%------------------------------------------------------------------------------
% 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    : Combinable_Wands/0003_CombinableWands/prob_00204_007357__7614796_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1441 ( 534 unt; 164 typ;   0 def)
%            Number of atoms       : 3853 (1351 equ;   0 cnn)
%            Maximal formula atoms :   21 (   3 avg)
%            Number of connectives : 11152 ( 448   ~;  53   |; 307   &;8563   @)
%                                         (   0 <=>;1781  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   7 avg)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  :  867 ( 867   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  152 ( 151 usr;  24 con; 0-5 aty)
%            Number of variables   : 3525 ( 194   ^;3200   !; 131   ?;3525   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:08:57.223
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
    set_set_set_state: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    set_option_state: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    set_set_state: $tType ).

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

thf(ty_n_t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    option_state: $tType ).

thf(ty_n_t__Filter__Ofilter_It__PartialHeapSA__Ostate_J,type,
    filter_state: $tType ).

thf(ty_n_t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    set_state: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

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

thf(ty_n_t__PartialHeapSA__Ostate,type,
    state: $tType ).

thf(ty_n_t__PosRat__Oprat,type,
    prat: $tType ).

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

% Explicit typings (151)
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    boolea5298875108296682874_state: ( set_state > set_state > set_state ) > ( set_state > set_state > set_state ) > ( set_state > set_state ) > set_state > set_state > $o ).

thf(sy_c_CombinableWands_OR,type,
    r: state > state > state ).

thf(sy_c_CombinableWands_Ocwand,type,
    cwand: set_state > set_state > set_state ).

thf(sy_c_CombinableWands_Ointuitionistic,type,
    intuitionistic: set_state > $o ).

thf(sy_c_CombinableWands_Omultiply,type,
    multiply: prat > state > state ).

thf(sy_c_CombinableWands_Omultiply__sem__assertion,type,
    multip8064567061438756306ertion: prat > set_state > set_state ).

thf(sy_c_CombinableWands_Oscalable,type,
    scalable: state > state > $o ).

thf(sy_c_CombinableWands_Oscaled,type,
    scaled: state > set_state ).

thf(sy_c_CombinableWands_Owand,type,
    wand: set_state > set_state > set_state ).

thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
    cofinite_nat: filter_nat ).

thf(sy_c_Filter_Ocofinite_001t__PartialHeapSA__Ostate,type,
    cofinite_state: filter_state ).

thf(sy_c_Finite__Set_OFpow_001t__PartialHeapSA__Ostate,type,
    finite_Fpow_state: set_state > set_set_state ).

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

thf(sy_c_Finite__Set_Ocard_001t__PartialHeapSA__Ostate,type,
    finite_card_state: set_state > nat ).

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

thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
    finite5523153139673422903on_nat: set_option_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    finite3180955649987104801_state: set_option_state > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__PartialHeapSA__Ostate,type,
    finite_finite_state: set_state > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    finite4951987536711252743_state: set_set_state > $o ).

thf(sy_c_Fun_Ofun__upd_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up8843634000204221123_state: ( state > option_state ) > state > option_state > state > option_state ).

thf(sy_c_Fun_Ofun__upd_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    fun_upd_state_state: ( state > state ) > state > state > state > state ).

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_001t__Nat__Onat_001t__PartialHeapSA__Ostate,type,
    inj_on_nat_state: ( nat > state ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__Nat__Onat,type,
    inj_on_state_nat: ( state > nat ) > set_state > $o ).

thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    inj_on3577428053172332983_state: ( state > option_state ) > set_state > $o ).

thf(sy_c_Fun_Oinj__on_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    inj_on_state_state: ( state > state ) > set_state > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    the_in3035302284364921129_state: set_state > ( state > state ) > state > state ).

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__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__PartialHeapSA__Ostate_J,type,
    minus_3933957440811877961_state: set_state > set_state > set_state ).

