list_P6327159017948738492od_c_a ).

thf(sy_c_List_Orotate1_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    rotate1246884043934778121od_c_a: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a ).

thf(sy_c_List_Othose_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    those_5530478821918711090od_c_a: list_o7717683381156371842od_c_a > option503927706846959746od_c_a ).

thf(sy_c_Map_Odom_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_li1621849933323954258od_c_a: ( list_P6327159017948738492od_c_a > option503927706846959746od_c_a ) > set_li1159382662694783132od_c_a ).

thf(sy_c_Map_Odom_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001t__Nat__Onat,type,
    dom_li6393241031938617530_a_nat: ( list_P6327159017948738492od_c_a > option_nat ) > set_li1159382662694783132od_c_a ).

thf(sy_c_Map_Odom_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_001tf__a,type,
    dom_li8145836122603711636_c_a_a: ( list_P6327159017948738492od_c_a > option_a ) > set_li1159382662694783132od_c_a ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_na1185376985345782168od_c_a: ( nat > option503927706846959746od_c_a ) > set_nat ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Nat__Onat,type,
    dom_nat_nat: ( nat > option_nat ) > set_nat ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001tf__a,type,
    dom_nat_a: ( nat > option_a ) > set_nat ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_op4893529039689921932od_c_a: ( option503927706846959746od_c_a > option503927706846959746od_c_a ) > set_op7949082993927878370od_c_a ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_001t__Nat__Onat,type,
    dom_op3000714018489525760_a_nat: ( option503927706846959746od_c_a > option_nat ) > set_op7949082993927878370od_c_a ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_001tf__a,type,
    dom_op5089748838693076174_c_a_a: ( option503927706846959746od_c_a > option_a ) > set_op7949082993927878370od_c_a ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_op7568714055193396424od_c_a: ( option_nat > option503927706846959746od_c_a ) > set_option_nat ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
    dom_option_nat_nat: ( option_nat > option_nat ) > set_option_nat ).

thf(sy_c_Map_Odom_001t__Option__Ooption_It__Nat__Onat_J_001tf__a,type,
    dom_option_nat_a: ( option_nat > option_a ) > set_option_nat ).

thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_op9021813908095708874od_c_a: ( option_a > option503927706846959746od_c_a ) > set_option_a ).

thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001t__Nat__Onat,type,
    dom_option_a_nat: ( option_a > option_nat ) > set_option_a ).

thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001tf__a,type,
    dom_option_a_a: ( option_a > option_a ) > set_option_a ).

thf(sy_c_Map_Odom_001tf__a_001t__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    dom_a_4706468663598931670od_c_a: ( a > option7233216073105783496od_c_a ) > set_a ).

thf(sy_c_Map_Odom_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    dom_a_2450325921413825296od_c_a: ( a > option503927706846959746od_c_a ) > set_a ).

thf(sy_c_Map_Odom_001tf__a_001t__Nat__Onat,type,
    dom_a_nat: ( a > option_nat ) > set_a ).

thf(sy_c_Map_Odom_001tf__a_001tf__a,type,
    dom_a_a: ( a > option_a ) > set_a ).

thf(sy_c_Map_Oran_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
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thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    restri1700453863123333178od_c_a: ( nat > option503927706846959746od_c_a ) > set_nat > nat > option503927706846959746od_c_a ).

thf(sy_c_Map_Orestrict__map_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
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thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    size_s3386368156187063848od_c_a: list_P6327159017948738492od_c_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001tf__a,type,
    none_a: option_a ).

thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    some_l6142909491759833889od_c_a: list_P6327159017948738492od_c_a > option503927706846959746od_c_a ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001tf__a,type,
    some_a: a > option_a ).

thf(sy_c_Option_Othese_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
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thf(sy_c_Option_Othese_001t__Nat__Onat,type,
    these_nat: set_option_nat > set_nat ).

thf(sy_c_Option_Othese_001tf__a,type,
    these_a: set_option_a > set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
    bot_bot_option_a_o: option_a > $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__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
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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__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    bot_bo5009843511495006442on_nat: set_option_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
    bot_bot_set_option_a: set_option_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_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
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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__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_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

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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 ).

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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
    ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).

thf(sy_c_Set_OBex_001t__Nat__Onat,type,
    bex_nat: set_nat > ( nat > $o ) > $o ).

thf(sy_c_Set_OBex_001tf__a,type,
    bex_a: set_a > ( a > $o ) > $o ).

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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Nat__Onat_J,type,
    collect_option_nat: ( option_nat > $o ) > set_option_nat ).

thf(sy_c_Set_OCollect_001t__Option__Ooption_Itf__a_J,type,
    collect_option_a: ( option_a > $o ) > set_option_a ).

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

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thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
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thf(sy_c_Set_Ofilter_001t__Option__Ooption_It__Nat__Onat_J,type,
    filter_option_nat: ( option_nat > $o ) > set_option_nat > set_option_nat ).

thf(sy_c_Set_Ofilter_001t__Option__Ooption_Itf__a_J,type,
    filter_option_a: ( option_a > $o ) > set_option_a > set_option_a ).

thf(sy_c_Set_Ofilter_001tf__a,type,
    filter_a: ( a > $o ) > set_a > set_a ).

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

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thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Nat__Onat_J,type,
    insert_option_nat: option_nat > set_option_nat > set_option_nat ).

thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
    insert_option_a: option_a > set_option_a > set_option_a ).

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

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001tf__a,type,
    is_singleton_a: set_a > $o ).

thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
    remove_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oremove_001tf__a,type,
    remove_a: a > set_a > set_a ).

thf(sy_c_Set_Othe__elem_001tf__a,type,
    the_elem_a: set_a > a ).

thf(sy_c_State__Cover_Oreaching__paths__up__to__depth_001tf__a_001tf__b_001tf__c,type,
    state_6616341566432195646_a_b_c: fsm_a_b_c > set_a > set_a > ( a > option503927706846959746od_c_a ) > nat > a > option503927706846959746od_c_a ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_Util_Ois__prefix_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    is_pre701076807510392747od_c_a: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member7410604586820865893od_c_a: list_P6327159017948738492od_c_a > set_li1159382662694783132od_c_a > $o ).

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

thf(sy_c_member_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member2959255366155174571od_c_a: option503927706846959746od_c_a > set_op7949082993927878370od_c_a > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
    member_option_nat: option_nat > set_option_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
    member_option_a: option_a > set_option_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_path__assignments____,type,
    path_assignments: a > option503927706846959746od_c_a ).

thf(sy_v_q,type,
    q: a ).

% Relevant facts (1266)
thf(fact_0__092_060open_062_092_060And_062q_Ak_O_A_092_060lbrakk_062_092_060And_062q_Ap_O_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_A_092_060Longrightarrow_062_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_059_Aq_A_092_060in_062_A_123FSM_Oinitial_AM_125_A_092_060union_062_A_123_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Areaching__paths__up__to__depth_AM_A_123FSM_Oinitial_AM_125_A_123_125_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Ak_Aq_A_061_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_092_060close_062,axiom,
    ! [Q: a,K: nat] :
      ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
          ( ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) )
         => ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) ) )
     => ( ( member_a @ Q @ ( sup_sup_set_a @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a ) )
       => ( ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
            @ ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
            @ K
            @ Q )
          = ( fun_up7010226539926187649od_c_a
            @ ^ [X: a] : none_l592525953500355997od_c_a
            @ ( initial_a_b_c @ m )
            @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
            @ Q ) ) ) ) ).

% \<open>\<And>q k. \<lbrakk>\<And>q p. [FSM.initial M \<mapsto> []] q = Some p \<Longrightarrow> [FSM.initial M \<mapsto> []] q = Some p; q \<in> {FSM.initial M} \<union> {}\<rbrakk> \<Longrightarrow> reaching_paths_up_to_depth M {FSM.initial M} {} [FSM.initial M \<mapsto> []] k q = [FSM.initial M \<mapsto> []] q\<close>
thf(fact_1__092_060open_062_092_060And_062q_Ap_Ak_O_A_092_060lbrakk_062_092_060And_062q_Ap_O_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_A_092_060Longrightarrow_062_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_059_Areaching__paths__up__to__depth_AM_A_123FSM_Oinitial_AM_125_A_123_125_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Ak_Aq_A_061_ASome_Ap_092_060rbrakk_062_A_092_060Longrightarrow_062_Apath_AM_A_IFSM_Oinitial_AM_J_Ap_A_092_060and_062_Atarget_A_IFSM_Oinitial_AM_J_Ap_A_061_Aq_A_092_060and_062_Alength_Ap_A_092_060le_062_A0_A_L_Ak_092_060close_062,axiom,
    ! [K: nat,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
          ( ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) )
         => ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) ) )
     => ( ( ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
            @ ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
            @ K
            @ Q )
          = ( some_l6142909491759833889od_c_a @ P2 ) )
       => ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P2 )
          & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P2 )
            = Q )
          & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P2 ) @ ( plus_plus_nat @ zero_zero_nat @ K ) ) ) ) ) ).

% \<open>\<And>q p k. \<lbrakk>\<And>q p. [FSM.initial M \<mapsto> []] q = Some p \<Longrightarrow> [FSM.initial M \<mapsto> []] q = Some p; reaching_paths_up_to_depth M {FSM.initial M} {} [FSM.initial M \<mapsto> []] k q = Some p\<rbrakk> \<Longrightarrow> path M (FSM.initial M) p \<and> target (FSM.initial M) p = q \<and> length p \<le> 0 + k\<close>
thf(fact_2__092_060open_062_092_060And_062q_Ak_O_A_I_092_060And_062q_Ap_O_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_A_092_060Longrightarrow_062_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Aq_A_061_ASome_Ap_J_A_092_060Longrightarrow_062_A_Ireaching__paths__up__to__depth_AM_A_123FSM_Oinitial_AM_125_A_123_125_A_091FSM_Oinitial_AM_A_092_060mapsto_062_A_091_093_093_Ak_Aq_A_061_ANone_J_A_061_A_I_092_060nexists_062p_O_Apath_AM_A_IFSM_Oinitial_AM_J_Ap_A_092_060and_062_Atarget_A_IFSM_Oinitial_AM_J_Ap_A_061_Aq_A_092_060and_062_Alength_Ap_A_092_060le_062_A0_A_L_Ak_J_092_060close_062,axiom,
    ! [K: nat,Q: a] :
      ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
          ( ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) )
         => ( ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
              @ Q2 )
            = ( some_l6142909491759833889od_c_a @ P ) ) )
     => ( ( ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
            @ ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
            @ K
            @ Q )
          = none_l592525953500355997od_c_a )
        = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
                  = Q )
                & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ ( plus_plus_nat @ zero_zero_nat @ K ) ) ) ) ) ) ).

% \<open>\<And>q k. (\<And>q p. [FSM.initial M \<mapsto> []] q = Some p \<Longrightarrow> [FSM.initial M \<mapsto> []] q = Some p) \<Longrightarrow> (reaching_paths_up_to_depth M {FSM.initial M} {} [FSM.initial M \<mapsto> []] k q = None) = (\<nexists>p. path M (FSM.initial M) p \<and> target (FSM.initial M) p = q \<and> length p \<le> 0 + k)\<close>
thf(fact_3_c3,axiom,
    ! [Q: a] :
      ( ( ( fun_up5326097583011407431od_c_a
          @ ^ [X: a] : none_l3348010773085364707od_c_a
          @ ( initial_a_b_c @ m )
          @ ( some_l3194232794420892007od_c_a @ nil_op474191768008379010od_c_a )
          @ Q )
        = none_l3348010773085364707od_c_a )
      = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
              & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
                = Q )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ zero_zero_nat ) ) ) ) ).

% c3
thf(fact_4_c3,axiom,
    ! [Q: a] :
      ( ( ( fun_up7010226539926187649od_c_a
          @ ^ [X: a] : none_l592525953500355997od_c_a
          @ ( initial_a_b_c @ m )
          @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
          @ Q )
        = none_l592525953500355997od_c_a )
      = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
              & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
                = Q )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ zero_zero_nat ) ) ) ) ).

% c3
thf(fact_5_c4,axiom,
    ! [Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( ( fun_up7010226539926187649od_c_a
          @ ^ [X: a] : none_l592525953500355997od_c_a
          @ ( initial_a_b_c @ m )
          @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a )
          @ Q )
        = ( some_l6142909491759833889od_c_a @ P2 ) )
     => ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P2 )
        & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P2 )
          = Q )
        & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P2 ) @ zero_zero_nat ) ) ) ).

% c4
thf(fact_6_path__assignments__def,axiom,
    ( path_assignments
    = ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
      @ ( fun_up7010226539926187649od_c_a
        @ ^ [X: a] : none_l592525953500355997od_c_a
        @ ( initial_a_b_c @ m )
        @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
      @ ( minus_minus_nat @ ( size_a_b_c @ m ) @ one_one_nat ) ) ) ).

% path_assignments_def
thf(fact_7_empty__upd__none,axiom,
    ! [X2: nat] :
      ( ( fun_up4606601786420396787od_c_a
        @ ^ [X: nat] : none_l592525953500355997od_c_a
        @ X2
        @ none_l592525953500355997od_c_a )
      = ( ^ [X: nat] : none_l592525953500355997od_c_a ) ) ).

% empty_upd_none
thf(fact_8_empty__upd__none,axiom,
    ! [X2: a] :
      ( ( fun_up7010226539926187649od_c_a
        @ ^ [X: a] : none_l592525953500355997od_c_a
        @ X2
        @ none_l592525953500355997od_c_a )
      = ( ^ [X: a] : none_l592525953500355997od_c_a ) ) ).

% empty_upd_none
thf(fact_9_singleton__conv,axiom,
    ! [A: option503927706846959746od_c_a] :
      ( ( collec8119457264603017453od_c_a
        @ ^ [X: option503927706846959746od_c_a] : ( X = A ) )
      = ( insert3566037597566277202od_c_a @ A @ bot_bo3379513543401233742od_c_a ) ) ).

% singleton_conv
thf(fact_10_singleton__conv,axiom,
    ! [A: option_a] :
      ( ( collect_option_a
        @ ^ [X: option_a] : ( X = A ) )
      = ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).

% singleton_conv
thf(fact_11_singleton__conv,axiom,
    ! [A: list_P6327159017948738492od_c_a] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X: list_P6327159017948738492od_c_a] : ( X = A ) )
      = ( insert4789241225314331020od_c_a @ A @ bot_bo6236370880139903240od_c_a ) ) ).

% singleton_conv
thf(fact_12_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_13_singleton__conv,axiom,
    ! [A: a] :
      ( ( collect_a
        @ ^ [X: a] : ( X = A ) )
      = ( insert_a @ A @ bot_bot_set_a ) ) ).

% singleton_conv
thf(fact_14_singleton__conv2,axiom,
    ! [A: option503927706846959746od_c_a] :
      ( ( collec8119457264603017453od_c_a
        @ ( ^ [Y: option503927706846959746od_c_a,Z: option503927706846959746od_c_a] : ( Y = Z )
          @ A ) )
      = ( insert3566037597566277202od_c_a @ A @ bot_bo3379513543401233742od_c_a ) ) ).

% singleton_conv2
thf(fact_15_singleton__conv2,axiom,
    ! [A: option_a] :
      ( ( collect_option_a
        @ ( ^ [Y: option_a,Z: option_a] : ( Y = Z )
          @ A ) )
      = ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).

% singleton_conv2
thf(fact_16_singleton__conv2,axiom,
    ! [A: list_P6327159017948738492od_c_a] :
      ( ( collec6273869032445462695od_c_a
        @ ( ^ [Y: list_P6327159017948738492od_c_a,Z: list_P6327159017948738492od_c_a] : ( Y = Z )
          @ A ) )
      = ( insert4789241225314331020od_c_a @ A @ bot_bo6236370880139903240od_c_a ) ) ).

% singleton_conv2
thf(fact_17_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y: nat,Z: nat] : ( Y = Z )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_18_singleton__conv2,axiom,
    ! [A: a] :
      ( ( collect_a
        @ ( ^ [Y: a,Z: a] : ( Y = Z )
          @ A ) )
      = ( insert_a @ A @ bot_bot_set_a ) ) ).

% singleton_conv2
thf(fact_19_target__nil,axiom,
    ! [Q: a] :
      ( ( target_a_b_c @ Q @ nil_Pr1342775757158464060od_c_a )
      = Q ) ).

% target_nil
thf(fact_20_not__None__eq,axiom,
    ! [X2: option_a] :
      ( ( X2 != none_a )
      = ( ? [Y2: a] :
            ( X2
            = ( some_a @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_21_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
      = ( ? [Y2: nat] :
            ( X2
            = ( some_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_22_not__None__eq,axiom,
    ! [X2: option503927706846959746od_c_a] :
      ( ( X2 != none_l592525953500355997od_c_a )
      = ( ? [Y2: list_P6327159017948738492od_c_a] :
            ( X2
            = ( some_l6142909491759833889od_c_a @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_23_not__Some__eq,axiom,
    ! [X2: option_a] :
      ( ( ! [Y2: a] :
            ( X2
           != ( some_a @ Y2 ) ) )
      = ( X2 = none_a ) ) ).

% not_Some_eq
thf(fact_24_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ( ! [Y2: nat] :
            ( X2
           != ( some_nat @ Y2 ) ) )
      = ( X2 = none_nat ) ) ).

% not_Some_eq
thf(fact_25_not__Some__eq,axiom,
    ! [X2: option503927706846959746od_c_a] :
      ( ( ! [Y2: list_P6327159017948738492od_c_a] :
            ( X2
           != ( some_l6142909491759833889od_c_a @ Y2 ) ) )
      = ( X2 = none_l592525953500355997od_c_a ) ) ).

% not_Some_eq
thf(fact_26_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_27_singletonI,axiom,
    ! [A: option503927706846959746od_c_a] : ( member2959255366155174571od_c_a @ A @ ( insert3566037597566277202od_c_a @ A @ bot_bo3379513543401233742od_c_a ) ) ).

% singletonI
thf(fact_28_singletonI,axiom,
    ! [A: option_nat] : ( member_option_nat @ A @ ( insert_option_nat @ A @ bot_bo5009843511495006442on_nat ) ) ).

% singletonI
thf(fact_29_singletonI,axiom,
    ! [A: option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).

% singletonI
thf(fact_30_singletonI,axiom,
    ! [A: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ A @ ( insert4789241225314331020od_c_a @ A @ bot_bo6236370880139903240od_c_a ) ) ).

% singletonI
thf(fact_31_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_32_singletonI,axiom,
    ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_33_c5,axiom,
    ( ( dom_a_4706468663598931670od_c_a
      @ ( fun_up5326097583011407431od_c_a
        @ ^ [X: a] : none_l3348010773085364707od_c_a
        @ ( initial_a_b_c @ m )
        @ ( some_l3194232794420892007od_c_a @ nil_op474191768008379010od_c_a ) ) )
    = ( sup_sup_set_a @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a ) ) ).

% c5
thf(fact_34_c5,axiom,
    ( ( dom_a_2450325921413825296od_c_a
      @ ( fun_up7010226539926187649od_c_a
        @ ^ [X: a] : none_l592525953500355997od_c_a
        @ ( initial_a_b_c @ m )
        @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
    = ( sup_sup_set_a @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a ) ) ).

% c5
thf(fact_35_map__upd__nonempty,axiom,
    ! [T: nat > option503927706846959746od_c_a,K: nat,X2: list_P6327159017948738492od_c_a] :
      ( ( fun_up4606601786420396787od_c_a @ T @ K @ ( some_l6142909491759833889od_c_a @ X2 ) )
     != ( ^ [X: nat] : none_l592525953500355997od_c_a ) ) ).

% map_upd_nonempty
thf(fact_36_map__upd__nonempty,axiom,
    ! [T: a > option503927706846959746od_c_a,K: a,X2: list_P6327159017948738492od_c_a] :
      ( ( fun_up7010226539926187649od_c_a @ T @ K @ ( some_l6142909491759833889od_c_a @ X2 ) )
     != ( ^ [X: a] : none_l592525953500355997od_c_a ) ) ).

% map_upd_nonempty
thf(fact_37__C_K_C,axiom,
    ( ( filter_a
      @ ^ [Q3: a] :
          ( ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
            @ ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ ( initial_a_b_c @ m )
              @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
            @ ( minus_minus_nat @ ( size_a_b_c @ m ) @ one_one_nat )
            @ Q3 )
         != none_l592525953500355997od_c_a )
      @ ( states_a_b_c @ m ) )
    = ( filter_a
      @ ^ [Q3: a] :
          ( ( path_assignments @ Q3 )
         != none_l592525953500355997od_c_a )
      @ ( states_a_b_c @ m ) ) ) ).

% "*"
thf(fact_38_diff__shunt__var,axiom,
    ! [X2: set_option_a,Y3: set_option_a] :
      ( ( ( minus_1574173051537231627tion_a @ X2 @ Y3 )
        = bot_bot_set_option_a )
      = ( ord_le1955136853071979460tion_a @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_39_diff__shunt__var,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( minus_4060711634664891779od_c_a @ X2 @ Y3 )
        = bot_bo6236370880139903240od_c_a )
      = ( ord_le2998388488506175548od_c_a @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_40_diff__shunt__var,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X2 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_41_diff__shunt__var,axiom,
    ! [X2: nat > $o,Y3: nat > $o] :
      ( ( ( minus_minus_nat_o @ X2 @ Y3 )
        = bot_bot_nat_o )
      = ( ord_less_eq_nat_o @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_42_diff__shunt__var,axiom,
    ! [X2: a > $o,Y3: a > $o] :
      ( ( ( minus_minus_a_o @ X2 @ Y3 )
        = bot_bot_a_o )
      = ( ord_less_eq_a_o @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_43_diff__shunt__var,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( ( minus_minus_set_a @ X2 @ Y3 )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_44_Diff__eq__empty__iff,axiom,
    ! [A2: set_option_a,B: set_option_a] :
      ( ( ( minus_1574173051537231627tion_a @ A2 @ B )
        = bot_bot_set_option_a )
      = ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_45_Diff__eq__empty__iff,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ( minus_4060711634664891779od_c_a @ A2 @ B )
        = bot_bo6236370880139903240od_c_a )
      = ( ord_le2998388488506175548od_c_a @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_46_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_47_Diff__eq__empty__iff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( minus_minus_set_a @ A2 @ B )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ A2 @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_48_Diff__cancel,axiom,
    ! [A2: set_option_a] :
      ( ( minus_1574173051537231627tion_a @ A2 @ A2 )
      = bot_bot_set_option_a ) ).

% Diff_cancel
thf(fact_49_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_50_Diff__cancel,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( minus_4060711634664891779od_c_a @ A2 @ A2 )
      = bot_bo6236370880139903240od_c_a ) ).

% Diff_cancel
thf(fact_51_Diff__cancel,axiom,
    ! [A2: set_a] :
      ( ( minus_minus_set_a @ A2 @ A2 )
      = bot_bot_set_a ) ).

% Diff_cancel
thf(fact_52_empty__Diff,axiom,
    ! [A2: set_option_a] :
      ( ( minus_1574173051537231627tion_a @ bot_bot_set_option_a @ A2 )
      = bot_bot_set_option_a ) ).

% empty_Diff
thf(fact_53_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_54_empty__Diff,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( minus_4060711634664891779od_c_a @ bot_bo6236370880139903240od_c_a @ A2 )
      = bot_bo6236370880139903240od_c_a ) ).

% empty_Diff
thf(fact_55_empty__Diff,axiom,
    ! [A2: set_a] :
      ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
      = bot_bot_set_a ) ).

% empty_Diff
thf(fact_56_Diff__empty,axiom,
    ! [A2: set_option_a] :
      ( ( minus_1574173051537231627tion_a @ A2 @ bot_bot_set_option_a )
      = A2 ) ).

% Diff_empty
thf(fact_57_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_58_Diff__empty,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( minus_4060711634664891779od_c_a @ A2 @ bot_bo6236370880139903240od_c_a )
      = A2 ) ).

% Diff_empty
thf(fact_59_Diff__empty,axiom,
    ! [A2: set_a] :
      ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
      = A2 ) ).

% Diff_empty
thf(fact_60_empty__subsetI,axiom,
    ! [A2: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A2 ) ).

% empty_subsetI
thf(fact_61_empty__subsetI,axiom,
    ! [A2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ bot_bo6236370880139903240od_c_a @ A2 ) ).

% empty_subsetI
thf(fact_62_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_63_empty__subsetI,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).

% empty_subsetI
thf(fact_64_subset__empty,axiom,
    ! [A2: set_option_a] :
      ( ( ord_le1955136853071979460tion_a @ A2 @ bot_bot_set_option_a )
      = ( A2 = bot_bot_set_option_a ) ) ).

% subset_empty
thf(fact_65_subset__empty,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ bot_bo6236370880139903240od_c_a )
      = ( A2 = bot_bo6236370880139903240od_c_a ) ) ).

% subset_empty
thf(fact_66_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_67_subset__empty,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
      = ( A2 = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_68_empty__Collect__eq,axiom,
    ! [P4: option_a > $o] :
      ( ( bot_bot_set_option_a
        = ( collect_option_a @ P4 ) )
      = ( ! [X: option_a] :
            ~ ( P4 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
    ! [P4: list_P6327159017948738492od_c_a > $o] :
      ( ( bot_bo6236370880139903240od_c_a
        = ( collec6273869032445462695od_c_a @ P4 ) )
      = ( ! [X: list_P6327159017948738492od_c_a] :
            ~ ( P4 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
    ! [P4: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P4 ) )
      = ( ! [X: nat] :
            ~ ( P4 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_71_empty__Collect__eq,axiom,
    ! [P4: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P4 ) )
      = ( ! [X: a] :
            ~ ( P4 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_72_Collect__empty__eq,axiom,
    ! [P4: option_a > $o] :
      ( ( ( collect_option_a @ P4 )
        = bot_bot_set_option_a )
      = ( ! [X: option_a] :
            ~ ( P4 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
    ! [P4: list_P6327159017948738492od_c_a > $o] :
      ( ( ( collec6273869032445462695od_c_a @ P4 )
        = bot_bo6236370880139903240od_c_a )
      = ( ! [X: list_P6327159017948738492od_c_a] :
            ~ ( P4 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
    ! [P4: nat > $o] :
      ( ( ( collect_nat @ P4 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P4 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_75_Collect__empty__eq,axiom,
    ! [P4: a > $o] :
      ( ( ( collect_a @ P4 )
        = bot_bot_set_a )
      = ( ! [X: a] :
            ~ ( P4 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_76_all__not__in__conv,axiom,
    ! [A2: set_op7949082993927878370od_c_a] :
      ( ( ! [X: option503927706846959746od_c_a] :
            ~ ( member2959255366155174571od_c_a @ X @ A2 ) )
      = ( A2 = bot_bo3379513543401233742od_c_a ) ) ).

% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
    ! [A2: set_option_nat] :
      ( ( ! [X: option_nat] :
            ~ ( member_option_nat @ X @ A2 ) )
      = ( A2 = bot_bo5009843511495006442on_nat ) ) ).

% all_not_in_conv
thf(fact_78_all__not__in__conv,axiom,
    ! [A2: set_option_a] :
      ( ( ! [X: option_a] :
            ~ ( member_option_a @ X @ A2 ) )
      = ( A2 = bot_bot_set_option_a ) ) ).

% all_not_in_conv
thf(fact_79_all__not__in__conv,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( ! [X: list_P6327159017948738492od_c_a] :
            ~ ( member7410604586820865893od_c_a @ X @ A2 ) )
      = ( A2 = bot_bo6236370880139903240od_c_a ) ) ).

% all_not_in_conv
thf(fact_80_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_81_all__not__in__conv,axiom,
    ! [A2: set_a] :
      ( ( ! [X: a] :
            ~ ( member_a @ X @ A2 ) )
      = ( A2 = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_82_empty__iff,axiom,
    ! [C: option503927706846959746od_c_a] :
      ~ ( member2959255366155174571od_c_a @ C @ bot_bo3379513543401233742od_c_a ) ).

% empty_iff
thf(fact_83_empty__iff,axiom,
    ! [C: option_nat] :
      ~ ( member_option_nat @ C @ bot_bo5009843511495006442on_nat ) ).

% empty_iff
thf(fact_84_empty__iff,axiom,
    ! [C: option_a] :
      ~ ( member_option_a @ C @ bot_bot_set_option_a ) ).

% empty_iff
thf(fact_85_empty__iff,axiom,
    ! [C: list_P6327159017948738492od_c_a] :
      ~ ( member7410604586820865893od_c_a @ C @ bot_bo6236370880139903240od_c_a ) ).

% empty_iff
thf(fact_86_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_87_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_88_insert__Diff1,axiom,
    ! [X2: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ X2 @ B )
     => ( ( minus_1221259694836641097od_c_a @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) @ B )
        = ( minus_1221259694836641097od_c_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_89_insert__Diff1,axiom,
    ! [X2: option_nat,B: set_option_nat,A2: set_option_nat] :
      ( ( member_option_nat @ X2 @ B )
     => ( ( minus_5999362281193037231on_nat @ ( insert_option_nat @ X2 @ A2 ) @ B )
        = ( minus_5999362281193037231on_nat @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_90_insert__Diff1,axiom,
    ! [X2: option_a,B: set_option_a,A2: set_option_a] :
      ( ( member_option_a @ X2 @ B )
     => ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X2 @ A2 ) @ B )
        = ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_91_insert__Diff1,axiom,
    ! [X2: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ B )
     => ( ( minus_4060711634664891779od_c_a @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) @ B )
        = ( minus_4060711634664891779od_c_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_92_insert__Diff1,axiom,
    ! [X2: a,B: set_a,A2: set_a] :
      ( ( member_a @ X2 @ B )
     => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
        = ( minus_minus_set_a @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_93_insert__Diff1,axiom,
    ! [X2: nat,B: set_nat,A2: set_nat] :
      ( ( member_nat @ X2 @ B )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
        = ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% insert_Diff1
thf(fact_94_Diff__insert0,axiom,
    ! [X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ~ ( member2959255366155174571od_c_a @ X2 @ A2 )
     => ( ( minus_1221259694836641097od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ B ) )
        = ( minus_1221259694836641097od_c_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_95_Diff__insert0,axiom,
    ! [X2: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ~ ( member_option_nat @ X2 @ A2 )
     => ( ( minus_5999362281193037231on_nat @ A2 @ ( insert_option_nat @ X2 @ B ) )
        = ( minus_5999362281193037231on_nat @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_96_Diff__insert0,axiom,
    ! [X2: option_a,A2: set_option_a,B: set_option_a] :
      ( ~ ( member_option_a @ X2 @ A2 )
     => ( ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X2 @ B ) )
        = ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_97_Diff__insert0,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ~ ( member7410604586820865893od_c_a @ X2 @ A2 )
     => ( ( minus_4060711634664891779od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ B ) )
        = ( minus_4060711634664891779od_c_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_98_Diff__insert0,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B ) )
        = ( minus_minus_set_a @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_99_Diff__insert0,axiom,
    ! [X2: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
        = ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% Diff_insert0
thf(fact_100_insert__subset,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) @ B )
      = ( ( member7410604586820865893od_c_a @ X2 @ B )
        & ( ord_le2998388488506175548od_c_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_101_insert__subset,axiom,
    ! [X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) @ B )
      = ( ( member2959255366155174571od_c_a @ X2 @ B )
        & ( ord_le7915732482681745026od_c_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_102_insert__subset,axiom,
    ! [X2: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ( ord_le6937355464348597430on_nat @ ( insert_option_nat @ X2 @ A2 ) @ B )
      = ( ( member_option_nat @ X2 @ B )
        & ( ord_le6937355464348597430on_nat @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_103_insert__subset,axiom,
    ! [X2: option_a,A2: set_option_a,B: set_option_a] :
      ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X2 @ A2 ) @ B )
      = ( ( member_option_a @ X2 @ B )
        & ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_104_insert__subset,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B )
      = ( ( member_a @ X2 @ B )
        & ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_105_insert__subset,axiom,
    ! [X2: nat,A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
      = ( ( member_nat @ X2 @ B )
        & ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).

% insert_subset
thf(fact_106_insert__absorb2,axiom,
    ! [X2: nat,A2: set_nat] :
      ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
      = ( insert_nat @ X2 @ A2 ) ) ).

% insert_absorb2
thf(fact_107_insert__absorb2,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( insert4789241225314331020od_c_a @ X2 @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) )
      = ( insert4789241225314331020od_c_a @ X2 @ A2 ) ) ).

% insert_absorb2
thf(fact_108_insert__absorb2,axiom,
    ! [X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( insert3566037597566277202od_c_a @ X2 @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) )
      = ( insert3566037597566277202od_c_a @ X2 @ A2 ) ) ).

% insert_absorb2
thf(fact_109_insert__absorb2,axiom,
    ! [X2: option_a,A2: set_option_a] :
      ( ( insert_option_a @ X2 @ ( insert_option_a @ X2 @ A2 ) )
      = ( insert_option_a @ X2 @ A2 ) ) ).

% insert_absorb2
thf(fact_110_insert__absorb2,axiom,
    ! [X2: a,A2: set_a] :
      ( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
      = ( insert_a @ X2 @ A2 ) ) ).

% insert_absorb2
thf(fact_111_insert__iff,axiom,
    ! [A: list_P6327159017948738492od_c_a,B2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ A @ ( insert4789241225314331020od_c_a @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member7410604586820865893od_c_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_112_insert__iff,axiom,
    ! [A: option503927706846959746od_c_a,B2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ A @ ( insert3566037597566277202od_c_a @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member2959255366155174571od_c_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_113_insert__iff,axiom,
    ! [A: option_nat,B2: option_nat,A2: set_option_nat] :
      ( ( member_option_nat @ A @ ( insert_option_nat @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_option_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_114_insert__iff,axiom,
    ! [A: option_a,B2: option_a,A2: set_option_a] :
      ( ( member_option_a @ A @ ( insert_option_a @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_option_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_115_insert__iff,axiom,
    ! [A: a,B2: a,A2: set_a] :
      ( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_116_insert__iff,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
      = ( ( A = B2 )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_117_insertCI,axiom,
    ! [A: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,B2: list_P6327159017948738492od_c_a] :
      ( ( ~ ( member7410604586820865893od_c_a @ A @ B )
       => ( A = B2 ) )
     => ( member7410604586820865893od_c_a @ A @ ( insert4789241225314331020od_c_a @ B2 @ B ) ) ) ).

% insertCI
thf(fact_118_insertCI,axiom,
    ! [A: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,B2: option503927706846959746od_c_a] :
      ( ( ~ ( member2959255366155174571od_c_a @ A @ B )
       => ( A = B2 ) )
     => ( member2959255366155174571od_c_a @ A @ ( insert3566037597566277202od_c_a @ B2 @ B ) ) ) ).

% insertCI
thf(fact_119_insertCI,axiom,
    ! [A: option_nat,B: set_option_nat,B2: option_nat] :
      ( ( ~ ( member_option_nat @ A @ B )
       => ( A = B2 ) )
     => ( member_option_nat @ A @ ( insert_option_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_120_insertCI,axiom,
    ! [A: option_a,B: set_option_a,B2: option_a] :
      ( ( ~ ( member_option_a @ A @ B )
       => ( A = B2 ) )
     => ( member_option_a @ A @ ( insert_option_a @ B2 @ B ) ) ) ).

% insertCI
thf(fact_121_insertCI,axiom,
    ! [A: a,B: set_a,B2: a] :
      ( ( ~ ( member_a @ A @ B )
       => ( A = B2 ) )
     => ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).

% insertCI
thf(fact_122_insertCI,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A @ B )
       => ( A = B2 ) )
     => ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_123_UnCI,axiom,
    ! [C: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( ~ ( member2959255366155174571od_c_a @ C @ B )
       => ( member2959255366155174571od_c_a @ C @ A2 ) )
     => ( member2959255366155174571od_c_a @ C @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) ) ) ).

% UnCI
thf(fact_124_UnCI,axiom,
    ! [C: option_nat,B: set_option_nat,A2: set_option_nat] :
      ( ( ~ ( member_option_nat @ C @ B )
       => ( member_option_nat @ C @ A2 ) )
     => ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A2 @ B ) ) ) ).

% UnCI
thf(fact_125_UnCI,axiom,
    ! [C: option_a,B: set_option_a,A2: set_option_a] :
      ( ( ~ ( member_option_a @ C @ B )
       => ( member_option_a @ C @ A2 ) )
     => ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).

% UnCI
thf(fact_126_UnCI,axiom,
    ! [C: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ~ ( member7410604586820865893od_c_a @ C @ B )
       => ( member7410604586820865893od_c_a @ C @ A2 ) )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ) ).

% UnCI
thf(fact_127_UnCI,axiom,
    ! [C: nat,B: set_nat,A2: set_nat] :
      ( ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ A2 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).

% UnCI
thf(fact_128_UnCI,axiom,
    ! [C: a,B: set_a,A2: set_a] :
      ( ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ A2 ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).

% UnCI
thf(fact_129_Un__iff,axiom,
    ! [C: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ C @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) )
      = ( ( member2959255366155174571od_c_a @ C @ A2 )
        | ( member2959255366155174571od_c_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_130_Un__iff,axiom,
    ! [C: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A2 @ B ) )
      = ( ( member_option_nat @ C @ A2 )
        | ( member_option_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_131_Un__iff,axiom,
    ! [C: option_a,A2: set_option_a,B: set_option_a] :
      ( ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) )
      = ( ( member_option_a @ C @ A2 )
        | ( member_option_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_132_Un__iff,axiom,
    ! [C: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
      = ( ( member7410604586820865893od_c_a @ C @ A2 )
        | ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_133_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
      = ( ( member_nat @ C @ A2 )
        | ( member_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_134_Un__iff,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
      = ( ( member_a @ C @ A2 )
        | ( member_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_135_Un__subset__iff,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B ) @ C2 )
      = ( ( ord_le2998388488506175548od_c_a @ A2 @ C2 )
        & ( ord_le2998388488506175548od_c_a @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_136_Un__subset__iff,axiom,
    ! [A2: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A2 @ C2 )
        & ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_137_Un__subset__iff,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
      = ( ( ord_less_eq_set_a @ A2 @ C2 )
        & ( ord_less_eq_set_a @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_138_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
      = ( sup_sup_set_nat @ A2 @ B ) ) ).

% Un_Diff_cancel
thf(fact_139_Un__Diff__cancel,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ A2 @ ( minus_4060711634664891779od_c_a @ B @ A2 ) )
      = ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ).

% Un_Diff_cancel
thf(fact_140_Un__Diff__cancel,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
      = ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_Diff_cancel
thf(fact_141_Un__Diff__cancel2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
      = ( sup_sup_set_nat @ B @ A2 ) ) ).

% Un_Diff_cancel2
thf(fact_142_Un__Diff__cancel2,axiom,
    ! [B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ ( minus_4060711634664891779od_c_a @ B @ A2 ) @ A2 )
      = ( sup_su500200128730103920od_c_a @ B @ A2 ) ) ).

% Un_Diff_cancel2
thf(fact_143_Un__Diff__cancel2,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
      = ( sup_sup_set_a @ B @ A2 ) ) ).

% Un_Diff_cancel2
thf(fact_144_option_Oinject,axiom,
    ! [X22: a,Y22: a] :
      ( ( ( some_a @ X22 )
        = ( some_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_145_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_146_option_Oinject,axiom,
    ! [X22: list_P6327159017948738492od_c_a,Y22: list_P6327159017948738492od_c_a] :
      ( ( ( some_l6142909491759833889od_c_a @ X22 )
        = ( some_l6142909491759833889od_c_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_147_member__filter,axiom,
    ! [X2: list_P6327159017948738492od_c_a,P4: list_P6327159017948738492od_c_a > $o,A2: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( filter8200269361223028333od_c_a @ P4 @ A2 ) )
      = ( ( member7410604586820865893od_c_a @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_148_member__filter,axiom,
    ! [X2: option503927706846959746od_c_a,P4: option503927706846959746od_c_a > $o,A2: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ X2 @ ( filter8108757734611162547od_c_a @ P4 @ A2 ) )
      = ( ( member2959255366155174571od_c_a @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_149_member__filter,axiom,
    ! [X2: option_nat,P4: option_nat > $o,A2: set_option_nat] :
      ( ( member_option_nat @ X2 @ ( filter_option_nat @ P4 @ A2 ) )
      = ( ( member_option_nat @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_150_member__filter,axiom,
    ! [X2: option_a,P4: option_a > $o,A2: set_option_a] :
      ( ( member_option_a @ X2 @ ( filter_option_a @ P4 @ A2 ) )
      = ( ( member_option_a @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_151_member__filter,axiom,
    ! [X2: a,P4: a > $o,A2: set_a] :
      ( ( member_a @ X2 @ ( filter_a @ P4 @ A2 ) )
      = ( ( member_a @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_152_member__filter,axiom,
    ! [X2: nat,P4: nat > $o,A2: set_nat] :
      ( ( member_nat @ X2 @ ( filter_nat @ P4 @ A2 ) )
      = ( ( member_nat @ X2 @ A2 )
        & ( P4 @ X2 ) ) ) ).

% member_filter
thf(fact_153_insert__Diff__single,axiom,
    ! [A: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( insert3566037597566277202od_c_a @ A @ ( minus_1221259694836641097od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ A @ bot_bo3379513543401233742od_c_a ) ) )
      = ( insert3566037597566277202od_c_a @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_154_insert__Diff__single,axiom,
    ! [A: option_a,A2: set_option_a] :
      ( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
      = ( insert_option_a @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_155_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_156_insert__Diff__single,axiom,
    ! [A: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( insert4789241225314331020od_c_a @ A @ ( minus_4060711634664891779od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ A @ bot_bo6236370880139903240od_c_a ) ) )
      = ( insert4789241225314331020od_c_a @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_157_insert__Diff__single,axiom,
    ! [A: a,A2: set_a] :
      ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
      = ( insert_a @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_158_singleton__insert__inj__eq_H,axiom,
    ! [A: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B2: option503927706846959746od_c_a] :
      ( ( ( insert3566037597566277202od_c_a @ A @ A2 )
        = ( insert3566037597566277202od_c_a @ B2 @ bot_bo3379513543401233742od_c_a ) )
      = ( ( A = B2 )
        & ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ B2 @ bot_bo3379513543401233742od_c_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_159_singleton__insert__inj__eq_H,axiom,
    ! [A: option_a,A2: set_option_a,B2: option_a] :
      ( ( ( insert_option_a @ A @ A2 )
        = ( insert_option_a @ B2 @ bot_bot_set_option_a ) )
      = ( ( A = B2 )
        & ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_160_singleton__insert__inj__eq_H,axiom,
    ! [A: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B2: list_P6327159017948738492od_c_a] :
      ( ( ( insert4789241225314331020od_c_a @ A @ A2 )
        = ( insert4789241225314331020od_c_a @ B2 @ bot_bo6236370880139903240od_c_a ) )
      = ( ( A = B2 )
        & ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ B2 @ bot_bo6236370880139903240od_c_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_161_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B2: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_162_singleton__insert__inj__eq_H,axiom,
    ! [A: a,A2: set_a,B2: a] :
      ( ( ( insert_a @ A @ A2 )
        = ( insert_a @ B2 @ bot_bot_set_a ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_163_singleton__insert__inj__eq,axiom,
    ! [B2: option503927706846959746od_c_a,A: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( ( insert3566037597566277202od_c_a @ B2 @ bot_bo3379513543401233742od_c_a )
        = ( insert3566037597566277202od_c_a @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ B2 @ bot_bo3379513543401233742od_c_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_164_singleton__insert__inj__eq,axiom,
    ! [B2: option_a,A: option_a,A2: set_option_a] :
      ( ( ( insert_option_a @ B2 @ bot_bot_set_option_a )
        = ( insert_option_a @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ bot_bot_set_option_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_165_singleton__insert__inj__eq,axiom,
    ! [B2: list_P6327159017948738492od_c_a,A: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ( insert4789241225314331020od_c_a @ B2 @ bot_bo6236370880139903240od_c_a )
        = ( insert4789241225314331020od_c_a @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ B2 @ bot_bo6236370880139903240od_c_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_166_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_167_singleton__insert__inj__eq,axiom,
    ! [B2: a,A: a,A2: set_a] :
      ( ( ( insert_a @ B2 @ bot_bot_set_a )
        = ( insert_a @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_168_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_169_mem__Collect__eq,axiom,
    ! [A: option503927706846959746od_c_a,P4: option503927706846959746od_c_a > $o] :
      ( ( member2959255366155174571od_c_a @ A @ ( collec8119457264603017453od_c_a @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_170_mem__Collect__eq,axiom,
    ! [A: option_nat,P4: option_nat > $o] :
      ( ( member_option_nat @ A @ ( collect_option_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_171_mem__Collect__eq,axiom,
    ! [A: option_a,P4: option_a > $o] :
      ( ( member_option_a @ A @ ( collect_option_a @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
    ! [A: list_P6327159017948738492od_c_a,P4: list_P6327159017948738492od_c_a > $o] :
      ( ( member7410604586820865893od_c_a @ A @ ( collec6273869032445462695od_c_a @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_173_mem__Collect__eq,axiom,
    ! [A: nat,P4: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_174_mem__Collect__eq,axiom,
    ! [A: a,P4: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P4 ) )
      = ( P4 @ A ) ) ).

% mem_Collect_eq
thf(fact_175_Collect__mem__eq,axiom,
    ! [A2: set_op7949082993927878370od_c_a] :
      ( ( collec8119457264603017453od_c_a
        @ ^ [X: option503927706846959746od_c_a] : ( member2959255366155174571od_c_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
    ! [A2: set_option_nat] :
      ( ( collect_option_nat
        @ ^ [X: option_nat] : ( member_option_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
    ! [A2: set_option_a] :
      ( ( collect_option_a
        @ ^ [X: option_a] : ( member_option_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_178_Collect__mem__eq,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_179_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_180_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_181_Collect__cong,axiom,
    ! [P4: nat > $o,Q4: nat > $o] :
      ( ! [X3: nat] :
          ( ( P4 @ X3 )
          = ( Q4 @ X3 ) )
     => ( ( collect_nat @ P4 )
        = ( collect_nat @ Q4 ) ) ) ).

% Collect_cong
thf(fact_182_Collect__cong,axiom,
    ! [P4: list_P6327159017948738492od_c_a > $o,Q4: list_P6327159017948738492od_c_a > $o] :
      ( ! [X3: list_P6327159017948738492od_c_a] :
          ( ( P4 @ X3 )
          = ( Q4 @ X3 ) )
     => ( ( collec6273869032445462695od_c_a @ P4 )
        = ( collec6273869032445462695od_c_a @ Q4 ) ) ) ).

% Collect_cong
thf(fact_183_Collect__cong,axiom,
    ! [P4: a > $o,Q4: a > $o] :
      ( ! [X3: a] :
          ( ( P4 @ X3 )
          = ( Q4 @ X3 ) )
     => ( ( collect_a @ P4 )
        = ( collect_a @ Q4 ) ) ) ).

% Collect_cong
thf(fact_184_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_185_Un__empty,axiom,
    ! [A2: set_option_a,B: set_option_a] :
      ( ( ( sup_sup_set_option_a @ A2 @ B )
        = bot_bot_set_option_a )
      = ( ( A2 = bot_bot_set_option_a )
        & ( B = bot_bot_set_option_a ) ) ) ).

% Un_empty
thf(fact_186_Un__empty,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ( sup_su500200128730103920od_c_a @ A2 @ B )
        = bot_bo6236370880139903240od_c_a )
      = ( ( A2 = bot_bo6236370880139903240od_c_a )
        & ( B = bot_bo6236370880139903240od_c_a ) ) ) ).

% Un_empty
thf(fact_187_Un__empty,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_188_Un__empty,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( sup_sup_set_a @ A2 @ B )
        = bot_bot_set_a )
      = ( ( A2 = bot_bot_set_a )
        & ( B = bot_bot_set_a ) ) ) ).

% Un_empty
thf(fact_189_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_190_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_191_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_192_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_193_Un__insert__right,axiom,
    ! [A2: set_op7949082993927878370od_c_a,A: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( sup_su7734108422889136310od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ A @ B ) )
      = ( insert3566037597566277202od_c_a @ A @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) ) ) ).

% Un_insert_right
thf(fact_194_Un__insert__right,axiom,
    ! [A2: set_option_a,A: option_a,B: set_option_a] :
      ( ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ A @ B ) )
      = ( insert_option_a @ A @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).

% Un_insert_right
thf(fact_195_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).

% Un_insert_right
thf(fact_196_Un__insert__right,axiom,
    ! [A2: set_li1159382662694783132od_c_a,A: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ A @ B ) )
      = ( insert4789241225314331020od_c_a @ A @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ) ).

% Un_insert_right
thf(fact_197_Un__insert__right,axiom,
    ! [A2: set_a,A: a,B: set_a] :
      ( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
      = ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).

% Un_insert_right
thf(fact_198_Un__insert__left,axiom,
    ! [A: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,C2: set_op7949082993927878370od_c_a] :
      ( ( sup_su7734108422889136310od_c_a @ ( insert3566037597566277202od_c_a @ A @ B ) @ C2 )
      = ( insert3566037597566277202od_c_a @ A @ ( sup_su7734108422889136310od_c_a @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_199_Un__insert__left,axiom,
    ! [A: option_a,B: set_option_a,C2: set_option_a] :
      ( ( sup_sup_set_option_a @ ( insert_option_a @ A @ B ) @ C2 )
      = ( insert_option_a @ A @ ( sup_sup_set_option_a @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_200_Un__insert__left,axiom,
    ! [A: nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B ) @ C2 )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_201_Un__insert__left,axiom,
    ! [A: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ ( insert4789241225314331020od_c_a @ A @ B ) @ C2 )
      = ( insert4789241225314331020od_c_a @ A @ ( sup_su500200128730103920od_c_a @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_202_Un__insert__left,axiom,
    ! [A: a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C2 )
      = ( insert_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_203_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_204_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_205_fsm__initial,axiom,
    ! [M2: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M2 ) @ ( states_a_b_c @ M2 ) ) ).

% fsm_initial
thf(fact_206_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_207_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_208_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_209_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_210_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_211_dom__eq__empty__conv,axiom,
    ! [F: a > option_a] :
      ( ( ( dom_a_a @ F )
        = bot_bot_set_a )
      = ( F
        = ( ^ [X: a] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_212_dom__eq__empty__conv,axiom,
    ! [F: option_a > option503927706846959746od_c_a] :
      ( ( ( dom_op9021813908095708874od_c_a @ F )
        = bot_bot_set_option_a )
      = ( F
        = ( ^ [X: option_a] : none_l592525953500355997od_c_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_213_dom__eq__empty__conv,axiom,
    ! [F: option_a > option_a] :
      ( ( ( dom_option_a_a @ F )
        = bot_bot_set_option_a )
      = ( F
        = ( ^ [X: option_a] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_214_dom__eq__empty__conv,axiom,
    ! [F: list_P6327159017948738492od_c_a > option503927706846959746od_c_a] :
      ( ( ( dom_li1621849933323954258od_c_a @ F )
        = bot_bo6236370880139903240od_c_a )
      = ( F
        = ( ^ [X: list_P6327159017948738492od_c_a] : none_l592525953500355997od_c_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_215_dom__eq__empty__conv,axiom,
    ! [F: list_P6327159017948738492od_c_a > option_a] :
      ( ( ( dom_li8145836122603711636_c_a_a @ F )
        = bot_bo6236370880139903240od_c_a )
      = ( F
        = ( ^ [X: list_P6327159017948738492od_c_a] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_216_dom__eq__empty__conv,axiom,
    ! [F: nat > option503927706846959746od_c_a] :
      ( ( ( dom_na1185376985345782168od_c_a @ F )
        = bot_bot_set_nat )
      = ( F
        = ( ^ [X: nat] : none_l592525953500355997od_c_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_217_dom__eq__empty__conv,axiom,
    ! [F: nat > option_a] :
      ( ( ( dom_nat_a @ F )
        = bot_bot_set_nat )
      = ( F
        = ( ^ [X: nat] : none_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_218_dom__eq__empty__conv,axiom,
    ! [F: a > option503927706846959746od_c_a] :
      ( ( ( dom_a_2450325921413825296od_c_a @ F )
        = bot_bot_set_a )
      = ( F
        = ( ^ [X: a] : none_l592525953500355997od_c_a ) ) ) ).