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_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    uminus472742206872269241_state: set_state > set_state ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

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

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    inf_inf_set_state: set_state > set_state > set_state ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    semila3858001251800985922_state: ( set_state > set_state > set_state ) > set_state > ( set_state > set_state > $o ) > ( set_state > set_state > $o ) > $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__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__PartialHeapSA__Ostate_J,type,
    sup_sup_set_state: set_state > set_state > set_state ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    sup_su4188871578264421970_state: set_set_state > set_set_state > set_set_state ).

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

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__PartialHeapSA__Ostate_001t__Nat__Onat,type,
    lattic8930993470425071781te_nat: ( state > nat ) > set_state > state ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
    lattic5238388535129920115in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    lattic4879230916095660051_state: set_set_state > set_state ).

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

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    lattic1454283544731368441_state: set_set_state > set_state ).

thf(sy_c_Map_Odom_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    dom_state_state: ( state > option_state ) > set_state ).

thf(sy_c_Map_Omap__add_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    map_add_state_state: ( state > option_state ) > ( state > option_state ) > state > option_state ).

thf(sy_c_Map_Orestrict__map_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    restri2287918369865870758_state: ( state > option_state ) > set_state > state > option_state ).

thf(sy_c_Mask_Onull_001t__Nat__Onat,type,
    null_nat: nat ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__PartialHeapSA__Ostate,type,
    none_state: option_state ).

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

thf(sy_c_Option_Ooption_OSome_001t__PartialHeapSA__Ostate,type,
    some_state: state > option_state ).

thf(sy_c_Option_Ooption_Oset__option_001t__PartialHeapSA__Ostate,type,
    set_option_state2: option_state > set_state ).

thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

thf(sy_c_Option_Ooption_Othe_001t__PartialHeapSA__Ostate,type,
    the_state: option_state > state ).

thf(sy_c_Option_Othese_001t__PartialHeapSA__Ostate,type,
    these_state: set_option_state > set_state ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
    bot_bot_state_o: state > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__PartialHeapSA__Ostate_J,type,
    bot_bot_filter_state: filter_state ).

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__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    bot_bo710180891245420500_state: set_option_state ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    bot_bot_set_state: set_state ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    bot_bo2271482359692755898_state: set_set_state ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
    bot_bo7389043141884672880_state: set_set_set_state ).

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_It__PartialHeapSA__Ostate_J,type,
    ord_less_set_state: set_state > set_state > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    ord_le7116032884704190368_state: set_option_state > set_option_state > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    ord_le2494988322063910608_state: set_state > set_state > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    ord_le5175021213330142598_state: set_set_state > set_set_state > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
    ord_le3154822624800881980_state: set_set_set_state > set_set_set_state > $o ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    order_2642746146112740183_state: ( set_state > $o ) > set_state ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
    top_top_state_o: state > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    top_to7666338855062656496_state: set_option_state ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    top_top_set_state: set_state ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    top_to5262587396890829782_state: set_set_state ).

thf(sy_c_PackageLogic_Opackage__logic_001t__PartialHeapSA__Ostate,type,
    package_logic_state: ( state > state > option_state ) > ( state > state ) > state > ( state > $o ) > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Obool__conj_001t__PartialHeapSA__Ostate,type,
    packag1330488657391168505_state: ( state > $o ) > ( state > $o ) > state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Ointuitionistic_001t__PartialHeapSA__Ostate,type,
    packag8361946002163212404_state: ( state > state > option_state ) > ( state > $o ) > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Omono__pruner_001t__PartialHeapSA__Ostate,type,
    packag2456304381420842418_state: ( state > state > option_state ) > ( state > $o ) > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Omono__pure__cond_001t__PartialHeapSA__Ostate,type,
    packag6595153354952283300_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Omono__transformer_001t__PartialHeapSA__Ostate,type,
    packag4817583915586052229_state: ( state > state > option_state ) > state > ( state > state ) > $o ).

thf(sy_c_PackageLogic_Opackage__logic__axioms_001t__PartialHeapSA__Ostate,type,
    packag2647621270594721818_state: ( state > state > option_state ) > state > ( state > $o ) > $o ).