% dom_eq_empty_conv
thf(fact_219_fun__upd__None__if__notin__dom,axiom,
    ! [K: nat,M: nat > option503927706846959746od_c_a] :
      ( ~ ( member_nat @ K @ ( dom_na1185376985345782168od_c_a @ M ) )
     => ( ( fun_up4606601786420396787od_c_a @ M @ K @ none_l592525953500355997od_c_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_220_fun__upd__None__if__notin__dom,axiom,
    ! [K: a,M: a > option503927706846959746od_c_a] :
      ( ~ ( member_a @ K @ ( dom_a_2450325921413825296od_c_a @ M ) )
     => ( ( fun_up7010226539926187649od_c_a @ M @ K @ none_l592525953500355997od_c_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_221_fun__upd__None__if__notin__dom,axiom,
    ! [K: a,M: a > option_a] :
      ( ~ ( member_a @ K @ ( dom_a_a @ M ) )
     => ( ( fun_upd_a_option_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_222_fun__upd__None__if__notin__dom,axiom,
    ! [K: nat,M: nat > option_a] :
      ( ~ ( member_nat @ K @ ( dom_nat_a @ M ) )
     => ( ( fun_upd_nat_option_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_223_fun__upd__None__if__notin__dom,axiom,
    ! [K: option_nat,M: option_nat > option_a] :
      ( ~ ( member_option_nat @ K @ ( dom_option_nat_a @ M ) )
     => ( ( fun_up1391842941490748133tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_224_fun__upd__None__if__notin__dom,axiom,
    ! [K: option_a,M: option_a > option_a] :
      ( ~ ( member_option_a @ K @ ( dom_option_a_a @ M ) )
     => ( ( fun_up1079276522633388797tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_225_fun__upd__None__if__notin__dom,axiom,
    ! [K: list_P6327159017948738492od_c_a,M: list_P6327159017948738492od_c_a > option_a] :
      ( ~ ( member7410604586820865893od_c_a @ K @ ( dom_li8145836122603711636_c_a_a @ M ) )
     => ( ( fun_up4729603247160217221tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_226_fun__upd__None__if__notin__dom,axiom,
    ! [K: option_nat,M: option_nat > option503927706846959746od_c_a] :
      ( ~ ( member_option_nat @ K @ ( dom_op7568714055193396424od_c_a @ M ) )
     => ( ( fun_up4491647651108784803od_c_a @ M @ K @ none_l592525953500355997od_c_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_227_fun__upd__None__if__notin__dom,axiom,
    ! [K: option_a,M: option_a > option503927706846959746od_c_a] :
      ( ~ ( member_option_a @ K @ ( dom_op9021813908095708874od_c_a @ M ) )
     => ( ( fun_up2061198598404480699od_c_a @ M @ K @ none_l592525953500355997od_c_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_228_fun__upd__None__if__notin__dom,axiom,
    ! [K: option503927706846959746od_c_a,M: option503927706846959746od_c_a > option_a] :
      ( ~ ( member2959255366155174571od_c_a @ K @ ( dom_op5089748838693076174_c_a_a @ M ) )
     => ( ( fun_up4228323085938402111tion_a @ M @ K @ none_a )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_229_dom__empty,axiom,
    ( ( dom_a_a
      @ ^ [X: a] : none_a )
    = bot_bot_set_a ) ).

% dom_empty
thf(fact_230_dom__empty,axiom,
    ( ( dom_op9021813908095708874od_c_a
      @ ^ [X: option_a] : none_l592525953500355997od_c_a )
    = bot_bot_set_option_a ) ).

% dom_empty
thf(fact_231_dom__empty,axiom,
    ( ( dom_option_a_a
      @ ^ [X: option_a] : none_a )
    = bot_bot_set_option_a ) ).

% dom_empty
thf(fact_232_dom__empty,axiom,
    ( ( dom_li1621849933323954258od_c_a
      @ ^ [X: list_P6327159017948738492od_c_a] : none_l592525953500355997od_c_a )
    = bot_bo6236370880139903240od_c_a ) ).

% dom_empty
thf(fact_233_dom__empty,axiom,
    ( ( dom_li8145836122603711636_c_a_a
      @ ^ [X: list_P6327159017948738492od_c_a] : none_a )
    = bot_bo6236370880139903240od_c_a ) ).

% dom_empty
thf(fact_234_dom__empty,axiom,
    ( ( dom_na1185376985345782168od_c_a
      @ ^ [X: nat] : none_l592525953500355997od_c_a )
    = bot_bot_set_nat ) ).

% dom_empty
thf(fact_235_dom__empty,axiom,
    ( ( dom_nat_a
      @ ^ [X: nat] : none_a )
    = bot_bot_set_nat ) ).

% dom_empty
thf(fact_236_dom__empty,axiom,
    ( ( dom_a_2450325921413825296od_c_a
      @ ^ [X: a] : none_l592525953500355997od_c_a )
    = bot_bot_set_a ) ).

% dom_empty
thf(fact_237_dom__fun__upd,axiom,
    ! [Y3: option503927706846959746od_c_a,F: option503927706846959746od_c_a > option503927706846959746od_c_a,X2: option503927706846959746od_c_a] :
      ( ( ( Y3 = none_l592525953500355997od_c_a )
       => ( ( dom_op4893529039689921932od_c_a @ ( fun_up3067012645132738557od_c_a @ F @ X2 @ Y3 ) )
          = ( minus_1221259694836641097od_c_a @ ( dom_op4893529039689921932od_c_a @ F ) @ ( insert3566037597566277202od_c_a @ X2 @ bot_bo3379513543401233742od_c_a ) ) ) )
      & ( ( Y3 != none_l592525953500355997od_c_a )
       => ( ( dom_op4893529039689921932od_c_a @ ( fun_up3067012645132738557od_c_a @ F @ X2 @ Y3 ) )
          = ( insert3566037597566277202od_c_a @ X2 @ ( dom_op4893529039689921932od_c_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_238_dom__fun__upd,axiom,
    ! [Y3: option_a,F: option503927706846959746od_c_a > option_a,X2: option503927706846959746od_c_a] :
      ( ( ( Y3 = none_a )
       => ( ( dom_op5089748838693076174_c_a_a @ ( fun_up4228323085938402111tion_a @ F @ X2 @ Y3 ) )
          = ( minus_1221259694836641097od_c_a @ ( dom_op5089748838693076174_c_a_a @ F ) @ ( insert3566037597566277202od_c_a @ X2 @ bot_bo3379513543401233742od_c_a ) ) ) )
      & ( ( Y3 != none_a )
       => ( ( dom_op5089748838693076174_c_a_a @ ( fun_up4228323085938402111tion_a @ F @ X2 @ Y3 ) )
          = ( insert3566037597566277202od_c_a @ X2 @ ( dom_op5089748838693076174_c_a_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_239_dom__fun__upd,axiom,
    ! [Y3: option503927706846959746od_c_a,F: option_a > option503927706846959746od_c_a,X2: option_a] :
      ( ( ( Y3 = none_l592525953500355997od_c_a )
       => ( ( dom_op9021813908095708874od_c_a @ ( fun_up2061198598404480699od_c_a @ F @ X2 @ Y3 ) )
          = ( minus_1574173051537231627tion_a @ ( dom_op9021813908095708874od_c_a @ F ) @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) )
      & ( ( Y3 != none_l592525953500355997od_c_a )
       => ( ( dom_op9021813908095708874od_c_a @ ( fun_up2061198598404480699od_c_a @ F @ X2 @ Y3 ) )
          = ( insert_option_a @ X2 @ ( dom_op9021813908095708874od_c_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_240_dom__fun__upd,axiom,
    ! [Y3: option_a,F: option_a > option_a,X2: option_a] :
      ( ( ( Y3 = none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X2 @ Y3 ) )
          = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) )
      & ( ( Y3 != none_a )
       => ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X2 @ Y3 ) )
          = ( insert_option_a @ X2 @ ( dom_option_a_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_241_dom__fun__upd,axiom,
    ! [Y3: option_a,F: a > option_a,X2: a] :
      ( ( ( Y3 = none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X2 @ Y3 ) )
          = ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
      & ( ( Y3 != none_a )
       => ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X2 @ Y3 ) )
          = ( insert_a @ X2 @ ( dom_a_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_242_dom__fun__upd,axiom,
    ! [Y3: option503927706846959746od_c_a,F: nat > option503927706846959746od_c_a,X2: nat] :
      ( ( ( Y3 = none_l592525953500355997od_c_a )
       => ( ( dom_na1185376985345782168od_c_a @ ( fun_up4606601786420396787od_c_a @ F @ X2 @ Y3 ) )
          = ( minus_minus_set_nat @ ( dom_na1185376985345782168od_c_a @ F ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
      & ( ( Y3 != none_l592525953500355997od_c_a )
       => ( ( dom_na1185376985345782168od_c_a @ ( fun_up4606601786420396787od_c_a @ F @ X2 @ Y3 ) )
          = ( insert_nat @ X2 @ ( dom_na1185376985345782168od_c_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_243_dom__fun__upd,axiom,
    ! [Y3: option_a,F: nat > option_a,X2: nat] :
      ( ( ( Y3 = none_a )
       => ( ( dom_nat_a @ ( fun_upd_nat_option_a @ F @ X2 @ Y3 ) )
          = ( minus_minus_set_nat @ ( dom_nat_a @ F ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
      & ( ( Y3 != none_a )
       => ( ( dom_nat_a @ ( fun_upd_nat_option_a @ F @ X2 @ Y3 ) )
          = ( insert_nat @ X2 @ ( dom_nat_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_244_dom__fun__upd,axiom,
    ! [Y3: option503927706846959746od_c_a,F: list_P6327159017948738492od_c_a > option503927706846959746od_c_a,X2: list_P6327159017948738492od_c_a] :
      ( ( ( Y3 = none_l592525953500355997od_c_a )
       => ( ( dom_li1621849933323954258od_c_a @ ( fun_up230880734466166339od_c_a @ F @ X2 @ Y3 ) )
          = ( minus_4060711634664891779od_c_a @ ( dom_li1621849933323954258od_c_a @ F ) @ ( insert4789241225314331020od_c_a @ X2 @ bot_bo6236370880139903240od_c_a ) ) ) )
      & ( ( Y3 != none_l592525953500355997od_c_a )
       => ( ( dom_li1621849933323954258od_c_a @ ( fun_up230880734466166339od_c_a @ F @ X2 @ Y3 ) )
          = ( insert4789241225314331020od_c_a @ X2 @ ( dom_li1621849933323954258od_c_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_245_dom__fun__upd,axiom,
    ! [Y3: option_a,F: list_P6327159017948738492od_c_a > option_a,X2: list_P6327159017948738492od_c_a] :
      ( ( ( Y3 = none_a )
       => ( ( dom_li8145836122603711636_c_a_a @ ( fun_up4729603247160217221tion_a @ F @ X2 @ Y3 ) )
          = ( minus_4060711634664891779od_c_a @ ( dom_li8145836122603711636_c_a_a @ F ) @ ( insert4789241225314331020od_c_a @ X2 @ bot_bo6236370880139903240od_c_a ) ) ) )
      & ( ( Y3 != none_a )
       => ( ( dom_li8145836122603711636_c_a_a @ ( fun_up4729603247160217221tion_a @ F @ X2 @ Y3 ) )
          = ( insert4789241225314331020od_c_a @ X2 @ ( dom_li8145836122603711636_c_a_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_246_dom__fun__upd,axiom,
    ! [Y3: option503927706846959746od_c_a,F: a > option503927706846959746od_c_a,X2: a] :
      ( ( ( Y3 = none_l592525953500355997od_c_a )
       => ( ( dom_a_2450325921413825296od_c_a @ ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 ) )
          = ( minus_minus_set_a @ ( dom_a_2450325921413825296od_c_a @ F ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
      & ( ( Y3 != none_l592525953500355997od_c_a )
       => ( ( dom_a_2450325921413825296od_c_a @ ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 ) )
          = ( insert_a @ X2 @ ( dom_a_2450325921413825296od_c_a @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_247_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_248_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
      ( ( A2
        = ( sup_sup_set_nat @ K @ A ) )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_249_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: a > $o,K: a > $o,A: a > $o,B2: a > $o] :
      ( ( A2
        = ( sup_sup_a_o @ K @ A ) )
     => ( ( sup_sup_a_o @ A2 @ B2 )
        = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_250_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: nat > $o,K: nat > $o,A: nat > $o,B2: nat > $o] :
      ( ( A2
        = ( sup_sup_nat_o @ K @ A ) )
     => ( ( sup_sup_nat_o @ A2 @ B2 )
        = ( sup_sup_nat_o @ K @ ( sup_sup_nat_o @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_251_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ K @ A ) )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_252_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,K: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( A2
        = ( sup_su500200128730103920od_c_a @ K @ A ) )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ B2 )
        = ( sup_su500200128730103920od_c_a @ K @ ( sup_su500200128730103920od_c_a @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_253_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
      ( ( A2
        = ( sup_sup_set_a @ K @ A ) )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_254_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
      ( ( B
        = ( sup_sup_set_nat @ K @ B2 ) )
     => ( ( sup_sup_set_nat @ A @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_255_boolean__algebra__cancel_Osup2,axiom,
    ! [B: a > $o,K: a > $o,B2: a > $o,A: a > $o] :
      ( ( B
        = ( sup_sup_a_o @ K @ B2 ) )
     => ( ( sup_sup_a_o @ A @ B )
        = ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_256_boolean__algebra__cancel_Osup2,axiom,
    ! [B: nat > $o,K: nat > $o,B2: nat > $o,A: nat > $o] :
      ( ( B
        = ( sup_sup_nat_o @ K @ B2 ) )
     => ( ( sup_sup_nat_o @ A @ B )
        = ( sup_sup_nat_o @ K @ ( sup_sup_nat_o @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_257_boolean__algebra__cancel_Osup2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B
        = ( sup_sup_nat @ K @ B2 ) )
     => ( ( sup_sup_nat @ A @ B )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_258_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_li1159382662694783132od_c_a,K: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( B
        = ( sup_su500200128730103920od_c_a @ K @ B2 ) )
     => ( ( sup_su500200128730103920od_c_a @ A @ B )
        = ( sup_su500200128730103920od_c_a @ K @ ( sup_su500200128730103920od_c_a @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_259_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_a,K: set_a,B2: set_a,A: set_a] :
      ( ( B
        = ( sup_sup_set_a @ K @ B2 ) )
     => ( ( sup_sup_set_a @ A @ B )
        = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_260_UnE,axiom,
    ! [C: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ C @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) )
     => ( ~ ( member2959255366155174571od_c_a @ C @ A2 )
       => ( member2959255366155174571od_c_a @ C @ B ) ) ) ).

% UnE
thf(fact_261_UnE,axiom,
    ! [C: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A2 @ B ) )
     => ( ~ ( member_option_nat @ C @ A2 )
       => ( member_option_nat @ C @ B ) ) ) ).

% UnE
thf(fact_262_UnE,axiom,
    ! [C: option_a,A2: set_option_a,B: set_option_a] :
      ( ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) )
     => ( ~ ( member_option_a @ C @ A2 )
       => ( member_option_a @ C @ B ) ) ) ).

% UnE
thf(fact_263_UnE,axiom,
    ! [C: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
     => ( ~ ( member7410604586820865893od_c_a @ C @ A2 )
       => ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% UnE
thf(fact_264_UnE,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
     => ( ~ ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B ) ) ) ).

% UnE
thf(fact_265_UnE,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
     => ( ~ ( member_a @ C @ A2 )
       => ( member_a @ C @ B ) ) ) ).

% UnE
thf(fact_266_UnI1,axiom,
    ! [C: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ C @ A2 )
     => ( member2959255366155174571od_c_a @ C @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) ) ) ).

% UnI1
thf(fact_267_UnI1,axiom,
    ! [C: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ( member_option_nat @ C @ A2 )
     => ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A2 @ B ) ) ) ).

% UnI1
thf(fact_268_UnI1,axiom,
    ! [C: option_a,A2: set_option_a,B: set_option_a] :
      ( ( member_option_a @ C @ A2 )
     => ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).

% UnI1
thf(fact_269_UnI1,axiom,
    ! [C: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ A2 )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ) ).

% UnI1
thf(fact_270_UnI1,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).

% UnI1
thf(fact_271_UnI1,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ A2 )
     => ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).

% UnI1
thf(fact_272_UnI2,axiom,
    ! [C: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( member2959255366155174571od_c_a @ C @ B )
     => ( member2959255366155174571od_c_a @ C @ ( sup_su7734108422889136310od_c_a @ A2 @ B ) ) ) ).

% UnI2
thf(fact_273_UnI2,axiom,
    ! [C: option_nat,B: set_option_nat,A2: set_option_nat] :
      ( ( member_option_nat @ C @ B )
     => ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A2 @ B ) ) ) ).

% UnI2
thf(fact_274_UnI2,axiom,
    ! [C: option_a,B: set_option_a,A2: set_option_a] :
      ( ( member_option_a @ C @ B )
     => ( member_option_a @ C @ ( sup_sup_set_option_a @ A2 @ B ) ) ) ).

% UnI2
thf(fact_275_UnI2,axiom,
    ! [C: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ B )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ) ).

% UnI2
thf(fact_276_UnI2,axiom,
    ! [C: nat,B: set_nat,A2: set_nat] :
      ( ( member_nat @ C @ B )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).

% UnI2
thf(fact_277_UnI2,axiom,
    ! [C: a,B: set_a,A2: set_a] :
      ( ( member_a @ C @ B )
     => ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).

% UnI2
thf(fact_278_bex__Un,axiom,
    ! [A2: set_nat,B: set_nat,P4: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) )
            & ( P4 @ X ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P4 @ X ) )
        | ? [X: nat] :
            ( ( member_nat @ X @ B )
            & ( P4 @ X ) ) ) ) ).

% bex_Un
thf(fact_279_bex__Un,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,P4: list_P6327159017948738492od_c_a > $o] :
      ( ( ? [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
            & ( P4 @ X ) ) )
      = ( ? [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ A2 )
            & ( P4 @ X ) )
        | ? [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ B )
            & ( P4 @ X ) ) ) ) ).

% bex_Un
thf(fact_280_bex__Un,axiom,
    ! [A2: set_a,B: set_a,P4: a > $o] :
      ( ( ? [X: a] :
            ( ( member_a @ X @ ( sup_sup_set_a @ A2 @ B ) )
            & ( P4 @ X ) ) )
      = ( ? [X: a] :
            ( ( member_a @ X @ A2 )
            & ( P4 @ X ) )
        | ? [X: a] :
            ( ( member_a @ X @ B )
            & ( P4 @ X ) ) ) ) ).

% bex_Un
thf(fact_281_Un__Diff,axiom,
    ! [A2: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_282_Un__Diff,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( minus_4060711634664891779od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B ) @ C2 )
      = ( sup_su500200128730103920od_c_a @ ( minus_4060711634664891779od_c_a @ A2 @ C2 ) @ ( minus_4060711634664891779od_c_a @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_283_Un__Diff,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
      = ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C2 ) @ ( minus_minus_set_a @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_284_Un__mono,axiom,
    ! [A2: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,D: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ C2 )
     => ( ( ord_le2998388488506175548od_c_a @ B @ D )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B ) @ ( sup_su500200128730103920od_c_a @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_285_Un__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_286_Un__mono,axiom,
    ! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B @ D )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).

% Un_mono
thf(fact_287_ball__Un,axiom,
    ! [A2: set_nat,B: set_nat,P4: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B ) )
           => ( P4 @ X ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A2 )
           => ( P4 @ X ) )
        & ! [X: nat] :
            ( ( member_nat @ X @ B )
           => ( P4 @ X ) ) ) ) ).

% ball_Un
thf(fact_288_ball__Un,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,P4: list_P6327159017948738492od_c_a > $o] :
      ( ( ! [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
           => ( P4 @ X ) ) )
      = ( ! [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ A2 )
           => ( P4 @ X ) )
        & ! [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ B )
           => ( P4 @ X ) ) ) ) ).

% ball_Un
thf(fact_289_ball__Un,axiom,
    ! [A2: set_a,B: set_a,P4: a > $o] :
      ( ( ! [X: a] :
            ( ( member_a @ X @ ( sup_sup_set_a @ A2 @ B ) )
           => ( P4 @ X ) ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ A2 )
           => ( P4 @ X ) )
        & ! [X: a] :
            ( ( member_a @ X @ B )
           => ( P4 @ X ) ) ) ) ).

% ball_Un
thf(fact_290_Un__assoc,axiom,
    ! [A2: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_291_Un__assoc,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B ) @ C2 )
      = ( sup_su500200128730103920od_c_a @ A2 @ ( sup_su500200128730103920od_c_a @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_292_Un__assoc,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
      = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_293_Un__least,axiom,
    ! [A2: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ C2 )
     => ( ( ord_le2998388488506175548od_c_a @ B @ C2 )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_294_Un__least,axiom,
    ! [A2: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_295_Un__least,axiom,
    ! [A2: set_a,C2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C2 )
     => ( ( ord_less_eq_set_a @ B @ C2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_296_Un__absorb,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_297_Un__absorb,axiom,
    ! [A2: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_298_Un__absorb,axiom,
    ! [A2: set_a] :
      ( ( sup_sup_set_a @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_299_Un__upper1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ A2 @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ).

% Un_upper1
thf(fact_300_Un__upper1,axiom,
    ! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).

% Un_upper1
thf(fact_301_Un__upper1,axiom,
    ! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_upper1
thf(fact_302_Un__upper2,axiom,
    ! [B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ B @ ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ).

% Un_upper2
thf(fact_303_Un__upper2,axiom,
    ! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).

% Un_upper2
thf(fact_304_Un__upper2,axiom,
    ! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_upper2
thf(fact_305_Un__absorb1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_306_Un__absorb1,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( sup_sup_set_nat @ A2 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_307_Un__absorb1,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( sup_sup_set_a @ A2 @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_308_Un__absorb2,axiom,
    ! [B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ A2 )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ B )
        = A2 ) ) ).

% Un_absorb2
thf(fact_309_Un__absorb2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B )
        = A2 ) ) ).

% Un_absorb2
thf(fact_310_Un__absorb2,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B )
        = A2 ) ) ).

% Un_absorb2
thf(fact_311_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_312_Un__commute,axiom,
    ( sup_su500200128730103920od_c_a
    = ( ^ [A3: set_li1159382662694783132od_c_a,B3: set_li1159382662694783132od_c_a] : ( sup_su500200128730103920od_c_a @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_313_Un__commute,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_314_Set_Ofilter__def,axiom,
    ( filter8108757734611162547od_c_a
    = ( ^ [P5: option503927706846959746od_c_a > $o,A3: set_op7949082993927878370od_c_a] :
          ( collec8119457264603017453od_c_a
          @ ^ [A4: option503927706846959746od_c_a] :
              ( ( member2959255366155174571od_c_a @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_315_Set_Ofilter__def,axiom,
    ( filter_option_nat
    = ( ^ [P5: option_nat > $o,A3: set_option_nat] :
          ( collect_option_nat
          @ ^ [A4: option_nat] :
              ( ( member_option_nat @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_316_Set_Ofilter__def,axiom,
    ( filter_option_a
    = ( ^ [P5: option_a > $o,A3: set_option_a] :
          ( collect_option_a
          @ ^ [A4: option_a] :
              ( ( member_option_a @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_317_Set_Ofilter__def,axiom,
    ( filter8200269361223028333od_c_a
    = ( ^ [P5: list_P6327159017948738492od_c_a > $o,A3: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [A4: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_318_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P5: nat > $o,A3: set_nat] :
          ( collect_nat
          @ ^ [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_319_Set_Ofilter__def,axiom,
    ( filter_a
    = ( ^ [P5: a > $o,A3: set_a] :
          ( collect_a
          @ ^ [A4: a] :
              ( ( member_a @ A4 @ A3 )
              & ( P5 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_320_subset__UnE,axiom,
    ! [C2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ C2 @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
     => ~ ! [A5: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ A5 @ A2 )
           => ! [B4: set_li1159382662694783132od_c_a] :
                ( ( ord_le2998388488506175548od_c_a @ B4 @ B )
               => ( C2
                 != ( sup_su500200128730103920od_c_a @ A5 @ B4 ) ) ) ) ) ).

% subset_UnE
thf(fact_321_subset__UnE,axiom,
    ! [C2: set_nat,A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
     => ~ ! [A5: set_nat] :
            ( ( ord_less_eq_set_nat @ A5 @ A2 )
           => ! [B4: set_nat] :
                ( ( ord_less_eq_set_nat @ B4 @ B )
               => ( C2
                 != ( sup_sup_set_nat @ A5 @ B4 ) ) ) ) ) ).

% subset_UnE
thf(fact_322_subset__UnE,axiom,
    ! [C2: set_a,A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
     => ~ ! [A5: set_a] :
            ( ( ord_less_eq_set_a @ A5 @ A2 )
           => ! [B4: set_a] :
                ( ( ord_less_eq_set_a @ B4 @ B )
               => ( C2
                 != ( sup_sup_set_a @ A5 @ B4 ) ) ) ) ) ).

% subset_UnE
thf(fact_323_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_324_subset__Un__eq,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A3: set_li1159382662694783132od_c_a,B3: set_li1159382662694783132od_c_a] :
          ( ( sup_su500200128730103920od_c_a @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_325_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_326_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_327_Diff__partition,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ ( minus_4060711634664891779od_c_a @ B @ A2 ) )
        = B ) ) ).

% Diff_partition
thf(fact_328_Diff__partition,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
        = B ) ) ).