thf(sy_c_PartialHeapSA_Oadd__set,type,
    add_set: set_state > set_state > set_state ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__PartialHeapSA__Ostate,type,
    compat2278460363914054422_state: option_state > option_state > $o ).

thf(sy_c_PartialHeapSA_Ocore,type,
    core: state > state ).

thf(sy_c_PartialHeapSA_Odefined,type,
    defined: state > state > $o ).

thf(sy_c_PartialHeapSA_Ogreater,type,
    greater: state > state > $o ).

thf(sy_c_PartialHeapSA_Ogreater__set,type,
    greater_set: set_state > set_state > $o ).

thf(sy_c_PartialHeapSA_Ominus,type,
    minus: state > state > state ).

thf(sy_c_PartialHeapSA_Oplus,type,
    plus: state > state > option_state ).

thf(sy_c_PartialHeapSA_Ostable,type,
    stable: state > $o ).

thf(sy_c_PartialHeapSA_Ounit,type,
    unit: state ).

thf(sy_c_SepAlgebra_Osep__algebra_001t__PartialHeapSA__Ostate,type,
    sep_algebra_state: ( state > state > option_state ) > ( state > state ) > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Oadd__set_001t__PartialHeapSA__Ostate,type,
    sep_add_set_state: ( state > state > option_state ) > set_state > set_state > set_state ).

thf(sy_c_SepAlgebra_Osep__algebra_Oequiv_001t__PartialHeapSA__Ostate,type,
    sep_equiv_state: ( state > state > option_state ) > set_state > set_state > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Ogreater__set_001t__PartialHeapSA__Ostate,type,
    sep_gr7105985528888466643_state: ( state > state > option_state ) > set_state > set_state > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Omax__projection__prop_001t__PartialHeapSA__Ostate,type,
    sep_ma8214210560313151521_state: ( state > state > option_state ) > ( state > $o ) > ( state > state ) > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Ominus_001t__PartialHeapSA__Ostate,type,
    sep_minus_state: ( state > state > option_state ) > ( state > state ) > state > state > state ).

thf(sy_c_SepAlgebra_Osep__algebra_Omono__prop_001t__PartialHeapSA__Ostate,type,
    sep_mono_prop_state: ( state > state > option_state ) > ( state > $o ) > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Opure_001t__PartialHeapSA__Ostate,type,
    sep_pure_state: ( state > state > option_state ) > state > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Osetify_001t__PartialHeapSA__Ostate,type,
    sep_setify_state: ( state > $o ) > set_state > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Osplus_001t__PartialHeapSA__Ostate,type,
    sep_splus_state: ( state > state > option_state ) > option_state > option_state > option_state ).

thf(sy_c_SepAlgebra_Osep__algebra_Oup__closed_001t__PartialHeapSA__Ostate,type,
    sep_up_closed_state: ( state > state > option_state ) > set_state > $o ).

thf(sy_c_SepAlgebra_Osep__algebra_Oupper__closure_001t__PartialHeapSA__Ostate,type,
    sep_up1246176804924251236_state: ( state > state > option_state ) > set_state > set_state ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    collect_option_state: ( option_state > $o ) > set_option_state ).

thf(sy_c_Set_OCollect_001t__PartialHeapSA__Ostate,type,
    collect_state: ( state > $o ) > set_state ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    collect_set_state: ( set_state > $o ) > set_set_state ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    collec1217220441665645051_state: ( set_set_state > $o ) > set_set_set_state ).

thf(sy_c_Set_OPow_001t__PartialHeapSA__Ostate,type,
    pow_state: set_state > set_set_state ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
    image_nat_option_nat: ( nat > option_nat ) > set_nat > set_option_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__PartialHeapSA__Ostate,type,
    image_nat_state: ( nat > state ) > set_nat > set_state ).

thf(sy_c_Set_Oimage_001t__PartialHeapSA__Ostate_001t__Nat__Onat,type,
    image_state_nat: ( state > nat ) > set_state > set_nat ).

thf(sy_c_Set_Oimage_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    image_6076465424260689483_state: ( state > option_state ) > set_state > set_option_state ).

thf(sy_c_Set_Oimage_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    image_state_state: ( state > state ) > set_state > set_state ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__PartialHeapSA__Ostate_J_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    image_2476256681063834599_state: ( set_state > set_state ) > set_set_state > set_set_state ).

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

thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    insert_option_state: option_state > set_option_state > set_option_state ).

thf(sy_c_Set_Oinsert_001t__PartialHeapSA__Ostate,type,
    insert_state: state > set_state > set_state ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    insert_set_state: set_state > set_set_state > set_set_state ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    insert_set_set_state: set_set_state > set_set_set_state > set_set_set_state ).

thf(sy_c_Set_Ois__empty_001t__PartialHeapSA__Ostate,type,
    is_empty_state: set_state > $o ).

thf(sy_c_Set_Ois__singleton_001t__PartialHeapSA__Ostate,type,
    is_singleton_state: set_state > $o ).

thf(sy_c_Set_Opairwise_001t__PartialHeapSA__Ostate,type,
    pairwise_state: ( state > state > $o ) > set_state > $o ).

thf(sy_c_Set_Oremove_001t__PartialHeapSA__Ostate,type,
    remove_state: state > set_state > set_state ).

thf(sy_c_Set_Othe__elem_001t__PartialHeapSA__Ostate,type,
    the_elem_state: set_state > state ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__PartialHeapSA__Ostate,type,
    vimage_nat_state: ( nat > state ) > set_state > set_nat ).

thf(sy_c_Set_Ovimage_001t__PartialHeapSA__Ostate_001t__Nat__Onat,type,
    vimage_state_nat: ( state > nat ) > set_nat > set_state ).

thf(sy_c_Set_Ovimage_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    vimage_state_state: ( state > state ) > set_state > set_state ).

thf(sy_c_Zorn_Ochains_001t__PartialHeapSA__Ostate,type,
    chains_state: set_set_state > set_set_set_state ).

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

thf(sy_c_member_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    member_option_state: option_state > set_option_state > $o ).

thf(sy_c_member_001t__PartialHeapSA__Ostate,type,
    member_state: state > set_state > $o ).

thf(sy_c_member_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    member_set_state: set_state > set_set_state > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    member_set_set_state: set_set_state > set_set_set_state > $o ).

thf(sy_v_A,type,
    a: set_state ).

thf(sy_v_p,type,
    p: prat ).

thf(sy_v_x,type,
    x: state ).

% Relevant facts (1276)
thf(fact_0_assms,axiom,
    member_state @ x @ ( multip8064567061438756306ertion @ p @ a ) ).

% assms
thf(fact_1_intuitionistic__def,axiom,
    ( intuitionistic
    = ( ^ [A: set_state] :
        ! [A2: state,B: state] :
          ( ( ( greater @ A2 @ B )
            & ( member_state @ B @ A ) )
         => ( member_state @ A2 @ A ) ) ) ) ).

% intuitionistic_def
thf(fact_2_PartialSA_Osucc__refl,axiom,
    ! [A3: state] : ( greater @ A3 @ A3 ) ).

% PartialSA.succ_refl
thf(fact_3_PartialSA_Osucc__trans,axiom,
    ! [A3: state,B2: state,C: state] :
      ( ( greater @ A3 @ B2 )
     => ( ( greater @ B2 @ C )
       => ( greater @ A3 @ C ) ) ) ).

% PartialSA.succ_trans
thf(fact_4_PartialSA_Osucc__antisym,axiom,
    ! [A3: state,B2: state] :
      ( ( greater @ A3 @ B2 )
     => ( ( greater @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% PartialSA.succ_antisym
thf(fact_5_R__smaller,axiom,
    ! [W: state,A3: state] : ( greater @ W @ ( r @ A3 @ W ) ) ).

% R_smaller
thf(fact_6_w__in__scaled,axiom,
    ! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).