% Diff_partition
thf(fact_329_Diff__partition,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
        = B ) ) ).

% Diff_partition
thf(fact_330_Un__left__absorb,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
      = ( sup_sup_set_nat @ A2 @ B ) ) ).

% Un_left_absorb
thf(fact_331_Un__left__absorb,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ A2 @ ( sup_su500200128730103920od_c_a @ A2 @ B ) )
      = ( sup_su500200128730103920od_c_a @ A2 @ B ) ) ).

% Un_left_absorb
thf(fact_332_Un__left__absorb,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
      = ( sup_sup_set_a @ A2 @ B ) ) ).

% Un_left_absorb
thf(fact_333_Un__left__commute,axiom,
    ! [A2: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) )
      = ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_334_Un__left__commute,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( sup_su500200128730103920od_c_a @ A2 @ ( sup_su500200128730103920od_c_a @ B @ C2 ) )
      = ( sup_su500200128730103920od_c_a @ B @ ( sup_su500200128730103920od_c_a @ A2 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_335_Un__left__commute,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
      = ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_336_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_337_Diff__subset__conv,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( minus_4060711634664891779od_c_a @ A2 @ B ) @ C2 )
      = ( ord_le2998388488506175548od_c_a @ A2 @ ( sup_su500200128730103920od_c_a @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_338_Diff__subset__conv,axiom,
    ! [A2: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C2 )
      = ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_339_Diff__subset__conv,axiom,
    ! [A2: set_a,B: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C2 )
      = ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_340_subset__Diff__insert,axiom,
    ! [A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a,X2: option503927706846959746od_c_a,C2: set_op7949082993927878370od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ A2 @ ( minus_1221259694836641097od_c_a @ B @ ( insert3566037597566277202od_c_a @ X2 @ C2 ) ) )
      = ( ( ord_le7915732482681745026od_c_a @ A2 @ ( minus_1221259694836641097od_c_a @ B @ C2 ) )
        & ~ ( member2959255366155174571od_c_a @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_341_subset__Diff__insert,axiom,
    ! [A2: set_option_nat,B: set_option_nat,X2: option_nat,C2: set_option_nat] :
      ( ( ord_le6937355464348597430on_nat @ A2 @ ( minus_5999362281193037231on_nat @ B @ ( insert_option_nat @ X2 @ C2 ) ) )
      = ( ( ord_le6937355464348597430on_nat @ A2 @ ( minus_5999362281193037231on_nat @ B @ C2 ) )
        & ~ ( member_option_nat @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_342_subset__Diff__insert,axiom,
    ! [A2: set_option_a,B: set_option_a,X2: option_a,C2: set_option_a] :
      ( ( ord_le1955136853071979460tion_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ ( insert_option_a @ X2 @ C2 ) ) )
      = ( ( ord_le1955136853071979460tion_a @ A2 @ ( minus_1574173051537231627tion_a @ B @ C2 ) )
        & ~ ( member_option_a @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_343_subset__Diff__insert,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,X2: list_P6327159017948738492od_c_a,C2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( minus_4060711634664891779od_c_a @ B @ ( insert4789241225314331020od_c_a @ X2 @ C2 ) ) )
      = ( ( ord_le2998388488506175548od_c_a @ A2 @ ( minus_4060711634664891779od_c_a @ B @ C2 ) )
        & ~ ( member7410604586820865893od_c_a @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_344_subset__Diff__insert,axiom,
    ! [A2: set_a,B: set_a,X2: a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C2 ) )
        & ~ ( member_a @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_345_subset__Diff__insert,axiom,
    ! [A2: set_nat,B: set_nat,X2: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C2 ) )
        & ~ ( member_nat @ X2 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_346_bot__set__def,axiom,
    ( bot_bot_set_option_a
    = ( collect_option_a @ bot_bot_option_a_o ) ) ).

% bot_set_def
thf(fact_347_bot__set__def,axiom,
    ( bot_bo6236370880139903240od_c_a
    = ( collec6273869032445462695od_c_a @ bot_bo4941546593110769109_c_a_o ) ) ).

% bot_set_def
thf(fact_348_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_349_bot__set__def,axiom,
    ( bot_bot_set_a
    = ( collect_a @ bot_bot_a_o ) ) ).

% bot_set_def
thf(fact_350_Collect__disj__eq,axiom,
    ! [P4: nat > $o,Q4: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P4 @ X )
            | ( Q4 @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) ) ) ).

% Collect_disj_eq
thf(fact_351_Collect__disj__eq,axiom,
    ! [P4: list_P6327159017948738492od_c_a > $o,Q4: list_P6327159017948738492od_c_a > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X: list_P6327159017948738492od_c_a] :
            ( ( P4 @ X )
            | ( Q4 @ X ) ) )
      = ( sup_su500200128730103920od_c_a @ ( collec6273869032445462695od_c_a @ P4 ) @ ( collec6273869032445462695od_c_a @ Q4 ) ) ) ).

% Collect_disj_eq
thf(fact_352_Collect__disj__eq,axiom,
    ! [P4: a > $o,Q4: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( P4 @ X )
            | ( Q4 @ X ) ) )
      = ( sup_sup_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) ) ) ).

% Collect_disj_eq
thf(fact_353_Un__def,axiom,
    ( sup_su7734108422889136310od_c_a
    = ( ^ [A3: set_op7949082993927878370od_c_a,B3: set_op7949082993927878370od_c_a] :
          ( collec8119457264603017453od_c_a
          @ ^ [X: option503927706846959746od_c_a] :
              ( ( member2959255366155174571od_c_a @ X @ A3 )
              | ( member2959255366155174571od_c_a @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_354_Un__def,axiom,
    ( sup_su7692501479792361346on_nat
    = ( ^ [A3: set_option_nat,B3: set_option_nat] :
          ( collect_option_nat
          @ ^ [X: option_nat] :
              ( ( member_option_nat @ X @ A3 )
              | ( member_option_nat @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_355_Un__def,axiom,
    ( sup_sup_set_option_a
    = ( ^ [A3: set_option_a,B3: set_option_a] :
          ( collect_option_a
          @ ^ [X: option_a] :
              ( ( member_option_a @ X @ A3 )
              | ( member_option_a @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_356_Un__def,axiom,
    ( sup_su500200128730103920od_c_a
    = ( ^ [A3: set_li1159382662694783132od_c_a,B3: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [X: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ X @ A3 )
              | ( member7410604586820865893od_c_a @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_357_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              | ( member_nat @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_358_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A3 )
              | ( member_a @ X @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_359_domI,axiom,
    ! [M: a > option503927706846959746od_c_a,A: a,B2: list_P6327159017948738492od_c_a] :
      ( ( ( M @ A )
        = ( some_l6142909491759833889od_c_a @ B2 ) )
     => ( member_a @ A @ ( dom_a_2450325921413825296od_c_a @ M ) ) ) ).

% domI
thf(fact_360_domI,axiom,
    ! [M: nat > option503927706846959746od_c_a,A: nat,B2: list_P6327159017948738492od_c_a] :
      ( ( ( M @ A )
        = ( some_l6142909491759833889od_c_a @ B2 ) )
     => ( member_nat @ A @ ( dom_na1185376985345782168od_c_a @ M ) ) ) ).

% domI
thf(fact_361_domI,axiom,
    ! [M: a > option_a,A: a,B2: a] :
      ( ( ( M @ A )
        = ( some_a @ B2 ) )
     => ( member_a @ A @ ( dom_a_a @ M ) ) ) ).

% domI
thf(fact_362_domI,axiom,
    ! [M: nat > option_a,A: nat,B2: a] :
      ( ( ( M @ A )
        = ( some_a @ B2 ) )
     => ( member_nat @ A @ ( dom_nat_a @ M ) ) ) ).

% domI
thf(fact_363_domI,axiom,
    ! [M: a > option_nat,A: a,B2: nat] :
      ( ( ( M @ A )
        = ( some_nat @ B2 ) )
     => ( member_a @ A @ ( dom_a_nat @ M ) ) ) ).

% domI
thf(fact_364_domI,axiom,
    ! [M: nat > option_nat,A: nat,B2: nat] :
      ( ( ( M @ A )
        = ( some_nat @ B2 ) )
     => ( member_nat @ A @ ( dom_nat_nat @ M ) ) ) ).

% domI
thf(fact_365_domI,axiom,
    ! [M: option_nat > option_a,A: option_nat,B2: a] :
      ( ( ( M @ A )
        = ( some_a @ B2 ) )
     => ( member_option_nat @ A @ ( dom_option_nat_a @ M ) ) ) ).

% domI
thf(fact_366_domI,axiom,
    ! [M: option_a > option_a,A: option_a,B2: a] :
      ( ( ( M @ A )
        = ( some_a @ B2 ) )
     => ( member_option_a @ A @ ( dom_option_a_a @ M ) ) ) ).

% domI
thf(fact_367_domI,axiom,
    ! [M: option_nat > option_nat,A: option_nat,B2: nat] :
      ( ( ( M @ A )
        = ( some_nat @ B2 ) )
     => ( member_option_nat @ A @ ( dom_option_nat_nat @ M ) ) ) ).

% domI
thf(fact_368_domI,axiom,
    ! [M: option_a > option_nat,A: option_a,B2: nat] :
      ( ( ( M @ A )
        = ( some_nat @ B2 ) )
     => ( member_option_a @ A @ ( dom_option_a_nat @ M ) ) ) ).

% domI
thf(fact_369_domD,axiom,
    ! [A: a,M: a > option503927706846959746od_c_a] :
      ( ( member_a @ A @ ( dom_a_2450325921413825296od_c_a @ M ) )
     => ? [B5: list_P6327159017948738492od_c_a] :
          ( ( M @ A )
          = ( some_l6142909491759833889od_c_a @ B5 ) ) ) ).

% domD
thf(fact_370_domD,axiom,
    ! [A: nat,M: nat > option503927706846959746od_c_a] :
      ( ( member_nat @ A @ ( dom_na1185376985345782168od_c_a @ M ) )
     => ? [B5: list_P6327159017948738492od_c_a] :
          ( ( M @ A )
          = ( some_l6142909491759833889od_c_a @ B5 ) ) ) ).

% domD
thf(fact_371_domD,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_372_domD,axiom,
    ! [A: nat,M: nat > option_a] :
      ( ( member_nat @ A @ ( dom_nat_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_373_domD,axiom,
    ! [A: a,M: a > option_nat] :
      ( ( member_a @ A @ ( dom_a_nat @ M ) )
     => ? [B5: nat] :
          ( ( M @ A )
          = ( some_nat @ B5 ) ) ) ).

% domD
thf(fact_374_domD,axiom,
    ! [A: nat,M: nat > option_nat] :
      ( ( member_nat @ A @ ( dom_nat_nat @ M ) )
     => ? [B5: nat] :
          ( ( M @ A )
          = ( some_nat @ B5 ) ) ) ).

% domD
thf(fact_375_domD,axiom,
    ! [A: option_nat,M: option_nat > option_a] :
      ( ( member_option_nat @ A @ ( dom_option_nat_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_376_domD,axiom,
    ! [A: option_a,M: option_a > option_a] :
      ( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
     => ? [B5: a] :
          ( ( M @ A )
          = ( some_a @ B5 ) ) ) ).

% domD
thf(fact_377_domD,axiom,
    ! [A: option_nat,M: option_nat > option_nat] :
      ( ( member_option_nat @ A @ ( dom_option_nat_nat @ M ) )
     => ? [B5: nat] :
          ( ( M @ A )
          = ( some_nat @ B5 ) ) ) ).

% domD
thf(fact_378_domD,axiom,
    ! [A: option_a,M: option_a > option_nat] :
      ( ( member_option_a @ A @ ( dom_option_a_nat @ M ) )
     => ? [B5: nat] :
          ( ( M @ A )
          = ( some_nat @ B5 ) ) ) ).

% domD
thf(fact_379_domIff,axiom,
    ! [A: a,M: a > option503927706846959746od_c_a] :
      ( ( member_a @ A @ ( dom_a_2450325921413825296od_c_a @ M ) )
      = ( ( M @ A )
       != none_l592525953500355997od_c_a ) ) ).

% domIff
thf(fact_380_domIff,axiom,
    ! [A: nat,M: nat > option503927706846959746od_c_a] :
      ( ( member_nat @ A @ ( dom_na1185376985345782168od_c_a @ M ) )
      = ( ( M @ A )
       != none_l592525953500355997od_c_a ) ) ).

% domIff
thf(fact_381_domIff,axiom,
    ! [A: a,M: a > option_a] :
      ( ( member_a @ A @ ( dom_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_382_domIff,axiom,
    ! [A: nat,M: nat > option_a] :
      ( ( member_nat @ A @ ( dom_nat_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_383_domIff,axiom,
    ! [A: option_nat,M: option_nat > option_a] :
      ( ( member_option_nat @ A @ ( dom_option_nat_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_384_domIff,axiom,
    ! [A: option_a,M: option_a > option_a] :
      ( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_385_domIff,axiom,
    ! [A: list_P6327159017948738492od_c_a,M: list_P6327159017948738492od_c_a > option_a] :
      ( ( member7410604586820865893od_c_a @ A @ ( dom_li8145836122603711636_c_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_386_domIff,axiom,
    ! [A: option_nat,M: option_nat > option503927706846959746od_c_a] :
      ( ( member_option_nat @ A @ ( dom_op7568714055193396424od_c_a @ M ) )
      = ( ( M @ A )
       != none_l592525953500355997od_c_a ) ) ).

% domIff
thf(fact_387_domIff,axiom,
    ! [A: option_a,M: option_a > option503927706846959746od_c_a] :
      ( ( member_option_a @ A @ ( dom_op9021813908095708874od_c_a @ M ) )
      = ( ( M @ A )
       != none_l592525953500355997od_c_a ) ) ).

% domIff
thf(fact_388_domIff,axiom,
    ! [A: option503927706846959746od_c_a,M: option503927706846959746od_c_a > option_a] :
      ( ( member2959255366155174571od_c_a @ A @ ( dom_op5089748838693076174_c_a_a @ M ) )
      = ( ( M @ A )
       != none_a ) ) ).

% domIff
thf(fact_389_insert__Diff__if,axiom,
    ! [X2: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( ( member2959255366155174571od_c_a @ X2 @ B )
       => ( ( minus_1221259694836641097od_c_a @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) @ B )
          = ( minus_1221259694836641097od_c_a @ A2 @ B ) ) )
      & ( ~ ( member2959255366155174571od_c_a @ X2 @ B )
       => ( ( minus_1221259694836641097od_c_a @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) @ B )
          = ( insert3566037597566277202od_c_a @ X2 @ ( minus_1221259694836641097od_c_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_390_insert__Diff__if,axiom,
    ! [X2: option_nat,B: set_option_nat,A2: set_option_nat] :
      ( ( ( member_option_nat @ X2 @ B )
       => ( ( minus_5999362281193037231on_nat @ ( insert_option_nat @ X2 @ A2 ) @ B )
          = ( minus_5999362281193037231on_nat @ A2 @ B ) ) )
      & ( ~ ( member_option_nat @ X2 @ B )
       => ( ( minus_5999362281193037231on_nat @ ( insert_option_nat @ X2 @ A2 ) @ B )
          = ( insert_option_nat @ X2 @ ( minus_5999362281193037231on_nat @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_391_insert__Diff__if,axiom,
    ! [X2: option_a,B: set_option_a,A2: set_option_a] :
      ( ( ( member_option_a @ X2 @ B )
       => ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X2 @ A2 ) @ B )
          = ( minus_1574173051537231627tion_a @ A2 @ B ) ) )
      & ( ~ ( member_option_a @ X2 @ B )
       => ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X2 @ A2 ) @ B )
          = ( insert_option_a @ X2 @ ( minus_1574173051537231627tion_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_392_insert__Diff__if,axiom,
    ! [X2: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ( member7410604586820865893od_c_a @ X2 @ B )
       => ( ( minus_4060711634664891779od_c_a @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) @ B )
          = ( minus_4060711634664891779od_c_a @ A2 @ B ) ) )
      & ( ~ ( member7410604586820865893od_c_a @ X2 @ B )
       => ( ( minus_4060711634664891779od_c_a @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) @ B )
          = ( insert4789241225314331020od_c_a @ X2 @ ( minus_4060711634664891779od_c_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_393_insert__Diff__if,axiom,
    ! [X2: a,B: set_a,A2: set_a] :
      ( ( ( member_a @ X2 @ B )
       => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
          = ( minus_minus_set_a @ A2 @ B ) ) )
      & ( ~ ( member_a @ X2 @ B )
       => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
          = ( insert_a @ X2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_394_insert__Diff__if,axiom,
    ! [X2: nat,B: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X2 @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
          = ( minus_minus_set_nat @ A2 @ B ) ) )
      & ( ~ ( member_nat @ X2 @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
          = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_395_path__begin__state,axiom,
    ! [M2: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ Q @ P2 )
     => ( member_a @ Q @ ( states_a_b_c @ M2 ) ) ) ).

% path_begin_state
thf(fact_396_subset__insertI2,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,B2: list_P6327159017948738492od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B )
     => ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_397_subset__insertI2,axiom,
    ! [A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a,B2: option503927706846959746od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ A2 @ B )
     => ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_398_subset__insertI2,axiom,
    ! [A2: set_option_a,B: set_option_a,B2: option_a] :
      ( ( ord_le1955136853071979460tion_a @ A2 @ B )
     => ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_399_subset__insertI2,axiom,
    ! [A2: set_nat,B: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_400_subset__insertI2,axiom,
    ! [A2: set_a,B: set_a,B2: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_401_subset__insertI,axiom,
    ! [B: set_li1159382662694783132od_c_a,A: list_P6327159017948738492od_c_a] : ( ord_le2998388488506175548od_c_a @ B @ ( insert4789241225314331020od_c_a @ A @ B ) ) ).

% subset_insertI
thf(fact_402_subset__insertI,axiom,
    ! [B: set_op7949082993927878370od_c_a,A: option503927706846959746od_c_a] : ( ord_le7915732482681745026od_c_a @ B @ ( insert3566037597566277202od_c_a @ A @ B ) ) ).

% subset_insertI
thf(fact_403_subset__insertI,axiom,
    ! [B: set_option_a,A: option_a] : ( ord_le1955136853071979460tion_a @ B @ ( insert_option_a @ A @ B ) ) ).

% subset_insertI
thf(fact_404_subset__insertI,axiom,
    ! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).

% subset_insertI
thf(fact_405_subset__insertI,axiom,
    ! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).

% subset_insertI
thf(fact_406_subset__insert,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ~ ( member7410604586820865893od_c_a @ X2 @ A2 )
     => ( ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ B ) )
        = ( ord_le2998388488506175548od_c_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_407_subset__insert,axiom,
    ! [X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ~ ( member2959255366155174571od_c_a @ X2 @ A2 )
     => ( ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ B ) )
        = ( ord_le7915732482681745026od_c_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_408_subset__insert,axiom,
    ! [X2: option_nat,A2: set_option_nat,B: set_option_nat] :
      ( ~ ( member_option_nat @ X2 @ A2 )
     => ( ( ord_le6937355464348597430on_nat @ A2 @ ( insert_option_nat @ X2 @ B ) )
        = ( ord_le6937355464348597430on_nat @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_409_subset__insert,axiom,
    ! [X2: option_a,A2: set_option_a,B: set_option_a] :
      ( ~ ( member_option_a @ X2 @ A2 )
     => ( ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X2 @ B ) )
        = ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_410_subset__insert,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
        = ( ord_less_eq_set_a @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_411_subset__insert,axiom,
    ! [X2: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
        = ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).

% subset_insert
thf(fact_412_insert__mono,axiom,
    ! [C2: set_li1159382662694783132od_c_a,D: set_li1159382662694783132od_c_a,A: list_P6327159017948738492od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ C2 @ D )
     => ( ord_le2998388488506175548od_c_a @ ( insert4789241225314331020od_c_a @ A @ C2 ) @ ( insert4789241225314331020od_c_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_413_insert__mono,axiom,
    ! [C2: set_op7949082993927878370od_c_a,D: set_op7949082993927878370od_c_a,A: option503927706846959746od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ C2 @ D )
     => ( ord_le7915732482681745026od_c_a @ ( insert3566037597566277202od_c_a @ A @ C2 ) @ ( insert3566037597566277202od_c_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_414_insert__mono,axiom,
    ! [C2: set_option_a,D: set_option_a,A: option_a] :
      ( ( ord_le1955136853071979460tion_a @ C2 @ D )
     => ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ C2 ) @ ( insert_option_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_415_insert__mono,axiom,
    ! [C2: set_nat,D: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D ) ) ) ).

% insert_mono
thf(fact_416_insert__mono,axiom,
    ! [C2: set_a,D: set_a,A: a] :
      ( ( ord_less_eq_set_a @ C2 @ D )
     => ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).

% insert_mono
thf(fact_417_dom__minus,axiom,
    ! [F: option503927706846959746od_c_a > option503927706846959746od_c_a,X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( ( F @ X2 )
        = none_l592525953500355997od_c_a )
     => ( ( minus_1221259694836641097od_c_a @ ( dom_op4893529039689921932od_c_a @ F ) @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) )
        = ( minus_1221259694836641097od_c_a @ ( dom_op4893529039689921932od_c_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_418_dom__minus,axiom,
    ! [F: option_a > option503927706846959746od_c_a,X2: option_a,A2: set_option_a] :
      ( ( ( F @ X2 )
        = none_l592525953500355997od_c_a )
     => ( ( minus_1574173051537231627tion_a @ ( dom_op9021813908095708874od_c_a @ F ) @ ( insert_option_a @ X2 @ A2 ) )
        = ( minus_1574173051537231627tion_a @ ( dom_op9021813908095708874od_c_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_419_dom__minus,axiom,
    ! [F: option503927706846959746od_c_a > option_a,X2: option503927706846959746od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( ( F @ X2 )
        = none_a )
     => ( ( minus_1221259694836641097od_c_a @ ( dom_op5089748838693076174_c_a_a @ F ) @ ( insert3566037597566277202od_c_a @ X2 @ A2 ) )
        = ( minus_1221259694836641097od_c_a @ ( dom_op5089748838693076174_c_a_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_420_dom__minus,axiom,
    ! [F: option_a > option_a,X2: option_a,A2: set_option_a] :
      ( ( ( F @ X2 )
        = none_a )
     => ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X2 @ A2 ) )
        = ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_421_dom__minus,axiom,
    ! [F: a > option_a,X2: a,A2: set_a] :
      ( ( ( F @ X2 )
        = none_a )
     => ( ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X2 @ A2 ) )
        = ( minus_minus_set_a @ ( dom_a_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_422_dom__minus,axiom,
    ! [F: nat > option503927706846959746od_c_a,X2: nat,A2: set_nat] :
      ( ( ( F @ X2 )
        = none_l592525953500355997od_c_a )
     => ( ( minus_minus_set_nat @ ( dom_na1185376985345782168od_c_a @ F ) @ ( insert_nat @ X2 @ A2 ) )
        = ( minus_minus_set_nat @ ( dom_na1185376985345782168od_c_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_423_dom__minus,axiom,
    ! [F: nat > option_a,X2: nat,A2: set_nat] :
      ( ( ( F @ X2 )
        = none_a )
     => ( ( minus_minus_set_nat @ ( dom_nat_a @ F ) @ ( insert_nat @ X2 @ A2 ) )
        = ( minus_minus_set_nat @ ( dom_nat_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_424_dom__minus,axiom,
    ! [F: list_P6327159017948738492od_c_a > option503927706846959746od_c_a,X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ( F @ X2 )
        = none_l592525953500355997od_c_a )
     => ( ( minus_4060711634664891779od_c_a @ ( dom_li1621849933323954258od_c_a @ F ) @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) )
        = ( minus_4060711634664891779od_c_a @ ( dom_li1621849933323954258od_c_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_425_dom__minus,axiom,
    ! [F: list_P6327159017948738492od_c_a > option_a,X2: list_P6327159017948738492od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ( F @ X2 )
        = none_a )
     => ( ( minus_4060711634664891779od_c_a @ ( dom_li8145836122603711636_c_a_a @ F ) @ ( insert4789241225314331020od_c_a @ X2 @ A2 ) )
        = ( minus_4060711634664891779od_c_a @ ( dom_li8145836122603711636_c_a_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_426_dom__minus,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,A2: set_a] :
      ( ( ( F @ X2 )
        = none_l592525953500355997od_c_a )
     => ( ( minus_minus_set_a @ ( dom_a_2450325921413825296od_c_a @ F ) @ ( insert_a @ X2 @ A2 ) )
        = ( minus_minus_set_a @ ( dom_a_2450325921413825296od_c_a @ F ) @ A2 ) ) ) ).

% dom_minus
thf(fact_427_Diff__single__insert,axiom,
    ! [A2: set_op7949082993927878370od_c_a,X2: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ ( minus_1221259694836641097od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ bot_bo3379513543401233742od_c_a ) ) @ B )
     => ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_428_Diff__single__insert,axiom,
    ! [A2: set_option_a,X2: option_a,B: set_option_a] :
      ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) @ B )
     => ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_429_Diff__single__insert,axiom,
    ! [A2: set_li1159382662694783132od_c_a,X2: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( minus_4060711634664891779od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ bot_bo6236370880139903240od_c_a ) ) @ B )
     => ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_430_Diff__single__insert,axiom,
    ! [A2: set_nat,X2: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_431_Diff__single__insert,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B )
     => ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) ) ) ).