% w_in_scaled
thf(fact_7_PartialSA_Ogreater__setI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ! [A5: state] :
          ( ( member_state @ A5 @ A4 )
         => ? [X: state] :
              ( ( member_state @ X @ B3 )
              & ( greater @ A5 @ X ) ) )
     => ( greater_set @ A4 @ B3 ) ) ).

% PartialSA.greater_setI
thf(fact_8_PartialSA_Ogreater__set__def,axiom,
    ( greater_set
    = ( ^ [A: set_state,B4: set_state] :
        ! [X2: state] :
          ( ( member_state @ X2 @ A )
         => ? [Y: state] :
              ( ( member_state @ Y @ B4 )
              & ( greater @ X2 @ Y ) ) ) ) ) ).

% PartialSA.greater_set_def
thf(fact_9_PartialSA_Osmaller__compatible,axiom,
    ! [A6: state,B2: state,A3: state] :
      ( ( defined @ A6 @ B2 )
     => ( ( greater @ A6 @ A3 )
       => ( defined @ A3 @ B2 ) ) ) ).

% PartialSA.smaller_compatible
thf(fact_10_PartialSA_Ounit__smaller,axiom,
    ! [Phi: state] : ( greater @ Phi @ unit ) ).

% PartialSA.unit_smaller
thf(fact_11_PartialSA_Ominus__default,axiom,
    ! [B2: state,A3: state] :
      ( ~ ( greater @ B2 @ A3 )
     => ( ( minus @ B2 @ A3 )
        = B2 ) ) ).

% PartialSA.minus_default
thf(fact_12_PartialSA_Ominus__smaller,axiom,
    ! [X3: state,A3: state] :
      ( ( greater @ X3 @ A3 )
     => ( greater @ X3 @ ( minus @ X3 @ A3 ) ) ) ).

% PartialSA.minus_smaller
thf(fact_13_PartialSA_Osucc__set__trans,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( greater_set @ A4 @ B3 )
     => ( ( greater_set @ B3 @ C2 )
       => ( greater_set @ A4 @ C2 ) ) ) ).

% PartialSA.succ_set_trans
thf(fact_14_PartialSA_Olarger__set__refl,axiom,
    ! [A4: set_state] : ( greater_set @ A4 @ A4 ) ).

% PartialSA.larger_set_refl
thf(fact_15_R__compatible__same,axiom,
    ! [A3: state,W: state] :
      ( ( defined @ A3 @ W )
     => ( ( r @ A3 @ W )
        = W ) ) ).

% R_compatible_same
thf(fact_16_PartialSA_Ogreater__minus__trans,axiom,
    ! [Y2: state,X3: state,A3: state] :
      ( ( greater @ Y2 @ X3 )
     => ( ( greater @ X3 @ A3 )
       => ( greater @ ( minus @ Y2 @ A3 ) @ ( minus @ X3 @ A3 ) ) ) ) ).

% PartialSA.greater_minus_trans
thf(fact_17_scalable__def,axiom,
    ( scalable
    = ( ^ [W2: state,A2: state] :
        ! [X2: state] :
          ( ( member_state @ X2 @ ( scaled @ W2 ) )
         => ~ ( defined @ A2 @ X2 ) ) ) ) ).

% scalable_def
thf(fact_18_PartialSA_Osmaller__compatible__core,axiom,
    ! [Y2: state,X3: state] :
      ( ( greater @ Y2 @ X3 )
     => ( defined @ X3 @ ( core @ Y2 ) ) ) ).

% PartialSA.smaller_compatible_core
thf(fact_19_PartialSA_Obigger__set,axiom,
    ! [A7: set_state,A4: set_state,B3: set_state] :
      ( ( greater_set @ A7 @ A4 )
     => ( greater_set @ ( add_set @ A7 @ B3 ) @ ( add_set @ A4 @ B3 ) ) ) ).

% PartialSA.bigger_set
thf(fact_20_PartialSA_Osub__bigger,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( greater_set @ A4 @ B3 ) ) ).