% Diff_single_insert
thf(fact_432_subset__insert__iff,axiom,
    ! [A2: set_op7949082993927878370od_c_a,X2: option503927706846959746od_c_a,B: set_op7949082993927878370od_c_a] :
      ( ( ord_le7915732482681745026od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ B ) )
      = ( ( ( member2959255366155174571od_c_a @ X2 @ A2 )
         => ( ord_le7915732482681745026od_c_a @ ( minus_1221259694836641097od_c_a @ A2 @ ( insert3566037597566277202od_c_a @ X2 @ bot_bo3379513543401233742od_c_a ) ) @ B ) )
        & ( ~ ( member2959255366155174571od_c_a @ X2 @ A2 )
         => ( ord_le7915732482681745026od_c_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_433_subset__insert__iff,axiom,
    ! [A2: set_option_nat,X2: option_nat,B: set_option_nat] :
      ( ( ord_le6937355464348597430on_nat @ A2 @ ( insert_option_nat @ X2 @ B ) )
      = ( ( ( member_option_nat @ X2 @ A2 )
         => ( ord_le6937355464348597430on_nat @ ( minus_5999362281193037231on_nat @ A2 @ ( insert_option_nat @ X2 @ bot_bo5009843511495006442on_nat ) ) @ B ) )
        & ( ~ ( member_option_nat @ X2 @ A2 )
         => ( ord_le6937355464348597430on_nat @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_434_subset__insert__iff,axiom,
    ! [A2: set_option_a,X2: option_a,B: set_option_a] :
      ( ( ord_le1955136853071979460tion_a @ A2 @ ( insert_option_a @ X2 @ B ) )
      = ( ( ( member_option_a @ X2 @ A2 )
         => ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) @ B ) )
        & ( ~ ( member_option_a @ X2 @ A2 )
         => ( ord_le1955136853071979460tion_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_435_subset__insert__iff,axiom,
    ! [A2: set_li1159382662694783132od_c_a,X2: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ B ) )
      = ( ( ( member7410604586820865893od_c_a @ X2 @ A2 )
         => ( ord_le2998388488506175548od_c_a @ ( minus_4060711634664891779od_c_a @ A2 @ ( insert4789241225314331020od_c_a @ X2 @ bot_bo6236370880139903240od_c_a ) ) @ B ) )
        & ( ~ ( member7410604586820865893od_c_a @ X2 @ A2 )
         => ( ord_le2998388488506175548od_c_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_436_subset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
      = ( ( ( member_nat @ X2 @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) )
        & ( ~ ( member_nat @ X2 @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_437_subset__insert__iff,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
      = ( ( ( member_a @ X2 @ A2 )
         => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
        & ( ~ ( member_a @ X2 @ A2 )
         => ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_438_dom__def,axiom,
    ( dom_na1185376985345782168od_c_a
    = ( ^ [M3: nat > option503927706846959746od_c_a] :
          ( collect_nat
          @ ^ [A4: nat] :
              ( ( M3 @ A4 )
             != none_l592525953500355997od_c_a ) ) ) ) ).

% dom_def
thf(fact_439_dom__def,axiom,
    ( dom_li1621849933323954258od_c_a
    = ( ^ [M3: list_P6327159017948738492od_c_a > option503927706846959746od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [A4: list_P6327159017948738492od_c_a] :
              ( ( M3 @ A4 )
             != none_l592525953500355997od_c_a ) ) ) ) ).

% dom_def
thf(fact_440_dom__def,axiom,
    ( dom_a_a
    = ( ^ [M3: a > option_a] :
          ( collect_a
          @ ^ [A4: a] :
              ( ( M3 @ A4 )
             != none_a ) ) ) ) ).

% dom_def
thf(fact_441_dom__def,axiom,
    ( dom_nat_a
    = ( ^ [M3: nat > option_a] :
          ( collect_nat
          @ ^ [A4: nat] :
              ( ( M3 @ A4 )
             != none_a ) ) ) ) ).

% dom_def
thf(fact_442_dom__def,axiom,
    ( dom_li8145836122603711636_c_a_a
    = ( ^ [M3: list_P6327159017948738492od_c_a > option_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [A4: list_P6327159017948738492od_c_a] :
              ( ( M3 @ A4 )
             != none_a ) ) ) ) ).

% dom_def
thf(fact_443_dom__def,axiom,
    ( dom_a_2450325921413825296od_c_a
    = ( ^ [M3: a > option503927706846959746od_c_a] :
          ( collect_a
          @ ^ [A4: a] :
              ( ( M3 @ A4 )
             != none_l592525953500355997od_c_a ) ) ) ) ).

% dom_def
thf(fact_444_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_445_insert__dom,axiom,
    ! [F: nat > option_nat,X2: nat,Y3: nat] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( insert_nat @ X2 @ ( dom_nat_nat @ F ) )
        = ( dom_nat_nat @ F ) ) ) ).

% insert_dom
thf(fact_446_insert__dom,axiom,
    ! [F: list_P6327159017948738492od_c_a > option_nat,X2: list_P6327159017948738492od_c_a,Y3: nat] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( insert4789241225314331020od_c_a @ X2 @ ( dom_li6393241031938617530_a_nat @ F ) )
        = ( dom_li6393241031938617530_a_nat @ F ) ) ) ).

% insert_dom
thf(fact_447_insert__dom,axiom,
    ! [F: option503927706846959746od_c_a > option_nat,X2: option503927706846959746od_c_a,Y3: nat] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( insert3566037597566277202od_c_a @ X2 @ ( dom_op3000714018489525760_a_nat @ F ) )
        = ( dom_op3000714018489525760_a_nat @ F ) ) ) ).

% insert_dom
thf(fact_448_insert__dom,axiom,
    ! [F: option_a > option_nat,X2: option_a,Y3: nat] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( insert_option_a @ X2 @ ( dom_option_a_nat @ F ) )
        = ( dom_option_a_nat @ F ) ) ) ).

% insert_dom
thf(fact_449_insert__dom,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( F @ X2 )
        = ( some_l6142909491759833889od_c_a @ Y3 ) )
     => ( ( insert_a @ X2 @ ( dom_a_2450325921413825296od_c_a @ F ) )
        = ( dom_a_2450325921413825296od_c_a @ F ) ) ) ).

% insert_dom
thf(fact_450_Diff__insert__absorb,axiom,
    ! [X2: nat,A2: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_451_Diff__insert__absorb,axiom,
    ! [X2: a,A2: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_452_Diff__insert2,axiom,
    ! [A2: set_a,A: a,B: set_a] :
      ( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).

% Diff_insert2
thf(fact_453_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_454_insert__Diff,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_455_Diff__insert,axiom,
    ! [A2: set_a,A: a,B: set_a] :
      ( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).

% Diff_insert
thf(fact_456_subset__singleton__iff,axiom,
    ! [X4: set_a,A: a] :
      ( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
      = ( ( X4 = bot_bot_set_a )
        | ( X4
          = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_457_subset__singletonD,axiom,
    ! [A2: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
     => ( ( A2 = bot_bot_set_a )
        | ( A2
          = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_458_path__target__is__state,axiom,
    ! [M2: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ Q @ P2 )
     => ( member_a @ ( target_a_b_c @ Q @ P2 ) @ ( states_a_b_c @ M2 ) ) ) ).

% path_target_is_state
thf(fact_459_path__nil__elim,axiom,
    ! [M2: fsm_a_b_c,Q: a] :
      ( ( path_a_b_c @ M2 @ Q @ nil_Pr1342775757158464060od_c_a )
     => ( member_a @ Q @ ( states_a_b_c @ M2 ) ) ) ).

% path_nil_elim
thf(fact_460_nil,axiom,
    ! [Q: a,M2: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M2 ) )
     => ( path_a_b_c @ M2 @ Q @ nil_Pr1342775757158464060od_c_a ) ) ).

% nil
thf(fact_461_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_a] :
      ( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_462_Un__empty__right,axiom,
    ! [A2: set_a] :
      ( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
      = A2 ) ).

% Un_empty_right
thf(fact_463_Un__empty__left,axiom,
    ! [B: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ B )
      = B ) ).

% Un_empty_left
thf(fact_464_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_465_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_466_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_467_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_468_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_469_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_470_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_471_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_472_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_473_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_474_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_475_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_476_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_477_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_478_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_479_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_480_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_481_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_482_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_483_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_484_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_485_insert__def,axiom,
    ( insert_a
    = ( ^ [A4: a] :
          ( sup_sup_set_a
          @ ( collect_a
            @ ^ [X: a] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_486_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_487_ex__in__conv,axiom,
    ! [A2: set_a] :
      ( ( ? [X: a] : ( member_a @ X @ A2 ) )
      = ( A2 != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_488_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_489_equals0I,axiom,
    ! [A2: set_a] :
      ( ! [Y4: a] :
          ~ ( member_a @ Y4 @ A2 )
     => ( A2 = bot_bot_set_a ) ) ).

% equals0I
thf(fact_490_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_491_equals0D,axiom,
    ! [A2: set_a,A: a] :
      ( ( A2 = bot_bot_set_a )
     => ~ ( member_a @ A @ A2 ) ) ).

% equals0D
thf(fact_492_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_493_emptyE,axiom,
    ! [A: a] :
      ~ ( member_a @ A @ bot_bot_set_a ) ).

% emptyE
thf(fact_494_mk__disjoint__insert,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ? [B6: set_a] :
          ( ( A2
            = ( insert_a @ A @ B6 ) )
          & ~ ( member_a @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_495_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ? [B6: set_nat] :
          ( ( A2
            = ( insert_nat @ A @ B6 ) )
          & ~ ( member_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_496_insert__commute,axiom,
    ! [X2: a,Y3: a,A2: set_a] :
      ( ( insert_a @ X2 @ ( insert_a @ Y3 @ A2 ) )
      = ( insert_a @ Y3 @ ( insert_a @ X2 @ A2 ) ) ) ).

% insert_commute
thf(fact_497_insert__eq__iff,axiom,
    ! [A: a,A2: set_a,B2: a,B: set_a] :
      ( ~ ( member_a @ A @ A2 )
     => ( ~ ( member_a @ B2 @ B )
       => ( ( ( insert_a @ A @ A2 )
            = ( insert_a @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_a] :
                  ( ( A2
                    = ( insert_a @ B2 @ C3 ) )
                  & ~ ( member_a @ B2 @ C3 )
                  & ( B
                    = ( insert_a @ A @ C3 ) )
                  & ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_498_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B2 @ B )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B2 @ B ) )
          = ( ( ( A = B2 )
             => ( A2 = B ) )
            & ( ( A != B2 )
             => ? [C3: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B2 @ C3 ) )
                  & ~ ( member_nat @ B2 @ C3 )
                  & ( B
                    = ( insert_nat @ A @ C3 ) )
                  & ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_499_insert__absorb,axiom,
    ! [A: a,A2: set_a] :
      ( ( member_a @ A @ A2 )
     => ( ( insert_a @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_500_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_501_insert__ident,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ~ ( member_a @ X2 @ B )
       => ( ( ( insert_a @ X2 @ A2 )
            = ( insert_a @ X2 @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_502_insert__ident,axiom,
    ! [X2: nat,A2: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ~ ( member_nat @ X2 @ B )
       => ( ( ( insert_nat @ X2 @ A2 )
            = ( insert_nat @ X2 @ B ) )
          = ( A2 = B ) ) ) ) ).

% insert_ident
thf(fact_503_Set_Oset__insert,axiom,
    ! [X2: a,A2: set_a] :
      ( ( member_a @ X2 @ A2 )
     => ~ ! [B6: set_a] :
            ( ( A2
              = ( insert_a @ X2 @ B6 ) )
           => ( member_a @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_504_Set_Oset__insert,axiom,
    ! [X2: nat,A2: set_nat] :
      ( ( member_nat @ X2 @ A2 )
     => ~ ! [B6: set_nat] :
            ( ( A2
              = ( insert_nat @ X2 @ B6 ) )
           => ( member_nat @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_505_insertI2,axiom,
    ! [A: a,B: set_a,B2: a] :
      ( ( member_a @ A @ B )
     => ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).

% insertI2
thf(fact_506_insertI2,axiom,
    ! [A: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A @ B )
     => ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_507_insertI1,axiom,
    ! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).

% insertI1
thf(fact_508_insertI1,axiom,
    ! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).

% insertI1
thf(fact_509_insertE,axiom,
    ! [A: a,B2: a,A2: set_a] :
      ( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_a @ A @ A2 ) ) ) ).

% insertE
thf(fact_510_insertE,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
     => ( ( A != B2 )
       => ( member_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_511_size__neq__size__imp__neq,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( size_s3386368156187063848od_c_a @ X2 )
       != ( size_s3386368156187063848od_c_a @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_512_size__neq__size__imp__neq,axiom,
    ! [X2: char,Y3: char] :
      ( ( ( size_size_char @ X2 )
       != ( size_size_char @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_513_Nat_Oex__has__greatest__nat,axiom,
    ! [P4: nat > $o,K: nat,B2: nat] :
      ( ( P4 @ K )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X3: nat] :
            ( ( P4 @ X3 )
            & ! [Y5: nat] :
                ( ( P4 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_514_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_515_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_516_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_517_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_518_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_519_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_520_singleton__Un__iff,axiom,
    ! [X2: a,A2: set_a,B: set_a] :
      ( ( ( insert_a @ X2 @ bot_bot_set_a )
        = ( sup_sup_set_a @ A2 @ B ) )
      = ( ( ( A2 = bot_bot_set_a )
          & ( B
            = ( insert_a @ X2 @ bot_bot_set_a ) ) )
        | ( ( A2
            = ( insert_a @ X2 @ bot_bot_set_a ) )
          & ( B = bot_bot_set_a ) )
        | ( ( A2
            = ( insert_a @ X2 @ bot_bot_set_a ) )
          & ( B
            = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_521_Un__singleton__iff,axiom,
    ! [A2: set_a,B: set_a,X2: a] :
      ( ( ( sup_sup_set_a @ A2 @ B )
        = ( insert_a @ X2 @ bot_bot_set_a ) )
      = ( ( ( A2 = bot_bot_set_a )
          & ( B
            = ( insert_a @ X2 @ bot_bot_set_a ) ) )
        | ( ( A2
            = ( insert_a @ X2 @ bot_bot_set_a ) )
          & ( B = bot_bot_set_a ) )
        | ( ( A2
            = ( insert_a @ X2 @ bot_bot_set_a ) )
          & ( B
            = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_522_insert__is__Un,axiom,
    ( insert_a
    = ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).

% insert_is_Un
thf(fact_523_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_524_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_525_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_526_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_527_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_528_dom__eq__singleton__conv,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a] :
      ( ( ( dom_a_2450325921413825296od_c_a @ F )
        = ( insert_a @ X2 @ bot_bot_set_a ) )
      = ( ? [V: list_P6327159017948738492od_c_a] :
            ( F
            = ( fun_up7010226539926187649od_c_a
              @ ^ [X: a] : none_l592525953500355997od_c_a
              @ X2
              @ ( some_l6142909491759833889od_c_a @ V ) ) ) ) ) ).

% dom_eq_singleton_conv
thf(fact_529_Set_Oempty__def,axiom,
    ( bot_bot_set_a
    = ( collect_a
      @ ^ [X: a] : $false ) ) ).

% Set.empty_def
thf(fact_530_insert__Collect,axiom,
    ! [A: a,P4: a > $o] :
      ( ( insert_a @ A @ ( collect_a @ P4 ) )
      = ( collect_a
        @ ^ [U: a] :
            ( ( U != A )
           => ( P4 @ U ) ) ) ) ).

% insert_Collect
thf(fact_531_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A4 )
              | ( member_nat @ X @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_532_insert__compr,axiom,
    ( insert_a
    = ( ^ [A4: a,B3: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( X = A4 )
              | ( member_a @ X @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_533_singleton__inject,axiom,
    ! [A: a,B2: a] :
      ( ( ( insert_a @ A @ bot_bot_set_a )
        = ( insert_a @ B2 @ bot_bot_set_a ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_534_insert__not__empty,axiom,
    ! [A: a,A2: set_a] :
      ( ( insert_a @ A @ A2 )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_535_doubleton__eq__iff,axiom,
    ! [A: a,B2: a,C: a,D2: a] :
      ( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
        = ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
      = ( ( ( A = C )
          & ( B2 = D2 ) )
        | ( ( A = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_536_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_537_singleton__iff,axiom,
    ! [B2: a,A: a] :
      ( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_538_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_539_singletonD,axiom,
    ! [B2: a,A: a] :
      ( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_540_reaching__paths__up__to__depth_Osimps_I1_J,axiom,
    ! [M2: fsm_a_b_c,Nexts: set_a,Dones: set_a,Assignment: a > option503927706846959746od_c_a] :
      ( ( state_6616341566432195646_a_b_c @ M2 @ Nexts @ Dones @ Assignment @ zero_zero_nat )
      = Assignment ) ).

% reaching_paths_up_to_depth.simps(1)
thf(fact_541_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_542_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_543_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_544_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_545_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_546_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_547_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_548_combine__options__cases,axiom,
    ! [X2: option503927706846959746od_c_a,P4: option503927706846959746od_c_a > option503927706846959746od_c_a > $o,Y3: option503927706846959746od_c_a] :
      ( ( ( X2 = none_l592525953500355997od_c_a )
       => ( P4 @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_l592525953500355997od_c_a )
         => ( P4 @ X2 @ Y3 ) )
       => ( ! [A6: list_P6327159017948738492od_c_a,B5: list_P6327159017948738492od_c_a] :
              ( ( X2
                = ( some_l6142909491759833889od_c_a @ A6 ) )
             => ( ( Y3
                  = ( some_l6142909491759833889od_c_a @ B5 ) )
               => ( P4 @ X2 @ Y3 ) ) )
         => ( P4 @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_549_split__option__all,axiom,
    ( ( ^ [P6: option503927706846959746od_c_a > $o] :
        ! [X5: option503927706846959746od_c_a] : ( P6 @ X5 ) )
    = ( ^ [P5: option503927706846959746od_c_a > $o] :
          ( ( P5 @ none_l592525953500355997od_c_a )
          & ! [X: list_P6327159017948738492od_c_a] : ( P5 @ ( some_l6142909491759833889od_c_a @ X ) ) ) ) ) ).

% split_option_all
thf(fact_550_split__option__ex,axiom,
    ( ( ^ [P6: option503927706846959746od_c_a > $o] :
        ? [X5: option503927706846959746od_c_a] : ( P6 @ X5 ) )
    = ( ^ [P5: option503927706846959746od_c_a > $o] :
          ( ( P5 @ none_l592525953500355997od_c_a )
          | ? [X: list_P6327159017948738492od_c_a] : ( P5 @ ( some_l6142909491759833889od_c_a @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_551_option_Oexhaust,axiom,
    ! [Y3: option503927706846959746od_c_a] :
      ( ( Y3 != none_l592525953500355997od_c_a )
     => ~ ! [X23: list_P6327159017948738492od_c_a] :
            ( Y3
           != ( some_l6142909491759833889od_c_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_552_option_OdiscI,axiom,
    ! [Option: option503927706846959746od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( Option
        = ( some_l6142909491759833889od_c_a @ X22 ) )
     => ( Option != none_l592525953500355997od_c_a ) ) ).

% option.discI
thf(fact_553_option_Odistinct_I1_J,axiom,
    ! [X22: list_P6327159017948738492od_c_a] :
      ( none_l592525953500355997od_c_a
     != ( some_l6142909491759833889od_c_a @ X22 ) ) ).

% option.distinct(1)
thf(fact_554_map__upd__Some__unfold,axiom,
    ! [M: a > option503927706846959746od_c_a,A: a,B2: list_P6327159017948738492od_c_a,X2: a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( fun_up7010226539926187649od_c_a @ M @ A @ ( some_l6142909491759833889od_c_a @ B2 ) @ X2 )
        = ( some_l6142909491759833889od_c_a @ Y3 ) )
      = ( ( ( X2 = A )
          & ( B2 = Y3 ) )
        | ( ( X2 != A )
          & ( ( M @ X2 )
            = ( some_l6142909491759833889od_c_a @ Y3 ) ) ) ) ) ).

% map_upd_Some_unfold
thf(fact_555_map__upd__triv,axiom,
    ! [T: a > option503927706846959746od_c_a,K: a,X2: list_P6327159017948738492od_c_a] :
      ( ( ( T @ K )
        = ( some_l6142909491759833889od_c_a @ X2 ) )
     => ( ( fun_up7010226539926187649od_c_a @ T @ K @ ( some_l6142909491759833889od_c_a @ X2 ) )
        = T ) ) ).

% map_upd_triv
thf(fact_556_map__upd__eqD1,axiom,
    ! [M: a > option503927706846959746od_c_a,A: a,X2: list_P6327159017948738492od_c_a,N: a > option503927706846959746od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( fun_up7010226539926187649od_c_a @ M @ A @ ( some_l6142909491759833889od_c_a @ X2 ) )
        = ( fun_up7010226539926187649od_c_a @ N @ A @ ( some_l6142909491759833889od_c_a @ Y3 ) ) )
     => ( X2 = Y3 ) ) ).

% map_upd_eqD1
thf(fact_557_Collect__conv__if2,axiom,
    ! [P4: a > $o,A: a] :
      ( ( ( P4 @ A )
       => ( ( collect_a
            @ ^ [X: a] :
                ( ( A = X )
                & ( P4 @ X ) ) )
          = ( insert_a @ A @ bot_bot_set_a ) ) )
      & ( ~ ( P4 @ A )
       => ( ( collect_a
            @ ^ [X: a] :
                ( ( A = X )
                & ( P4 @ X ) ) )
          = bot_bot_set_a ) ) ) ).

% Collect_conv_if2
thf(fact_558_Collect__conv__if,axiom,
    ! [P4: a > $o,A: a] :
      ( ( ( P4 @ A )
       => ( ( collect_a
            @ ^ [X: a] :
                ( ( X = A )
                & ( P4 @ X ) ) )
          = ( insert_a @ A @ bot_bot_set_a ) ) )
      & ( ~ ( P4 @ A )
       => ( ( collect_a
            @ ^ [X: a] :
                ( ( X = A )
                & ( P4 @ X ) ) )
          = bot_bot_set_a ) ) ) ).

% Collect_conv_if
thf(fact_559_length__0__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% length_0_conv
thf(fact_560_diff__add__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_561_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_562_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_563_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_564_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_565_add__le__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_566_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_567_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_568_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_569_subsetI,axiom,
    ! [A2: set_a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ( member_a @ X3 @ B ) )
     => ( ord_less_eq_set_a @ A2 @ B ) ) ).

% subsetI
thf(fact_570_subsetI,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A2 @ B ) ) ).

% subsetI
thf(fact_571_DiffI,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ A2 )
     => ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_572_DiffI,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).

% DiffI
thf(fact_573_Diff__iff,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
      = ( ( member_a @ C @ A2 )
        & ~ ( member_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_574_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_575_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_576_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_577_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_578_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_579_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y3 ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_580_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y3 )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_581_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_582_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ B2 @ A ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_583_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_584_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_585_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_586_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_587_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_588_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_589_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_590_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_591_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_592_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_593_DiffE,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ~ ( ( member_a @ C @ A2 )
         => ( member_a @ C @ B ) ) ) ).

% DiffE
thf(fact_594_DiffE,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_595_DiffD1,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ( member_a @ C @ A2 ) ) ).

% DiffD1
thf(fact_596_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_597_DiffD2,axiom,
    ! [C: a,A2: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
     => ~ ( member_a @ C @ B ) ) ).

% DiffD2
thf(fact_598_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
     => ~ ( member_nat @ C @ B ) ) ).

% DiffD2
thf(fact_599_in__mono,axiom,
    ! [A2: set_a,B: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a @ X2 @ A2 )
       => ( member_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_600_in__mono,axiom,
    ! [A2: set_nat,B: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_nat @ X2 @ B ) ) ) ).

% in_mono
thf(fact_601_subsetD,axiom,
    ! [A2: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( member_a @ C @ A2 )
       => ( member_a @ C @ B ) ) ) ).

% subsetD
thf(fact_602_subsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_603_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
        ! [X: a] :
          ( ( member_a @ X @ A3 )
         => ( member_a @ X @ B3 ) ) ) ) ).

% subset_eq
thf(fact_604_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A3 )
         => ( member_nat @ X @ B3 ) ) ) ) ).

% subset_eq
thf(fact_605_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_606_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A3 )
         => ( member_nat @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_607_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ~ ( member_nat @ X @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_608_set__diff__eq,axiom,
    ( minus_minus_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A3 )
              & ~ ( member_a @ X @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_609_Collect__mono,axiom,
    ! [P4: a > $o,Q4: a > $o] :
      ( ! [X3: a] :
          ( ( P4 @ X3 )
         => ( Q4 @ X3 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) ) ) ).

% Collect_mono
thf(fact_610_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A3 )
            @ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_611_minus__set__def,axiom,
    ( minus_minus_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ( minus_minus_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 )
            @ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_612_Collect__subset,axiom,
    ! [A2: set_nat,P4: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P4 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_613_Collect__subset,axiom,
    ! [A2: set_a,P4: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ A2 )
            & ( P4 @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_614_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X: a] : ( member_a @ X @ A3 )
          @ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_615_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A3 )
          @ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_616_Collect__mono__iff,axiom,
    ! [P4: a > $o,Q4: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) )
      = ( ! [X: a] :
            ( ( P4 @ X )
           => ( Q4 @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_617_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A3 )
            @ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_618_sup__set__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( collect_a
          @ ( sup_sup_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 )
            @ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_619_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_620_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_621_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_622_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_623_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_624_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B7: nat] : ( plus_plus_nat @ B7 @ A4 ) ) ) ).

% add.commute
thf(fact_625_add_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_626_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_627_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_628_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_629_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_630_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_631_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_P6327159017948738492od_c_a] :
      ( ( size_s3386368156187063848od_c_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_632_neq__if__length__neq,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
       != ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_633_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_634_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_635_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_636_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_637_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_638_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_639_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_640_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_641_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ~ ! [C4: nat] :
            ( B2
           != ( plus_plus_nat @ A @ C4 ) ) ) ).

% less_eqE
thf(fact_642_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_643_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B7: nat] :
        ? [C5: nat] :
          ( B7
          = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_644_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_645_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_646_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_647_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_648_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_649_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_650_diff__diff__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_651_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_652_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_653_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_654_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_655_add__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_656_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_657_add__increasing2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_658_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_659_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_660_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_661_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_662_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_663_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% diff_add
thf(fact_664_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_665_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_666_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_667_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_668_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_669_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_670_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_672_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_673_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_674_list_Osize_I3_J,axiom,
    ( ( size_s3386368156187063848od_c_a @ nil_Pr1342775757158464060od_c_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_675_reachable__def,axiom,
    ( reachable_a_b_c
    = ( ^ [M4: fsm_a_b_c,Q3: a] :
        ? [P3: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ M4 @ ( initial_a_b_c @ M4 ) @ P3 )
          & ( ( target_a_b_c @ ( initial_a_b_c @ M4 ) @ P3 )
            = Q3 ) ) ) ) ).

% reachable_def
thf(fact_676_sup__bot__left,axiom,
    ! [X2: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_677_sup__bot__right,axiom,
    ! [X2: set_a] :
      ( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
      = X2 ) ).

% sup_bot_right
thf(fact_678_bot__eq__sup__iff,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ X2 @ Y3 ) )
      = ( ( X2 = bot_bot_set_a )
        & ( Y3 = bot_bot_set_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_679_sup__eq__bot__iff,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( ( sup_sup_set_a @ X2 @ Y3 )
        = bot_bot_set_a )
      = ( ( X2 = bot_bot_set_a )
        & ( Y3 = bot_bot_set_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_680_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( ( sup_sup_set_a @ A @ B2 )
        = bot_bot_set_a )
      = ( ( A = bot_bot_set_a )
        & ( B2 = bot_bot_set_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_681_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_682_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_683_sup_Oidem,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ A )
      = A ) ).

% sup.idem
thf(fact_684_sup__idem,axiom,
    ! [X2: set_a] :
      ( ( sup_sup_set_a @ X2 @ X2 )
      = X2 ) ).

% sup_idem
thf(fact_685_sup_Oleft__idem,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
      = ( sup_sup_set_a @ A @ B2 ) ) ).

% sup.left_idem
thf(fact_686_sup__left__idem,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
      = ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% sup_left_idem
thf(fact_687_sup_Oright__idem,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
      = ( sup_sup_set_a @ A @ B2 ) ) ).

% sup.right_idem
thf(fact_688_sup_Obounded__iff,axiom,
    ! [B2: set_a,C: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
      = ( ( ord_less_eq_set_a @ B2 @ A )
        & ( ord_less_eq_set_a @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_689_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_690_le__sup__iff,axiom,
    ! [X2: set_a,Y3: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_set_a @ X2 @ Z2 )
        & ( ord_less_eq_set_a @ Y3 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_691_le__sup__iff,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z2 )
      = ( ( ord_less_eq_nat @ X2 @ Z2 )
        & ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_692_sup__bot_Oright__neutral,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ bot_bot_set_a )
      = A ) ).

% sup_bot.right_neutral
thf(fact_693_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( bot_bot_set_a
        = ( sup_sup_set_a @ A @ B2 ) )
      = ( ( A = bot_bot_set_a )
        & ( B2 = bot_bot_set_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_694_sup__bot_Oleft__neutral,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_695_order__antisym__conv,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_696_linorder__le__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_le_cases
thf(fact_697_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_698_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_699_linorder__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_linear
thf(fact_700_order__eq__refl,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_701_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_702_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_703_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A4: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A4 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_704_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_705_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_706_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_707_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A4: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_708_linorder__wlog,axiom,
    ! [P4: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A6 @ B5 )
         => ( P4 @ A6 @ B5 ) )
     => ( ! [A6: nat,B5: nat] :
            ( ( P4 @ B5 @ A6 )
           => ( P4 @ A6 @ B5 ) )
       => ( P4 @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_709_order__trans,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_710_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_711_order__antisym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_712_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_713_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_714_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_715_le__cases3,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_716_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_717_inf__sup__aci_I8_J,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) )
      = ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% inf_sup_aci(8)
thf(fact_718_inf__sup__aci_I7_J,axiom,
    ! [X2: set_a,Y3: set_a,Z2: set_a] :
      ( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
      = ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_719_inf__sup__aci_I6_J,axiom,
    ! [X2: set_a,Y3: set_a,Z2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
      = ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_720_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_a
    = ( ^ [X: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_721_sup_Oassoc,axiom,
    ! [A: set_a,B2: set_a,C: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C )
      = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_722_sup__assoc,axiom,
    ! [X2: set_a,Y3: set_a,Z2: set_a] :
      ( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ Z2 )
      = ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_723_sup_Ocommute,axiom,
    ( sup_sup_set_a
    = ( ^ [A4: set_a,B7: set_a] : ( sup_sup_set_a @ B7 @ A4 ) ) ) ).

% sup.commute
thf(fact_724_sup__commute,axiom,
    ( sup_sup_set_a
    = ( ^ [X: set_a,Y2: set_a] : ( sup_sup_set_a @ Y2 @ X ) ) ) ).

% sup_commute
thf(fact_725_sup_Oleft__commute,axiom,
    ! [B2: set_a,A: set_a,C: set_a] :
      ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C ) )
      = ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_726_sup__left__commute,axiom,
    ! [X2: set_a,Y3: set_a,Z2: set_a] :
      ( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
      = ( sup_sup_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_727_bot_Oextremum__uniqueI,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
     => ( A = bot_bot_set_a ) ) ).

% bot.extremum_uniqueI
thf(fact_728_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_729_bot_Oextremum__unique,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% bot.extremum_unique
thf(fact_730_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_731_bot_Oextremum,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% bot.extremum
thf(fact_732_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_733_sup_OcoboundedI2,axiom,
    ! [C: set_a,B2: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ C @ B2 )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_734_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_735_sup_OcoboundedI1,axiom,
    ! [C: set_a,A: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C @ A )
     => ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_736_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_737_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B7: set_a] :
          ( ( sup_sup_set_a @ A4 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_738_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B7: nat] :
          ( ( sup_sup_nat @ A4 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_739_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B7: set_a,A4: set_a] :
          ( ( sup_sup_set_a @ A4 @ B7 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_740_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B7 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_741_sup_Ocobounded2,axiom,
    ! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_742_sup_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).

% sup.cobounded2
thf(fact_743_sup_Ocobounded1,axiom,
    ! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_744_sup_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).

% sup.cobounded1
thf(fact_745_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B7: set_a,A4: set_a] :
          ( A4
          = ( sup_sup_set_a @ A4 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_746_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_747_sup_OboundedI,axiom,
    ! [B2: set_a,A: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A )
     => ( ( ord_less_eq_set_a @ C @ A )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_748_sup_OboundedI,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_749_sup_OboundedE,axiom,
    ! [B2: set_a,C: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_set_a @ B2 @ A )
         => ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_750_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B2 @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_751_sup__absorb2,axiom,
    ! [X2: set_a,Y3: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y3 )
     => ( ( sup_sup_set_a @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_752_sup__absorb2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( sup_sup_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_753_sup__absorb1,axiom,
    ! [Y3: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y3 @ X2 )
     => ( ( sup_sup_set_a @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_754_sup__absorb1,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( sup_sup_nat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_755_sup_Oabsorb2,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B2 )
     => ( ( sup_sup_set_a @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_756_sup_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( sup_sup_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_757_sup_Oabsorb1,axiom,
    ! [B2: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A )
     => ( ( sup_sup_set_a @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_758_sup_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( sup_sup_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb1
thf(fact_759_sup__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y3: set_a] :
      ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_a,Y4: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ Y4 @ X3 )
             => ( ( ord_less_eq_set_a @ Z3 @ X3 )
               => ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_a @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_760_sup__unique,axiom,
    ! [F: nat > nat > nat,X2: nat,Y3: nat] :
      ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X3 )
             => ( ( ord_less_eq_nat @ Z3 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_761_sup_OorderI,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( A
        = ( sup_sup_set_a @ A @ B2 ) )
     => ( ord_less_eq_set_a @ B2 @ A ) ) ).

% sup.orderI
thf(fact_762_sup_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% sup.orderI
thf(fact_763_sup_OorderE,axiom,
    ! [B2: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A )
     => ( A
        = ( sup_sup_set_a @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_764_sup_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( A
        = ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.orderE
thf(fact_765_le__iff__sup,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X: set_a,Y2: set_a] :
          ( ( sup_sup_set_a @ X @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_766_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( sup_sup_nat @ X @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_767_sup__least,axiom,
    ! [Y3: set_a,X2: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ Y3 @ X2 )
     => ( ( ord_less_eq_set_a @ Z2 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z2 ) @ X2 ) ) ) ).

% sup_least
thf(fact_768_sup__least,axiom,
    ! [Y3: nat,X2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( ord_less_eq_nat @ Z2 @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z2 ) @ X2 ) ) ) ).

% sup_least
thf(fact_769_sup__mono,axiom,
    ! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A @ C )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_770_sup__mono,axiom,
    ! [A: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_771_sup_Omono,axiom,
    ! [C: set_a,A: set_a,D2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C @ A )
     => ( ( ord_less_eq_set_a @ D2 @ B2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_772_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).

% sup.mono
thf(fact_773_le__supI2,axiom,
    ! [X2: set_a,B2: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ B2 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_774_le__supI2,axiom,
    ! [X2: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ X2 @ B2 )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% le_supI2
thf(fact_775_le__supI1,axiom,
    ! [X2: set_a,A: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_776_le__supI1,axiom,
    ! [X2: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X2 @ A )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% le_supI1
thf(fact_777_sup__ge2,axiom,
    ! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_778_sup__ge2,axiom,
    ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_779_sup__ge1,axiom,
    ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_780_sup__ge1,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_781_le__supI,axiom,
    ! [A: set_a,X2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A @ X2 )
     => ( ( ord_less_eq_set_a @ B2 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_782_le__supI,axiom,
    ! [A: nat,X2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ X2 )
     => ( ( ord_less_eq_nat @ B2 @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_783_le__supE,axiom,
    ! [A: set_a,B2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_set_a @ A @ X2 )
         => ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_784_le__supE,axiom,
    ! [A: nat,B2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_nat @ A @ X2 )
         => ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_785_inf__sup__ord_I3_J,axiom,
    ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_786_inf__sup__ord_I3_J,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_787_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_788_inf__sup__ord_I4_J,axiom,
    ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_789_paths__up__to__length__set,axiom,
    ! [Q: a,M2: fsm_a_b_c,K: nat] :
      ( ( member_a @ Q @ ( states_a_b_c @ M2 ) )
     => ( ( paths_5655032219139660648_a_b_c @ M2 @ Q @ K )
        = ( collec6273869032445462695od_c_a
          @ ^ [P3: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ Q @ P3 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ K ) ) ) ) ) ).

% paths_up_to_length_set
thf(fact_790_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_791_sup__Un__eq,axiom,
    ! [R: set_a,S: set_a] :
      ( ( sup_sup_a_o
        @ ^ [X: a] : ( member_a @ X @ R )
        @ ^ [X: a] : ( member_a @ X @ S ) )
      = ( ^ [X: a] : ( member_a @ X @ ( sup_sup_set_a @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_792_reachable__k__0__initial,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( reachable_k_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ zero_zero_nat )
      = ( insert_a @ ( initial_a_b_c @ M2 ) @ bot_bot_set_a ) ) ).

% reachable_k_0_initial
thf(fact_793_restrict__to__reachable__states,axiom,
    ( restri9132545300209641082_a_b_c
    = ( ^ [M4: fsm_a_b_c] :
          ( filter_states_a_b_c @ M4
          @ ^ [Q3: a] :
              ( ( state_6616341566432195646_a_b_c @ M4 @ ( insert_a @ ( initial_a_b_c @ M4 ) @ bot_bot_set_a ) @ bot_bot_set_a
                @ ( fun_up7010226539926187649od_c_a
                  @ ^ [X: a] : none_l592525953500355997od_c_a
                  @ ( initial_a_b_c @ M4 )
                  @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
                @ ( minus_minus_nat @ ( size_a_b_c @ M4 ) @ one_one_nat )
                @ Q3 )
             != none_l592525953500355997od_c_a ) ) ) ) ).

% restrict_to_reachable_states
thf(fact_794_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( initial_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) )
      = ( initial_a_b_c @ M2 ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_795_filter__states__simps_I1_J,axiom,
    ! [P4: a > $o,M2: fsm_a_b_c] :
      ( ( P4 @ ( initial_a_b_c @ M2 ) )
     => ( ( initial_a_b_c @ ( filter_states_a_b_c @ M2 @ P4 ) )
        = ( initial_a_b_c @ M2 ) ) ) ).

% filter_states_simps(1)
thf(fact_796_filter__states__simps_I2_J,axiom,
    ! [P4: a > $o,M2: fsm_a_b_c] :
      ( ( P4 @ ( initial_a_b_c @ M2 ) )
     => ( ( states_a_b_c @ ( filter_states_a_b_c @ M2 @ P4 ) )
        = ( filter_a @ P4 @ ( states_a_b_c @ M2 ) ) ) ) ).

% filter_states_simps(2)
thf(fact_797_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_798_bot__empty__eq,axiom,
    ( bot_bot_a_o
    = ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).

% bot_empty_eq
thf(fact_799_pred__subset__eq,axiom,
    ! [R: set_a,S: set_a] :
      ( ( ord_less_eq_a_o
        @ ^ [X: a] : ( member_a @ X @ R )
        @ ^ [X: a] : ( member_a @ X @ S ) )
      = ( ord_less_eq_set_a @ R @ S ) ) ).

% pred_subset_eq
thf(fact_800_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_801_fun__upd__upd,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: option503927706846959746od_c_a,Z2: option503927706846959746od_c_a] :
      ( ( fun_up7010226539926187649od_c_a @ ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 ) @ X2 @ Z2 )
      = ( fun_up7010226539926187649od_c_a @ F @ X2 @ Z2 ) ) ).

% fun_upd_upd
thf(fact_802_fun__upd__triv,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a] :
      ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ ( F @ X2 ) )
      = F ) ).

% fun_upd_triv
thf(fact_803_fun__upd__apply,axiom,
    ( fun_up7010226539926187649od_c_a
    = ( ^ [F2: a > option503927706846959746od_c_a,X: a,Y2: option503927706846959746od_c_a,Z4: a] : ( if_opt5277955309491978056od_c_a @ ( Z4 = X ) @ Y2 @ ( F2 @ Z4 ) ) ) ) ).

% fun_upd_apply
thf(fact_804_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C5: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_805_fun__upd__idem__iff,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: option503927706846959746od_c_a] :
      ( ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 )
        = F )
      = ( ( F @ X2 )
        = Y3 ) ) ).

% fun_upd_idem_iff
thf(fact_806_fun__upd__twist,axiom,
    ! [A: a,C: a,M: a > option503927706846959746od_c_a,B2: option503927706846959746od_c_a,D2: option503927706846959746od_c_a] :
      ( ( A != C )
     => ( ( fun_up7010226539926187649od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ A @ B2 ) @ C @ D2 )
        = ( fun_up7010226539926187649od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ C @ D2 ) @ A @ B2 ) ) ) ).

% fun_upd_twist
thf(fact_807_fun__upd__other,axiom,
    ! [Z2: a,X2: a,F: a > option503927706846959746od_c_a,Y3: option503927706846959746od_c_a] :
      ( ( Z2 != X2 )
     => ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 @ Z2 )
        = ( F @ Z2 ) ) ) ).

% fun_upd_other
thf(fact_808_fun__upd__same,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: option503927706846959746od_c_a] :
      ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 @ X2 )
      = Y3 ) ).

% fun_upd_same
thf(fact_809_fun__upd__idem,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: option503927706846959746od_c_a] :
      ( ( ( F @ X2 )
        = Y3 )
     => ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 )
        = F ) ) ).

% fun_upd_idem
thf(fact_810_fun__upd__eqD,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a,Y3: option503927706846959746od_c_a,G: a > option503927706846959746od_c_a,Z2: option503927706846959746od_c_a] :
      ( ( ( fun_up7010226539926187649od_c_a @ F @ X2 @ Y3 )
        = ( fun_up7010226539926187649od_c_a @ G @ X2 @ Z2 ) )
     => ( Y3 = Z2 ) ) ).

% fun_upd_eqD
thf(fact_811_fun__upd__def,axiom,
    ( fun_up7010226539926187649od_c_a
    = ( ^ [F2: a > option503927706846959746od_c_a,A4: a,B7: option503927706846959746od_c_a,X: a] : ( if_opt5277955309491978056od_c_a @ ( X = A4 ) @ B7 @ ( F2 @ X ) ) ) ) ).

% fun_upd_def
thf(fact_812_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C5: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_813_Collect__empty__eq__bot,axiom,
    ! [P4: a > $o] :
      ( ( ( collect_a @ P4 )
        = bot_bot_set_a )
      = ( P4 = bot_bot_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_814_c1,axiom,
    ( ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a )
    = ( collect_a
      @ ^ [Q3: a] :
          ( ? [P3: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
              & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
                = Q3 )
              & ( ( size_s3386368156187063848od_c_a @ P3 )
                = zero_zero_nat ) )
          & ~ ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
                  = Q3 )
                & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ zero_zero_nat ) ) ) ) ) ).

% c1
thf(fact_815_is__singletonI,axiom,
    ! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).

% is_singletonI
thf(fact_816_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_817_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_818_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_819_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_820_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_821_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_822_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_823_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_824_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_825_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_826_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_827_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_828_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_829_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_830_c2,axiom,
    ( bot_bot_set_a
    = ( collect_a
      @ ^ [Q3: a] :
        ? [P3: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
          & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
            = Q3 )
          & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ zero_zero_nat ) ) ) ) ).

% c2
thf(fact_831_length__greater__0__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3386368156187063848od_c_a @ Xs ) )
      = ( Xs != nil_Pr1342775757158464060od_c_a ) ) ).

% length_greater_0_conv
thf(fact_832_length__induct,axiom,
    ! [P4: list_P6327159017948738492od_c_a > $o,Xs: list_P6327159017948738492od_c_a] :
      ( ! [Xs2: list_P6327159017948738492od_c_a] :
          ( ! [Ys2: list_P6327159017948738492od_c_a] :
              ( ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ Ys2 ) @ ( size_s3386368156187063848od_c_a @ Xs2 ) )
             => ( P4 @ Ys2 ) )
         => ( P4 @ Xs2 ) )
     => ( P4 @ Xs ) ) ).

% length_induct
thf(fact_833_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_834_less__imp__neq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_835_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_836_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_837_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_838_less__induct,axiom,
    ! [P4: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P4 @ Y5 ) )
         => ( P4 @ X3 ) )
     => ( P4 @ A ) ) ).

% less_induct
thf(fact_839_antisym__conv3,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_840_linorder__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_841_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_842_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_843_exists__least__iff,axiom,
    ( ( ^ [P6: nat > $o] :
        ? [X5: nat] : ( P6 @ X5 ) )
    = ( ^ [P5: nat > $o] :
        ? [N2: nat] :
          ( ( P5 @ N2 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ~ ( P5 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_844_linorder__less__wlog,axiom,
    ! [P4: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( ord_less_nat @ A6 @ B5 )
         => ( P4 @ A6 @ B5 ) )
     => ( ! [A6: nat] : ( P4 @ A6 @ A6 )
       => ( ! [A6: nat,B5: nat] :
              ( ( P4 @ B5 @ A6 )
             => ( P4 @ A6 @ B5 ) )
         => ( P4 @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_845_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_846_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_847_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_848_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_849_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_850_linorder__neqE,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_851_order__less__asym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_852_linorder__neq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
      = ( ( ord_less_nat @ X2 @ Y3 )
        | ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_853_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_854_order__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_855_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_856_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_857_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_858_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_859_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_860_order__less__not__sym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_861_order__less__imp__triv,axiom,
    ! [X2: nat,Y3: nat,P4: $o] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X2 )
       => P4 ) ) ).

% order_less_imp_triv
thf(fact_862_linorder__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_863_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_864_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_865_order__less__imp__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_866_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_867_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_868_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_869_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_870_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_871_nat__less__induct,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
             => ( P4 @ M5 ) )
         => ( P4 @ N3 ) )
     => ( P4 @ N ) ) ).

% nat_less_induct
thf(fact_872_infinite__descent,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P4 @ N3 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
              & ~ ( P4 @ M5 ) ) )
     => ( P4 @ N ) ) ).

% infinite_descent
thf(fact_873_linorder__neqE__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_874_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_875_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_876_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_877_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_878_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_879_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_880_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_881_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_882_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_883_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_884_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_885_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_886_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_887_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
          | ( M3 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_888_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_889_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M3 @ N2 )
          & ( M3 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_890_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_891_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_892_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_893_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_894_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_895_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_896_infinite__descent0,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ( P4 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P4 @ N3 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N3 )
                  & ~ ( P4 @ M5 ) ) ) )
       => ( P4 @ N ) ) ) ).

% infinite_descent0
thf(fact_897_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_a,B2: set_a,A: set_a] :
      ( ( ord_less_set_a @ C @ B2 )
     => ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_898_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_899_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_a,A: set_a,B2: set_a] :
      ( ( ord_less_set_a @ C @ A )
     => ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_900_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_901_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_a
    = ( ^ [B7: set_a,A4: set_a] :
          ( ( A4
            = ( sup_sup_set_a @ A4 @ B7 ) )
          & ( A4 != B7 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_902_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B7 ) )
          & ( A4 != B7 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_903_sup_Ostrict__boundedE,axiom,
    ! [B2: set_a,C: set_a,A: set_a] :
      ( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
     => ~ ( ( ord_less_set_a @ B2 @ A )
         => ~ ( ord_less_set_a @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_904_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_nat @ B2 @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_905_sup_Oabsorb4,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( ord_less_set_a @ A @ B2 )
     => ( ( sup_sup_set_a @ A @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_906_sup_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( sup_sup_nat @ A @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_907_sup_Oabsorb3,axiom,
    ! [B2: set_a,A: set_a] :
      ( ( ord_less_set_a @ B2 @ A )
     => ( ( sup_sup_set_a @ A @ B2 )
        = A ) ) ).

% sup.absorb3
thf(fact_908_sup_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( sup_sup_nat @ A @ B2 )
        = A ) ) ).

% sup.absorb3
thf(fact_909_less__supI2,axiom,
    ! [X2: set_a,B2: set_a,A: set_a] :
      ( ( ord_less_set_a @ X2 @ B2 )
     => ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% less_supI2
thf(fact_910_less__supI2,axiom,
    ! [X2: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ X2 @ B2 )
     => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI2
thf(fact_911_less__supI1,axiom,
    ! [X2: set_a,A: set_a,B2: set_a] :
      ( ( ord_less_set_a @ X2 @ A )
     => ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).

% less_supI1
thf(fact_912_less__supI1,axiom,
    ! [X2: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ X2 @ A )
     => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).

% less_supI1
thf(fact_913_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_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(5)
thf(fact_914_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: 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(2)
thf(fact_915_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_916_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_917_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_918_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_919_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_920_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_921_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_922_bot_Oextremum__strict,axiom,
    ! [A: set_a] :
      ~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).

% bot.extremum_strict
thf(fact_923_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_924_bot_Onot__eq__extremum,axiom,
    ! [A: set_a] :
      ( ( A != bot_bot_set_a )
      = ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).

% bot.not_eq_extremum
thf(fact_925_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_926_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_927_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_928_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_929_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_930_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_931_leD,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).

% leD
thf(fact_932_leI,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% leI
thf(fact_933_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_934_antisym__conv1,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_935_antisym__conv2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_936_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_937_not__le__imp__less,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ord_less_nat @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_938_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B7: nat] :
          ( ( ord_less_nat @ A4 @ B7 )
          | ( A4 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_939_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A4 @ B7 )
          & ( A4 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_940_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_941_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_942_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A4 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_943_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A4: nat] :
          ( ( ord_less_nat @ B7 @ A4 )
          | ( A4 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_944_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B7 @ A4 )
          & ( A4 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_945_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_946_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_947_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B7 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_948_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_949_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_950_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_nat @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% order_le_less
thf(fact_951_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ( X != Y2 ) ) ) ) ).

% order_less_le
thf(fact_952_linorder__not__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_953_linorder__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_954_order__less__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_955_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_956_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_957_order__le__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_958_order__less__le__trans,axiom,
    ! [X2: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_959_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_960_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_961_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_962_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_963_linorder__le__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_964_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_965_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_966_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_967_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_968_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_969_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_970_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_971_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_972_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_973_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ! [C4: nat] :
            ( ( B2
              = ( plus_plus_nat @ A @ C4 ) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_974_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_975_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_976_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_977_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_978_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_979_ex__least__nat__le,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ( P4 @ N )
     => ( ~ ( P4 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K3 )
               => ~ ( P4 @ I3 ) )
            & ( P4 @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_980_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_981_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_982_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M6: nat,N3: nat] :
          ( ( ord_less_nat @ M6 @ N3 )
         => ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_983_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_984_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_985_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_986_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_987_fsm__size__Suc,axiom,
    ! [M2: fsm_a_b_c] : ( ord_less_nat @ zero_zero_nat @ ( size_a_b_c @ M2 ) ) ).

% fsm_size_Suc
thf(fact_988_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ( member_nat @ Y4 @ A2 )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_989_is__singletonI_H,axiom,
    ! [A2: set_a] :
      ( ( A2 != bot_bot_set_a )
     => ( ! [X3: a,Y4: a] :
            ( ( member_a @ X3 @ A2 )
           => ( ( member_a @ Y4 @ A2 )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_a @ A2 ) ) ) ).

% is_singletonI'
thf(fact_990_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_991_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_992_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_993_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_994_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_995_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_996_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_997_nat__diff__split,axiom,
    ! [P4: nat > $o,A: nat,B2: nat] :
      ( ( P4 @ ( minus_minus_nat @ A @ B2 ) )
      = ( ( ( ord_less_nat @ A @ B2 )
         => ( P4 @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B2 @ D3 ) )
           => ( P4 @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_998_nat__diff__split__asm,axiom,
    ! [P4: nat > $o,A: nat,B2: nat] :
      ( ( P4 @ ( minus_minus_nat @ A @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B2 )
              & ~ ( P4 @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B2 @ D3 ) )
                & ~ ( P4 @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_999_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_1000_is__singletonE,axiom,
    ! [A2: set_a] :
      ( ( is_singleton_a @ A2 )
     => ~ ! [X3: a] :
            ( A2
           != ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).

% is_singletonE
thf(fact_1001_is__singleton__def,axiom,
    ( is_singleton_a
    = ( ^ [A3: set_a] :
        ? [X: a] :
          ( A3
          = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% is_singleton_def
thf(fact_1002_reaching__paths__up__to__depth__set_I3_J,axiom,
    ! [Nexts: set_a,M2: fsm_a_b_c,N: nat,Dones: set_a,Assignment: a > option503927706846959746od_c_a,Q: a,K: nat] :
      ( ( Nexts
        = ( collect_a
          @ ^ [Q3: a] :
              ( ? [P3: list_P6327159017948738492od_c_a] :
                  ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                    = Q3 )
                  & ( ( size_s3386368156187063848od_c_a @ P3 )
                    = N ) )
              & ~ ? [P3: list_P6327159017948738492od_c_a] :
                    ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                    & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                      = Q3 )
                    & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) ) )
     => ( ( Dones
          = ( collect_a
            @ ^ [Q3: a] :
              ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                  = Q3 )
                & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
       => ( ! [Q2: a] :
              ( ( ( Assignment @ Q2 )
                = none_l592525953500355997od_c_a )
              = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
                      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                      & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                        = Q2 )
                      & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
         => ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
                ( ( ( Assignment @ Q2 )
                  = ( some_l6142909491759833889od_c_a @ P ) )
               => ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P )
                    = Q2 )
                  & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P ) @ N ) ) )
           => ( ( ( dom_a_2450325921413825296od_c_a @ Assignment )
                = ( sup_sup_set_a @ Nexts @ Dones ) )
             => ( ( member_a @ Q @ ( sup_sup_set_a @ Nexts @ Dones ) )
               => ( ( state_6616341566432195646_a_b_c @ M2 @ Nexts @ Dones @ Assignment @ K @ Q )
                  = ( Assignment @ Q ) ) ) ) ) ) ) ) ).

% reaching_paths_up_to_depth_set(3)
thf(fact_1003_reaching__paths__up__to__depth__set_I2_J,axiom,
    ! [Nexts: set_a,M2: fsm_a_b_c,N: nat,Dones: set_a,Assignment: a > option503927706846959746od_c_a,K: nat,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( Nexts
        = ( collect_a
          @ ^ [Q3: a] :
              ( ? [P3: list_P6327159017948738492od_c_a] :
                  ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                    = Q3 )
                  & ( ( size_s3386368156187063848od_c_a @ P3 )
                    = N ) )
              & ~ ? [P3: list_P6327159017948738492od_c_a] :
                    ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                    & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                      = Q3 )
                    & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) ) )
     => ( ( Dones
          = ( collect_a
            @ ^ [Q3: a] :
              ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                  = Q3 )
                & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
       => ( ! [Q2: a] :
              ( ( ( Assignment @ Q2 )
                = none_l592525953500355997od_c_a )
              = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
                      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                      & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                        = Q2 )
                      & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
         => ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
                ( ( ( Assignment @ Q2 )
                  = ( some_l6142909491759833889od_c_a @ P ) )
               => ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P )
                    = Q2 )
                  & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P ) @ N ) ) )
           => ( ( ( dom_a_2450325921413825296od_c_a @ Assignment )
                = ( sup_sup_set_a @ Nexts @ Dones ) )
             => ( ( ( state_6616341566432195646_a_b_c @ M2 @ Nexts @ Dones @ Assignment @ K @ Q )
                  = ( some_l6142909491759833889od_c_a @ P2 ) )
               => ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P2 )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P2 )
                    = Q )
                  & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P2 ) @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ) ) ).

% reaching_paths_up_to_depth_set(2)
thf(fact_1004_reaching__paths__up__to__depth__set_I1_J,axiom,
    ! [Nexts: set_a,M2: fsm_a_b_c,N: nat,Dones: set_a,Assignment: a > option503927706846959746od_c_a,K: nat,Q: a] :
      ( ( Nexts
        = ( collect_a
          @ ^ [Q3: a] :
              ( ? [P3: list_P6327159017948738492od_c_a] :
                  ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                    = Q3 )
                  & ( ( size_s3386368156187063848od_c_a @ P3 )
                    = N ) )
              & ~ ? [P3: list_P6327159017948738492od_c_a] :
                    ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                    & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                      = Q3 )
                    & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) ) )
     => ( ( Dones
          = ( collect_a
            @ ^ [Q3: a] :
              ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                  = Q3 )
                & ( ord_less_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
       => ( ! [Q2: a] :
              ( ( ( Assignment @ Q2 )
                = none_l592525953500355997od_c_a )
              = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
                      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                      & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                        = Q2 )
                      & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N ) ) ) )
         => ( ! [Q2: a,P: list_P6327159017948738492od_c_a] :
                ( ( ( Assignment @ Q2 )
                  = ( some_l6142909491759833889od_c_a @ P ) )
               => ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P )
                  & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P )
                    = Q2 )
                  & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P ) @ N ) ) )
           => ( ( ( dom_a_2450325921413825296od_c_a @ Assignment )
                = ( sup_sup_set_a @ Nexts @ Dones ) )
             => ( ( ( state_6616341566432195646_a_b_c @ M2 @ Nexts @ Dones @ Assignment @ K @ Q )
                  = none_l592525953500355997od_c_a )
                = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
                        ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
                        & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 )
                          = Q )
                        & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ) ) ) ).

% reaching_paths_up_to_depth_set(1)
thf(fact_1005_reachable__k_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_a] :
      ( ( ( reachable_k_a_b_c @ X2 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( collect_a
          @ ^ [Uu: a] :
            ? [P3: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( target_a_b_c @ Xa @ P3 ) )
              & ( path_a_b_c @ X2 @ Xa @ P3 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ Xb ) ) ) ) ) ).

% reachable_k.elims
thf(fact_1006_reachable__k_Osimps,axiom,
    ( reachable_k_a_b_c
    = ( ^ [M4: fsm_a_b_c,Q3: a,N2: nat] :
          ( collect_a
          @ ^ [Uu: a] :
            ? [P3: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( target_a_b_c @ Q3 @ P3 ) )
              & ( path_a_b_c @ M4 @ Q3 @ P3 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ N2 ) ) ) ) ) ).

% reachable_k.simps
thf(fact_1007_fun__upd__None__restrict,axiom,
    ! [X2: nat,D: set_nat,M: nat > option503927706846959746od_c_a] :
      ( ( ( member_nat @ X2 @ D )
       => ( ( fun_up4606601786420396787od_c_a @ ( restri1700453863123333178od_c_a @ M @ D ) @ X2 @ none_l592525953500355997od_c_a )
          = ( restri1700453863123333178od_c_a @ M @ ( minus_minus_set_nat @ D @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X2 @ D )
       => ( ( fun_up4606601786420396787od_c_a @ ( restri1700453863123333178od_c_a @ M @ D ) @ X2 @ none_l592525953500355997od_c_a )
          = ( restri1700453863123333178od_c_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_1008_fun__upd__None__restrict,axiom,
    ! [X2: a,D: set_a,M: a > option503927706846959746od_c_a] :
      ( ( ( member_a @ X2 @ D )
       => ( ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ D ) @ X2 @ none_l592525953500355997od_c_a )
          = ( restri3348497847382419694od_c_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) )
      & ( ~ ( member_a @ X2 @ D )
       => ( ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ D ) @ X2 @ none_l592525953500355997od_c_a )
          = ( restri3348497847382419694od_c_a @ M @ D ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_1009_restrict__out,axiom,
    ! [X2: a,A2: set_a,M: a > option503927706846959746od_c_a] :
      ( ~ ( member_a @ X2 @ A2 )
     => ( ( restri3348497847382419694od_c_a @ M @ A2 @ X2 )
        = none_l592525953500355997od_c_a ) ) ).

% restrict_out
thf(fact_1010_restrict__out,axiom,
    ! [X2: nat,A2: set_nat,M: nat > option503927706846959746od_c_a] :
      ( ~ ( member_nat @ X2 @ A2 )
     => ( ( restri1700453863123333178od_c_a @ M @ A2 @ X2 )
        = none_l592525953500355997od_c_a ) ) ).

% restrict_out
thf(fact_1011_restrict__map__to__empty,axiom,
    ! [M: a > option503927706846959746od_c_a] :
      ( ( restri3348497847382419694od_c_a @ M @ bot_bot_set_a )
      = ( ^ [X: a] : none_l592525953500355997od_c_a ) ) ).

% restrict_map_to_empty
thf(fact_1012_fun__upd__restrict__conv,axiom,
    ! [X2: a,D: set_a,M: a > option503927706846959746od_c_a,Y3: option503927706846959746od_c_a] :
      ( ( member_a @ X2 @ D )
     => ( ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ D ) @ X2 @ Y3 )
        = ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ X2 @ Y3 ) ) ) ).

% fun_upd_restrict_conv
thf(fact_1013_restrict__fun__upd,axiom,
    ! [X2: a,D: set_a,M: a > option503927706846959746od_c_a,Y3: option503927706846959746od_c_a] :
      ( ( ( member_a @ X2 @ D )
       => ( ( restri3348497847382419694od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ X2 @ Y3 ) @ D )
          = ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ X2 @ Y3 ) ) )
      & ( ~ ( member_a @ X2 @ D )
       => ( ( restri3348497847382419694od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ X2 @ Y3 ) @ D )
          = ( restri3348497847382419694od_c_a @ M @ D ) ) ) ) ).

% restrict_fun_upd
thf(fact_1014_psubset__imp__ex__mem,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_set_a @ A2 @ B )
     => ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1015_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1016_not__psubset__empty,axiom,
    ! [A2: set_a] :
      ~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).

% not_psubset_empty
thf(fact_1017_restrict__map__def,axiom,
    ( restri3348497847382419694od_c_a
    = ( ^ [M3: a > option503927706846959746od_c_a,A3: set_a,X: a] : ( if_opt5277955309491978056od_c_a @ ( member_a @ X @ A3 ) @ ( M3 @ X ) @ none_l592525953500355997od_c_a ) ) ) ).

% restrict_map_def
thf(fact_1018_restrict__map__def,axiom,
    ( restri1700453863123333178od_c_a
    = ( ^ [M3: nat > option503927706846959746od_c_a,A3: set_nat,X: nat] : ( if_opt5277955309491978056od_c_a @ ( member_nat @ X @ A3 ) @ ( M3 @ X ) @ none_l592525953500355997od_c_a ) ) ) ).

% restrict_map_def
thf(fact_1019_restrict__map__insert,axiom,
    ! [F: a > option503927706846959746od_c_a,A: a,A2: set_a] :
      ( ( restri3348497847382419694od_c_a @ F @ ( insert_a @ A @ A2 ) )
      = ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ F @ A2 ) @ A @ ( F @ A ) ) ) ).

% restrict_map_insert
thf(fact_1020_bounded__Max__nat,axiom,
    ! [P4: nat > $o,X2: nat,M2: nat] :
      ( ( P4 @ X2 )
     => ( ! [X3: nat] :
            ( ( P4 @ X3 )
           => ( ord_less_eq_nat @ X3 @ M2 ) )
       => ~ ! [M6: nat] :
              ( ( P4 @ M6 )
             => ~ ! [X6: nat] :
                    ( ( P4 @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1021_psubset__insert__iff,axiom,
    ! [A2: set_nat,X2: nat,B: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
      = ( ( ( member_nat @ X2 @ B )
         => ( ord_less_set_nat @ A2 @ B ) )
        & ( ~ ( member_nat @ X2 @ B )
         => ( ( ( member_nat @ X2 @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) )
            & ( ~ ( member_nat @ X2 @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1022_psubset__insert__iff,axiom,
    ! [A2: set_a,X2: a,B: set_a] :
      ( ( ord_less_set_a @ A2 @ ( insert_a @ X2 @ B ) )
      = ( ( ( member_a @ X2 @ B )
         => ( ord_less_set_a @ A2 @ B ) )
        & ( ~ ( member_a @ X2 @ B )
         => ( ( ( member_a @ X2 @ A2 )
             => ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
            & ( ~ ( member_a @ X2 @ A2 )
             => ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1023_fun__upd__restrict,axiom,
    ! [M: a > option503927706846959746od_c_a,D: set_a,X2: a,Y3: option503927706846959746od_c_a] :
      ( ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ D ) @ X2 @ Y3 )
      = ( fun_up7010226539926187649od_c_a @ ( restri3348497847382419694od_c_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) @ X2 @ Y3 ) ) ).

% fun_upd_restrict
thf(fact_1024_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila2496817875450240012_set_a @ sup_sup_set_a @ bot_bot_set_a
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X )
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_1025_Euclid__induct,axiom,
    ! [P4: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( P4 @ A6 @ B5 )
          = ( P4 @ B5 @ A6 ) )
     => ( ! [A6: nat] : ( P4 @ A6 @ zero_zero_nat )
       => ( ! [A6: nat,B5: nat] :
              ( ( P4 @ A6 @ B5 )
             => ( P4 @ A6 @ ( plus_plus_nat @ A6 @ B5 ) ) )
         => ( P4 @ A @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_1026_nat__descend__induct,axiom,
    ! [N: nat,P4: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P4 @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K3 @ I3 )
                 => ( P4 @ I3 ) )
             => ( P4 @ K3 ) ) )
       => ( P4 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1027_psubsetD,axiom,
    ! [A2: set_a,B: set_a,C: a] :
      ( ( ord_less_set_a @ A2 @ B )
     => ( ( member_a @ C @ A2 )
       => ( member_a @ C @ B ) ) ) ).

% psubsetD
thf(fact_1028_psubsetD,axiom,
    ! [A2: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_1029_less__set__def,axiom,
    ( ord_less_set_a
    = ( ^ [A3: set_a,B3: set_a] :
          ( ord_less_a_o
          @ ^ [X: a] : ( member_a @ X @ A3 )
          @ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1030_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A3 )
          @ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).

% less_set_def
thf(fact_1031_insert__subsetI,axiom,
    ! [X2: a,A2: set_a,X4: set_a] :
      ( ( member_a @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ X4 @ A2 )
       => ( ord_less_eq_set_a @ ( insert_a @ X2 @ X4 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1032_insert__subsetI,axiom,
    ! [X2: nat,A2: set_nat,X4: set_nat] :
      ( ( member_nat @ X2 @ A2 )
     => ( ( ord_less_eq_set_nat @ X4 @ A2 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X4 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1033_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1034_subset__emptyI,axiom,
    ! [A2: set_a] :
      ( ! [X3: a] :
          ~ ( member_a @ X3 @ A2 )
     => ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).

% subset_emptyI
thf(fact_1035_ran__map__upd,axiom,
    ! [M: a > option503927706846959746od_c_a,A: a,B2: list_P6327159017948738492od_c_a] :
      ( ( ( M @ A )
        = none_l592525953500355997od_c_a )
     => ( ( ran_a_2612639472603350189od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ A @ ( some_l6142909491759833889od_c_a @ B2 ) ) )
        = ( insert4789241225314331020od_c_a @ B2 @ ( ran_a_2612639472603350189od_c_a @ M ) ) ) ) ).

% ran_map_upd
thf(fact_1036_Collect__restrict,axiom,
    ! [X4: set_nat,P4: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X4 )
            & ( P4 @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1037_Collect__restrict,axiom,
    ! [X4: set_a,P4: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ X4 )
            & ( P4 @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1038_prop__restrict,axiom,
    ! [X2: nat,Z5: set_nat,X4: set_nat,P4: nat > $o] :
      ( ( member_nat @ X2 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X4 )
                & ( P4 @ X ) ) ) )
       => ( P4 @ X2 ) ) ) ).

% prop_restrict
thf(fact_1039_prop__restrict,axiom,
    ! [X2: a,Z5: set_a,X4: set_a,P4: a > $o] :
      ( ( member_a @ X2 @ Z5 )
     => ( ( ord_less_eq_set_a @ Z5
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a @ X @ X4 )
                & ( P4 @ X ) ) ) )
       => ( P4 @ X2 ) ) ) ).

% prop_restrict
thf(fact_1040_ran__map__upd__Some,axiom,
    ! [M: a > option503927706846959746od_c_a,X2: a,Y3: list_P6327159017948738492od_c_a,Z2: list_P6327159017948738492od_c_a] :
      ( ( ( M @ X2 )
        = ( some_l6142909491759833889od_c_a @ Y3 ) )
     => ( ( inj_on8536872568654550389od_c_a @ M @ ( dom_a_2450325921413825296od_c_a @ M ) )
       => ( ~ ( member7410604586820865893od_c_a @ Z2 @ ( ran_a_2612639472603350189od_c_a @ M ) )
         => ( ( ran_a_2612639472603350189od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ X2 @ ( some_l6142909491759833889od_c_a @ Z2 ) ) )
            = ( sup_su500200128730103920od_c_a @ ( minus_4060711634664891779od_c_a @ ( ran_a_2612639472603350189od_c_a @ M ) @ ( insert4789241225314331020od_c_a @ Y3 @ bot_bo6236370880139903240od_c_a ) ) @ ( insert4789241225314331020od_c_a @ Z2 @ bot_bo6236370880139903240od_c_a ) ) ) ) ) ) ).

% ran_map_upd_Some
thf(fact_1041_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1042_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1043_inj__on__empty,axiom,
    ! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).

% inj_on_empty
thf(fact_1044_linorder__inj__onI_H,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( member_nat @ J2 @ A2 )
           => ( ( ord_less_nat @ I2 @ J2 )
             => ( ( F @ I2 )
               != ( F @ J2 ) ) ) ) )
     => ( inj_on_nat_nat @ F @ A2 ) ) ).

% linorder_inj_onI'
thf(fact_1045_linorder__inj__onI,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( ord_less_nat @ X3 @ Y4 )
         => ( ( member_nat @ X3 @ A2 )
           => ( ( member_nat @ Y4 @ A2 )
             => ( ( F @ X3 )
               != ( F @ Y4 ) ) ) ) )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ( member_nat @ Y4 @ A2 )
             => ( ( ord_less_eq_nat @ X3 @ Y4 )
                | ( ord_less_eq_nat @ Y4 @ X3 ) ) ) )
       => ( inj_on_nat_nat @ F @ A2 ) ) ) ).

% linorder_inj_onI
thf(fact_1046_inj__on__diff,axiom,
    ! [F: nat > nat,A2: set_nat,B: set_nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% inj_on_diff
thf(fact_1047_inj__on__subset,axiom,
    ! [F: nat > nat,A2: set_nat,B: set_nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( inj_on_nat_nat @ F @ B ) ) ) ).

% inj_on_subset
thf(fact_1048_subset__inj__on,axiom,
    ! [F: nat > nat,B: set_nat,A2: set_nat] :
      ( ( inj_on_nat_nat @ F @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( inj_on_nat_nat @ F @ A2 ) ) ) ).

% subset_inj_on
thf(fact_1049_inj__on__add,axiom,
    ! [A: nat,A2: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A2 ) ).

% inj_on_add
thf(fact_1050_inj__onD,axiom,
    ! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ( ( F @ X2 )
          = ( F @ Y3 ) )
       => ( ( member_nat @ X2 @ A2 )
         => ( ( member_nat @ Y3 @ A2 )
           => ( X2 = Y3 ) ) ) ) ) ).

% inj_onD
thf(fact_1051_inj__onI,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( member_nat @ Y4 @ A2 )
           => ( ( ( F @ X3 )
                = ( F @ Y4 ) )
             => ( X3 = Y4 ) ) ) )
     => ( inj_on_nat_nat @ F @ A2 ) ) ).

% inj_onI
thf(fact_1052_inj__on__def,axiom,
    ( inj_on_nat_nat
    = ( ^ [F2: nat > nat,A3: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A3 )
         => ! [Y2: nat] :
              ( ( member_nat @ Y2 @ A3 )
             => ( ( ( F2 @ X )
                  = ( F2 @ Y2 ) )
               => ( X = Y2 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_1053_inj__on__cong,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [A6: nat] :
          ( ( member_nat @ A6 @ A2 )
         => ( ( F @ A6 )
            = ( G @ A6 ) ) )
     => ( ( inj_on_nat_nat @ F @ A2 )
        = ( inj_on_nat_nat @ G @ A2 ) ) ) ).

% inj_on_cong
thf(fact_1054_inj__on__eq__iff,axiom,
    ! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ( member_nat @ X2 @ A2 )
       => ( ( member_nat @ Y3 @ A2 )
         => ( ( ( F @ X2 )
              = ( F @ Y3 ) )
            = ( X2 = Y3 ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_1055_inj__on__contraD,axiom,
    ! [F: nat > nat,A2: set_nat,X2: nat,Y3: nat] :
      ( ( inj_on_nat_nat @ F @ A2 )
     => ( ( X2 != Y3 )
       => ( ( member_nat @ X2 @ A2 )
         => ( ( member_nat @ Y3 @ A2 )
           => ( ( F @ X2 )
             != ( F @ Y3 ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_1056_inj__on__inverseI,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ ( F @ X3 ) )
            = X3 ) )
     => ( inj_on_nat_nat @ F @ A2 ) ) ).

% inj_on_inverseI
thf(fact_1057_inj__on__id2,axiom,
    ! [A2: set_nat] :
      ( inj_on_nat_nat
      @ ^ [X: nat] : X
      @ A2 ) ).

% inj_on_id2
thf(fact_1058_inj__Some,axiom,
    ! [A2: set_li1159382662694783132od_c_a] : ( inj_on3621648716004224823od_c_a @ some_l6142909491759833889od_c_a @ A2 ) ).

% inj_Some
thf(fact_1059_inj__on__add_H,axiom,
    ! [A: nat,A2: set_nat] :
      ( inj_on_nat_nat
      @ ^ [B7: nat] : ( plus_plus_nat @ B7 @ A )
      @ A2 ) ).

% inj_on_add'
thf(fact_1060_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1061_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_nat @ A @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_1062_verit__comp__simplify1_I3_J,axiom,
    ! [B8: nat,A7: nat] :
      ( ( ~ ( ord_less_eq_nat @ B8 @ A7 ) )
      = ( ord_less_nat @ A7 @ B8 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1063_complete__interval,axiom,
    ! [A: nat,B2: nat,P4: nat > $o] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( P4 @ A )
       => ( ~ ( P4 @ B2 )
         => ? [C4: nat] :
              ( ( ord_less_eq_nat @ A @ C4 )
              & ( ord_less_eq_nat @ C4 @ B2 )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C4 ) )
                 => ( P4 @ X6 ) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D4 ) )
                     => ( P4 @ X3 ) )
                 => ( ord_less_eq_nat @ D4 @ C4 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1064_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_1065_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_1066_inj__singleton,axiom,
    ! [A2: set_a] :
      ( inj_on_a_set_a
      @ ^ [X: a] : ( insert_a @ X @ bot_bot_set_a )
      @ A2 ) ).

% inj_singleton
thf(fact_1067_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N4 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N4 ) ) ).

% inj_on_diff_nat
thf(fact_1068_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_1069_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_1070_restrict__upd__same,axiom,
    ! [M: a > option503927706846959746od_c_a,X2: a,Y3: list_P6327159017948738492od_c_a] :
      ( ( restri3348497847382419694od_c_a @ ( fun_up7010226539926187649od_c_a @ M @ X2 @ ( some_l6142909491759833889od_c_a @ Y3 ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
      = ( restri3348497847382419694od_c_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% restrict_upd_same
thf(fact_1071_Gcd__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( gcd_Gcd_nat @ A2 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_1072_ComplI,axiom,
    ! [C: a,A2: set_a] :
      ( ~ ( member_a @ C @ A2 )
     => ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).

% ComplI
thf(fact_1073_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_1074_Compl__iff,axiom,
    ! [C: a,A2: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
      = ( ~ ( member_a @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_1075_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_1076_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_1077_Compl__Diff__eq,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A2 @ B ) )
      = ( sup_sup_set_a @ ( uminus_uminus_set_a @ A2 ) @ B ) ) ).

% Compl_Diff_eq
thf(fact_1078_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_1079_subset__Compl__singleton,axiom,
    ! [A2: set_a,B2: a] :
      ( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
      = ( ~ ( member_a @ B2 @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_1080_Collect__imp__eq,axiom,
    ! [P4: a > $o,Q4: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( P4 @ X )
           => ( Q4 @ X ) ) )
      = ( sup_sup_set_a @ ( uminus_uminus_set_a @ ( collect_a @ P4 ) ) @ ( collect_a @ Q4 ) ) ) ).

% Collect_imp_eq
thf(fact_1081_ComplD,axiom,
    ! [C: a,A2: set_a] :
      ( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
     => ~ ( member_a @ C @ A2 ) ) ).

% ComplD
thf(fact_1082_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_1083_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A3: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ~ ( member_nat @ X @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_1084_Compl__eq,axiom,
    ( uminus_uminus_set_a
    = ( ^ [A3: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ~ ( member_a @ X @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_1085_Collect__neg__eq,axiom,
    ! [P4: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ~ ( P4 @ X ) )
      = ( uminus_uminus_set_a @ ( collect_a @ P4 ) ) ) ).

% Collect_neg_eq
thf(fact_1086_subset__Compl__self__eq,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
      = ( A2 = bot_bot_set_a ) ) ).

% subset_Compl_self_eq
thf(fact_1087_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( ( member_nat @ one_one_nat @ N4 )
     => ( ( gcd_Gcd_nat @ N4 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_1088_Gcd__1,axiom,
    ! [A2: set_nat] :
      ( ( member_nat @ one_one_nat @ A2 )
     => ( ( gcd_Gcd_nat @ A2 )
        = one_one_nat ) ) ).

% Gcd_1
thf(fact_1089_Compl__insert,axiom,
    ! [X2: a,A2: set_a] :
      ( ( uminus_uminus_set_a @ ( insert_a @ X2 @ A2 ) )
      = ( minus_minus_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).

% Compl_insert
thf(fact_1090_restrict__complement__singleton__eq,axiom,
    ! [F: a > option503927706846959746od_c_a,X2: a] :
      ( ( restri3348497847382419694od_c_a @ F @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
      = ( fun_up7010226539926187649od_c_a @ F @ X2 @ none_l592525953500355997od_c_a ) ) ).

% restrict_complement_singleton_eq
thf(fact_1091_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A3: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1092_uminus__set__def,axiom,
    ( uminus_uminus_set_a
    = ( ^ [A3: set_a] :
          ( collect_a
          @ ( uminus_uminus_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1093_remove__def,axiom,
    ( remove_a
    = ( ^ [X: a,A3: set_a] : ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% remove_def
thf(fact_1094_add__state__simps_I5_J,axiom,
    ! [M2: fsm_a_b_c,Q: a] :
      ( ( states_a_b_c @ ( add_state_a_b_c @ M2 @ Q ) )
      = ( insert_a @ Q @ ( states_a_b_c @ M2 ) ) ) ).

% add_state_simps(5)
thf(fact_1095_member__remove,axiom,
    ! [X2: a,Y3: a,A2: set_a] :
      ( ( member_a @ X2 @ ( remove_a @ Y3 @ A2 ) )
      = ( ( member_a @ X2 @ A2 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_1096_member__remove,axiom,
    ! [X2: nat,Y3: nat,A2: set_nat] :
      ( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A2 ) )
      = ( ( member_nat @ X2 @ A2 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_1097_add__state__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c,Q: a] :
      ( ( initial_a_b_c @ ( add_state_a_b_c @ M2 @ Q ) )
      = ( initial_a_b_c @ M2 ) ) ).

% add_state_simps(1)
thf(fact_1098_arg__min__inj__eq,axiom,
    ! [F: nat > nat,P4: nat > $o,A: nat] :
      ( ( inj_on_nat_nat @ F @ ( collect_nat @ P4 ) )
     => ( ( P4 @ A )
       => ( ! [Y4: nat] :
              ( ( P4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ A ) @ ( F @ Y4 ) ) )
         => ( ( lattic8739620818006775868at_nat @ F @ P4 )
            = A ) ) ) ) ).

% arg_min_inj_eq
thf(fact_1099_arg__min__inj__eq,axiom,
    ! [F: a > nat,P4: a > $o,A: a] :
      ( ( inj_on_a_nat @ F @ ( collect_a @ P4 ) )
     => ( ( P4 @ A )
       => ( ! [Y4: a] :
              ( ( P4 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ A ) @ ( F @ Y4 ) ) )
         => ( ( lattic1189635703294652468_a_nat @ F @ P4 )
            = A ) ) ) ) ).

% arg_min_inj_eq
thf(fact_1100_cyclic__cycle__rev,axiom,
    ! [M2: fsm_a_b_c,P7: list_P6327159017948738492od_c_a,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P7 )
     => ( ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P7 )
          = Q )
       => ( ( path_a_b_c @ M2 @ Q @ P2 )
         => ( ( P2 != nil_Pr1342775757158464060od_c_a )
           => ( ( ( target_a_b_c @ Q @ P2 )
                = Q )
             => ~ ( acyclic_a_b_c @ M2 ) ) ) ) ) ) ).

% cyclic_cycle_rev
thf(fact_1101_cyclic__cycle,axiom,
    ! [M2: fsm_a_b_c] :
      ( ~ ( acyclic_a_b_c @ M2 )
     => ? [Q2: a,P: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ M2 @ Q2 @ P )
          & ( P != nil_Pr1342775757158464060od_c_a )
          & ( ( target_a_b_c @ Q2 @ P )
            = Q2 ) ) ) ).

% cyclic_cycle
thf(fact_1102_those_Osimps_I1_J,axiom,
    ( ( those_5530478821918711090od_c_a @ nil_op474191768008379010od_c_a )
    = ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) ) ).

% those.simps(1)
thf(fact_1103_reachable__k__states,axiom,
    ( reacha1620305530751930115_a_b_c
    = ( ^ [M4: fsm_a_b_c] : ( reachable_k_a_b_c @ M4 @ ( initial_a_b_c @ M4 ) @ ( minus_minus_nat @ ( size_a_b_c @ M4 ) @ one_one_nat ) ) ) ) ).

% reachable_k_states
thf(fact_1104_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_1105_reachable__states__intro,axiom,
    ! [M2: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P2 )
     => ( member_a @ ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P2 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ).

% reachable_states_intro
thf(fact_1106_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_1107_reachable__states__initial,axiom,
    ! [M2: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M2 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ).

% reachable_states_initial
thf(fact_1108_restrict__to__reachable__states__path,axiom,
    ! [Q: a,M2: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( path_a_b_c @ M2 @ Q @ P2 )
        = ( path_a_b_c @ ( restri9132545300209641082_a_b_c @ M2 ) @ Q @ P2 ) ) ) ).

% restrict_to_reachable_states_path
thf(fact_1109_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_1110_reachable__states__def,axiom,
    ( reacha1620305530751930115_a_b_c
    = ( ^ [M4: fsm_a_b_c] :
          ( collect_a
          @ ^ [Uu: a] :
            ? [P3: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( target_a_b_c @ ( initial_a_b_c @ M4 ) @ P3 ) )
              & ( path_a_b_c @ M4 @ ( initial_a_b_c @ M4 ) @ P3 ) ) ) ) ) ).

% reachable_states_def
thf(fact_1111_acyclic__paths__to__single__deadlock,axiom,
    ! [M2: fsm_a_b_c,Qd: a,Q: a] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ! [Q5: a] :
            ( ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( ( Q5 = Qd )
              | ~ ( deadlock_state_a_b_c @ M2 @ Q5 ) ) )
       => ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M2 ) )
         => ~ ! [P: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M2 @ Q @ P )
               => ( ( target_a_b_c @ Q @ P )
                 != Qd ) ) ) ) ) ).

% acyclic_paths_to_single_deadlock
thf(fact_1112_rotate1__length01,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs ) @ one_one_nat )
     => ( ( rotate1246884043934778121od_c_a @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_1113_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ( rotate1246884043934778121od_c_a @ Xs )
        = nil_Pr1342775757158464060od_c_a )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% rotate1_is_Nil_conv
thf(fact_1114_length__rotate1,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( size_s3386368156187063848od_c_a @ ( rotate1246884043934778121od_c_a @ Xs ) )
      = ( size_s3386368156187063848od_c_a @ Xs ) ) ).

% length_rotate1
thf(fact_1115_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1246884043934778121od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr1342775757158464060od_c_a ) ).

% rotate1.simps(1)
thf(fact_1116_states__initial__deadlock,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( deadlock_state_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
     => ( ( reacha1620305530751930115_a_b_c @ M2 )
        = ( insert_a @ ( initial_a_b_c @ M2 ) @ bot_bot_set_a ) ) ) ).

% states_initial_deadlock
thf(fact_1117_maximal__acyclic__paths__deadlock__targets,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ( maxima334016647137088875_a_b_c @ M2 )
        = ( collec6273869032445462695od_c_a
          @ ^ [P3: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P3 )
              & ( deadlock_state_a_b_c @ M2 @ ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P3 ) ) ) ) ) ) ).

% maximal_acyclic_paths_deadlock_targets
thf(fact_1118_path__to__deadlock__is__maximal,axiom,
    ! [M2: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P2 )
     => ( ( deadlock_state_a_b_c @ M2 @ ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P2 ) )
       => ~ ? [P8: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P8 )
              & ( is_pre701076807510392747od_c_a @ P2 @ P8 )
              & ( P2 != P8 ) ) ) ) ).

% path_to_deadlock_is_maximal
thf(fact_1119_maximal__path__target__deadlock,axiom,
    ! [M2: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P2 )
     => ( ~ ? [P9: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P9 )
              & ( is_pre701076807510392747od_c_a @ P2 @ P9 )
              & ( P2 != P9 ) )
       => ( deadlock_state_a_b_c @ M2 @ ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P2 ) ) ) ) ).

% maximal_path_target_deadlock
thf(fact_1120_is__prefix_Osimps_I1_J,axiom,
    ! [Uu2: list_P6327159017948738492od_c_a] : ( is_pre701076807510392747od_c_a @ nil_Pr1342775757158464060od_c_a @ Uu2 ) ).

% is_prefix.simps(1)
thf(fact_1121_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_a
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X )
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X )
    @ bot_bot_set_a ) ).

% bot.ordering_top_axioms
thf(fact_1122_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X )
    @ ^ [X: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_1123_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
thf(fact_1124_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( A != Top )
        = ( Less @ A @ Top ) ) ) ).

% ordering_top.not_eq_extremum
thf(fact_1125_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
        = ( A = Top ) ) ) ).

% ordering_top.extremum_unique
thf(fact_1126_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ~ ( Less @ Top @ A ) ) ).

% ordering_top.extremum_strict
thf(fact_1127_ordering__top_Oextremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( Less_eq @ A @ Top ) ) ).

% ordering_top.extremum
thf(fact_1128_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X )
    @ ^ [X: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1129_is__singleton__the__elem,axiom,
    ( is_singleton_a
    = ( ^ [A3: set_a] :
          ( A3
          = ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1130_subset__singleton__iff__Uniq,axiom,
    ! [A2: set_nat] :
      ( ( ? [A4: nat] : ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) )
      = ( uniq_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1131_subset__singleton__iff__Uniq,axiom,
    ! [A2: set_a] :
      ( ( ? [A4: a] : ( ord_less_eq_set_a @ A2 @ ( insert_a @ A4 @ bot_bot_set_a ) ) )
      = ( uniq_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1132_the__elem__eq,axiom,
    ! [X2: a] :
      ( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
      = X2 ) ).

% the_elem_eq
thf(fact_1133_inj__on__iff__Uniq,axiom,
    ( inj_on_nat_nat
    = ( ^ [F2: nat > nat,A3: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A3 )
         => ( uniq_nat
            @ ^ [Y2: nat] :
                ( ( member_nat @ Y2 @ A3 )
                & ( ( F2 @ X )
                  = ( F2 @ Y2 ) ) ) ) ) ) ) ).

% inj_on_iff_Uniq
thf(fact_1134_gen__length__def,axiom,
    ( gen_le1648446388249104713od_c_a
    = ( ^ [N2: nat,Xs3: list_P6327159017948738492od_c_a] : ( plus_plus_nat @ N2 @ ( size_s3386368156187063848od_c_a @ Xs3 ) ) ) ) ).

% gen_length_def
thf(fact_1135_these__insert__Some,axiom,
    ! [X2: a,A2: set_option_a] :
      ( ( these_a @ ( insert_option_a @ ( some_a @ X2 ) @ A2 ) )
      = ( insert_a @ X2 @ ( these_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_1136_these__insert__Some,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( these_2957773514865926521od_c_a @ ( insert3566037597566277202od_c_a @ ( some_l6142909491759833889od_c_a @ X2 ) @ A2 ) )
      = ( insert4789241225314331020od_c_a @ X2 @ ( these_2957773514865926521od_c_a @ A2 ) ) ) ).

% these_insert_Some
thf(fact_1137_these__empty,axiom,
    ( ( these_a @ bot_bot_set_option_a )
    = bot_bot_set_a ) ).

% these_empty
thf(fact_1138_these__insert__None,axiom,
    ! [A2: set_op7949082993927878370od_c_a] :
      ( ( these_2957773514865926521od_c_a @ ( insert3566037597566277202od_c_a @ none_l592525953500355997od_c_a @ A2 ) )
      = ( these_2957773514865926521od_c_a @ A2 ) ) ).

% these_insert_None
thf(fact_1139_in__these__eq,axiom,
    ! [X2: a,A2: set_option_a] :
      ( ( member_a @ X2 @ ( these_a @ A2 ) )
      = ( member_option_a @ ( some_a @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1140_in__these__eq,axiom,
    ! [X2: nat,A2: set_option_nat] :
      ( ( member_nat @ X2 @ ( these_nat @ A2 ) )
      = ( member_option_nat @ ( some_nat @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1141_in__these__eq,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A2: set_op7949082993927878370od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( these_2957773514865926521od_c_a @ A2 ) )
      = ( member2959255366155174571od_c_a @ ( some_l6142909491759833889od_c_a @ X2 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1142_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1648446388249104713od_c_a @ N @ nil_Pr1342775757158464060od_c_a )
      = N ) ).

% gen_length_code(1)
thf(fact_1143_these__empty__eq,axiom,
    ! [B: set_op7949082993927878370od_c_a] :
      ( ( ( these_2957773514865926521od_c_a @ B )
        = bot_bo6236370880139903240od_c_a )
      = ( ( B = bot_bo3379513543401233742od_c_a )
        | ( B
          = ( insert3566037597566277202od_c_a @ none_l592525953500355997od_c_a @ bot_bo3379513543401233742od_c_a ) ) ) ) ).

% these_empty_eq
thf(fact_1144_these__empty__eq,axiom,
    ! [B: set_option_a] :
      ( ( ( these_a @ B )
        = bot_bot_set_a )
      = ( ( B = bot_bot_set_option_a )
        | ( B
          = ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_empty_eq
thf(fact_1145_these__not__empty__eq,axiom,
    ! [B: set_op7949082993927878370od_c_a] :
      ( ( ( these_2957773514865926521od_c_a @ B )
       != bot_bo6236370880139903240od_c_a )
      = ( ( B != bot_bo3379513543401233742od_c_a )
        & ( B
         != ( insert3566037597566277202od_c_a @ none_l592525953500355997od_c_a @ bot_bo3379513543401233742od_c_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_1146_these__not__empty__eq,axiom,
    ! [B: set_option_a] :
      ( ( ( these_a @ B )
       != bot_bot_set_a )
      = ( ( B != bot_bot_set_option_a )
        & ( B
         != ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).

% these_not_empty_eq
thf(fact_1147_length__code,axiom,
    ( size_s3386368156187063848od_c_a
    = ( gen_le1648446388249104713od_c_a @ zero_zero_nat ) ) ).

% length_code
thf(fact_1148_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic8986249270076014136_set_a @ sup_sup_set_a
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ X )
    @ ^ [X: set_a,Y2: set_a] : ( ord_less_set_a @ Y2 @ X ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_1149_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ( lattic6009151579333465974et_nat @ sup_sup_nat
    @ ^ [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X )
    @ ^ [X: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X ) ) ).

% Sup_fin.semilattice_order_set_axioms
thf(fact_1150_completed__path_Osimps,axiom,
    ( completed_path_a_b_c
    = ( ^ [M4: fsm_a_b_c,Q3: a,P3: list_P6327159017948738492od_c_a] : ( deadlock_state_a_b_c @ M4 @ ( target_a_b_c @ Q3 @ P3 ) ) ) ) ).

% completed_path.simps
thf(fact_1151_completed__path_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P6327159017948738492od_c_a] :
      ( ~ ( completed_path_a_b_c @ X2 @ Xa @ Xb )
     => ~ ( deadlock_state_a_b_c @ X2 @ ( target_a_b_c @ Xa @ Xb ) ) ) ).

% completed_path.elims(3)
thf(fact_1152_completed__path_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P6327159017948738492od_c_a] :
      ( ( completed_path_a_b_c @ X2 @ Xa @ Xb )
     => ( deadlock_state_a_b_c @ X2 @ ( target_a_b_c @ Xa @ Xb ) ) ) ).

% completed_path.elims(2)
thf(fact_1153_completed__path_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_P6327159017948738492od_c_a,Y3: $o] :
      ( ( ( completed_path_a_b_c @ X2 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( deadlock_state_a_b_c @ X2 @ ( target_a_b_c @ Xa @ Xb ) ) ) ) ).

% completed_path.elims(1)
thf(fact_1154_maximal__acyclic__paths__code,axiom,
    ( maxima334016647137088875_a_b_c
    = ( ^ [M4: fsm_a_b_c] :
          ( filter8200269361223028333od_c_a
          @ ^ [P3: list_P6327159017948738492od_c_a] :
              ~ ? [X: list_P6327159017948738492od_c_a] :
                  ( ( member7410604586820865893od_c_a @ X @ ( acycli3978232057192500090_a_b_c @ M4 @ ( initial_a_b_c @ M4 ) @ ( minus_minus_nat @ ( size_a_b_c @ M4 ) @ one_one_nat ) ) )
                  & ( X != P3 )
                  & ( is_pre701076807510392747od_c_a @ P3 @ X ) )
          @ ( acycli3978232057192500090_a_b_c @ M4 @ ( initial_a_b_c @ M4 ) @ ( minus_minus_nat @ ( size_a_b_c @ M4 ) @ one_one_nat ) ) ) ) ) ).

% maximal_acyclic_paths_code
thf(fact_1155_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1156_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1157_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1158_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1159_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1160_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1161_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1162_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1163_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1164_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1165_bex__empty,axiom,
    ! [P4: a > $o] :
      ~ ? [X6: a] :
          ( ( member_a @ X6 @ bot_bot_set_a )
          & ( P4 @ X6 ) ) ).

% bex_empty
thf(fact_1166_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1167_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1168_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1169_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1170_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_1171_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_1172_minimal__fixpoint__helper_I1_J,axiom,
    ! [F: nat > nat,P4: nat > $o,K: nat,X2: nat] :
      ( ( F
        = ( ^ [X: nat] : ( if_nat @ ( P4 @ X ) @ X @ ( F @ ( suc @ X ) ) ) ) )
     => ( ! [X3: nat] :
            ( ( ord_less_eq_nat @ K @ X3 )
           => ( P4 @ X3 ) )
       => ( P4 @ ( F @ X2 ) ) ) ) ).

% minimal_fixpoint_helper(1)
thf(fact_1173_recursion__renaming__helper,axiom,
    ! [F1: nat > nat,P4: nat > $o,F22: nat > nat,K: nat] :
      ( ( F1
        = ( ^ [X: nat] : ( if_nat @ ( P4 @ X ) @ X @ ( F1 @ ( suc @ X ) ) ) ) )
     => ( ( F22
          = ( ^ [X: nat] : ( if_nat @ ( P4 @ X ) @ X @ ( F22 @ ( suc @ X ) ) ) ) )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ K @ X3 )
             => ( P4 @ X3 ) )
         => ( F1 = F22 ) ) ) ) ).

% recursion_renaming_helper
thf(fact_1174_minimal__fixpoint__helper_I2_J,axiom,
    ! [F: nat > nat,P4: nat > $o,K: nat,X2: nat,X7: nat] :
      ( ( F
        = ( ^ [X: nat] : ( if_nat @ ( P4 @ X ) @ X @ ( F @ ( suc @ X ) ) ) ) )
     => ( ! [X3: nat] :
            ( ( ord_less_eq_nat @ K @ X3 )
           => ( P4 @ X3 ) )
       => ( ( ord_less_eq_nat @ X2 @ X7 )
         => ( ( ord_less_nat @ X7 @ ( F @ X2 ) )
           => ~ ( P4 @ X7 ) ) ) ) ) ).

% minimal_fixpoint_helper(2)
thf(fact_1175_inj__Suc,axiom,
    ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).

% inj_Suc
thf(fact_1176_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1177_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1178_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1179_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1180_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1181_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1182_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1183_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1184_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1185_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1186_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1187_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1188_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1189_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1190_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1191_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1192_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1193_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1194_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1195_Ex__less__Suc,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P4 @ I4 ) ) )
      = ( ( P4 @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P4 @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1196_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1197_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1198_All__less__Suc,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P4 @ I4 ) ) )
      = ( ( P4 @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P4 @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_1199_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1200_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1201_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1202_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1203_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P4: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P4 @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P4 @ I2 @ J2 )
                 => ( ( P4 @ J2 @ K3 )
                   => ( P4 @ I2 @ K3 ) ) ) ) )
         => ( P4 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1204_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P4: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P4 @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P4 @ ( suc @ I2 ) )
               => ( P4 @ I2 ) ) )
         => ( P4 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1205_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1206_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1207_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1208_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1209_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1210_Bex__def,axiom,
    ( bex_a
    = ( ^ [A3: set_a,P5: a > $o] :
        ? [X: a] :
          ( ( member_a @ X @ A3 )
          & ( P5 @ X ) ) ) ) ).

% Bex_def
thf(fact_1211_Bex__def,axiom,
    ( bex_nat
    = ( ^ [A3: set_nat,P5: nat > $o] :
        ? [X: nat] :
          ( ( member_nat @ X @ A3 )
          & ( P5 @ X ) ) ) ) ).

% Bex_def
thf(fact_1212_Suc__inject,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y3 ) )
     => ( X2 = Y3 ) ) ).

% Suc_inject
thf(fact_1213_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1214_zero__induct__lemma,axiom,
    ! [P4: nat > $o,K: nat,I: nat] :
      ( ( P4 @ K )
     => ( ! [N3: nat] :
            ( ( P4 @ ( suc @ N3 ) )
           => ( P4 @ N3 ) )
       => ( P4 @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1215_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1216_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1217_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1218_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( R @ X3 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1219_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P4 @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P4 @ N3 )
               => ( P4 @ ( suc @ N3 ) ) ) )
         => ( P4 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1220_full__nat__induct,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
             => ( P4 @ M5 ) )
         => ( P4 @ N3 ) )
     => ( P4 @ N ) ) ).

% full_nat_induct
thf(fact_1221_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1222_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1223_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1224_Suc__le__D,axiom,
    ! [N: nat,M8: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
     => ? [M6: nat] :
          ( M8
          = ( suc @ M6 ) ) ) ).

% Suc_le_D
thf(fact_1225_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1226_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1227_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1228_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1229_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1230_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1231_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1232_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1233_nat__induct,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ( P4 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P4 @ N3 )
           => ( P4 @ ( suc @ N3 ) ) )
       => ( P4 @ N ) ) ) ).

% nat_induct
thf(fact_1234_diff__induct,axiom,
    ! [P4: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P4 @ X3 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P4 @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X3: nat,Y4: nat] :
              ( ( P4 @ X3 @ Y4 )
             => ( P4 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
         => ( P4 @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1235_zero__induct,axiom,
    ! [P4: nat > $o,K: nat] :
      ( ( P4 @ K )
     => ( ! [N3: nat] :
            ( ( P4 @ ( suc @ N3 ) )
           => ( P4 @ N3 ) )
       => ( P4 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1236_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1237_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1238_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1239_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% not0_implies_Suc
thf(fact_1240_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1241_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1242_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1243_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1244_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1245_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1246_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1247_inc__induct,axiom,
    ! [I: nat,J: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P4 @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P4 @ ( suc @ N3 ) )
                 => ( P4 @ N3 ) ) ) )
         => ( P4 @ I ) ) ) ) ).

% inc_induct
thf(fact_1248_dec__induct,axiom,
    ! [I: nat,J: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P4 @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P4 @ N3 )
                 => ( P4 @ ( suc @ N3 ) ) ) ) )
         => ( P4 @ J ) ) ) ) ).

% dec_induct
thf(fact_1249_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1250_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1251_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1252_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% gr0_implies_Suc
thf(fact_1253_All__less__Suc2,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P4 @ I4 ) ) )
      = ( ( P4 @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P4 @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1254_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1255_Ex__less__Suc2,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P4 @ I4 ) ) )
      = ( ( P4 @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P4 @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1256_Bex__def__raw,axiom,
    ( bex_a
    = ( ^ [A3: set_a,P5: a > $o] :
        ? [X: a] :
          ( ( member_a @ X @ A3 )
          & ( P5 @ X ) ) ) ) ).

% Bex_def_raw
thf(fact_1257_Bex__def__raw,axiom,
    ( bex_nat
    = ( ^ [A3: set_nat,P5: nat > $o] :
        ? [X: nat] :
          ( ( member_nat @ X @ A3 )
          & ( P5 @ X ) ) ) ) ).

% Bex_def_raw
thf(fact_1258_ex__least__nat__less,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ( P4 @ N )
     => ( ~ ( P4 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K3 )
               => ~ ( P4 @ I3 ) )
            & ( P4 @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1259_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1260_nat__induct__non__zero,axiom,
    ! [N: nat,P4: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P4 @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P4 @ N3 )
               => ( P4 @ ( suc @ N3 ) ) ) )
         => ( P4 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1261_option_Osize_I4_J,axiom,
    ! [X22: list_P6327159017948738492od_c_a] :
      ( ( size_s4346405003128473838od_c_a @ ( some_l6142909491759833889od_c_a @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1262_option_Osize_I3_J,axiom,
    ( ( size_s4346405003128473838od_c_a @ none_l592525953500355997od_c_a )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1263_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 ) )
             => ! [M6: nat] :
                  ( ( ord_less_eq_nat @ J2 @ M6 )
                 => ( ( F @ I2 )
                    = ( F @ M6 ) ) ) ) )
       => ( ! [I2: nat] : ( ord_less_eq_nat @ ( F @ I2 ) @ K )
         => ~ ! [X3: nat] :
                ( ( ( F @ ( suc @ X3 ) )
                  = ( F @ X3 ) )
               => ~ ( ord_less_eq_nat @ X3 @ ( minus_minus_nat @ K @ ( F @ zero_zero_nat ) ) ) ) ) ) ) ).

% monotone_function_with_limit_witness_helper
thf(fact_1264_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1265_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [P4: $o] :
      ( ( P4 = $true )
      | ( P4 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X2: option503927706846959746od_c_a,Y3: option503927706846959746od_c_a] :
      ( ( if_opt5277955309491978056od_c_a @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X2: option503927706846959746od_c_a,Y3: option503927706846959746od_c_a] :
      ( ( if_opt5277955309491978056od_c_a @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( ( state_6616341566432195646_a_b_c @ m @ ( insert_a @ ( initial_a_b_c @ m ) @ bot_bot_set_a ) @ bot_bot_set_a
        @ ( fun_up7010226539926187649od_c_a
          @ ^ [X: a] : none_l592525953500355997od_c_a
          @ ( initial_a_b_c @ m )
          @ ( some_l6142909491759833889od_c_a @ nil_Pr1342775757158464060od_c_a ) )
        @ ( minus_minus_nat @ ( size_a_b_c @ m ) @ one_one_nat )
        @ q )
      = none_l592525953500355997od_c_a )
    = ( ~ ? [P3: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P3 )
            & ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P3 )
              = q )
            & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P3 ) @ ( minus_minus_nat @ ( size_a_b_c @ m ) @ one_one_nat ) ) ) ) ) ).

%------------------------------------------------------------------------------