% PartialSA.sub_bigger
thf(fact_21_stable__unit,axiom,
    stable @ unit ).

% stable_unit
thf(fact_22_PartialSA_Omono__transformer__def,axiom,
    ! [T: state > state] :
      ( ( packag4817583915586052229_state @ plus @ unit @ T )
      = ( ! [Phi2: state,Phi3: state] :
            ( ( greater @ Phi3 @ Phi2 )
           => ( greater @ ( T @ Phi3 ) @ ( T @ Phi2 ) ) )
        & ( ( T @ unit )
          = unit ) ) ) ).

% PartialSA.mono_transformer_def
thf(fact_23_PartialSA_Ounit__core,axiom,
    ( ( core @ unit )
    = unit ) ).

% PartialSA.unit_core
thf(fact_24_PartialSA_Ominus__core,axiom,
    ! [A3: state,B2: state] :
      ( ( core @ ( minus @ A3 @ B2 ) )
      = ( core @ A3 ) ) ).

% PartialSA.minus_core
thf(fact_25_PartialSA_Ominus__core__weaker,axiom,
    ! [A3: state,B2: state] :
      ( ( core @ ( minus @ A3 @ B2 ) )
      = ( minus @ ( core @ A3 ) @ ( core @ B2 ) ) ) ).

% PartialSA.minus_core_weaker
thf(fact_26_PartialSA_Obigger__singleton,axiom,
    ! [Phi4: state,Phi: state] :
      ( ( greater @ Phi4 @ Phi )
     => ( greater_set @ ( insert_state @ Phi4 @ bot_bot_set_state ) @ ( insert_state @ Phi @ bot_bot_set_state ) ) ) ).

% PartialSA.bigger_singleton
thf(fact_27_PartialSA_Ocore__mono,axiom,
    ! [A3: state,B2: state] :
      ( ( greater @ A3 @ B2 )
     => ( greater @ ( core @ A3 ) @ ( core @ B2 ) ) ) ).

% PartialSA.core_mono
thf(fact_28_PartialSA_Ominus__sum,axiom,
    ! [A3: state,B2: state,C: state,X3: state] :
      ( ( ( some_state @ A3 )
        = ( plus @ B2 @ C ) )
     => ( ( greater @ X3 @ A3 )
       => ( ( minus @ X3 @ A3 )
          = ( minus @ ( minus @ X3 @ B2 ) @ C ) ) ) ) ).

% PartialSA.minus_sum
thf(fact_29_PartialSA_Oadd__set__asso,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( add_set @ ( add_set @ A4 @ B3 ) @ C2 )
      = ( add_set @ A4 @ ( add_set @ B3 @ C2 ) ) ) ).

% PartialSA.add_set_asso
thf(fact_30_PartialSA_Oadd__set__elem,axiom,
    ! [Phi: state,A4: set_state,B3: set_state] :
      ( ( member_state @ Phi @ ( add_set @ A4 @ B3 ) )
      = ( ? [A2: state,B: state] :
            ( ( ( some_state @ Phi )
              = ( plus @ A2 @ B ) )
            & ( member_state @ A2 @ A4 )
            & ( member_state @ B @ B3 ) ) ) ) ).

% PartialSA.add_set_elem
thf(fact_31_PartialSA_Oextract__core,axiom,
    ! [B2: state,A3: state,X3: state] :
      ( ( ( ( some_state @ B2 )
          = ( plus @ A3 @ X3 ) )
        & ( greater @ X3 @ ( core @ B2 ) ) )
     => ( ( core @ X3 )
        = ( core @ B2 ) ) ) ).

% PartialSA.extract_core
thf(fact_32_PartialSA_Ominus__exists,axiom,
    ! [B2: state,A3: state] :
      ( ( greater @ B2 @ A3 )
     => ? [X4: state] :
          ( ( ( some_state @ B2 )
            = ( plus @ A3 @ X4 ) )
          & ( greater @ X4 @ ( core @ B2 ) ) ) ) ).

% PartialSA.minus_exists
thf(fact_33_PartialSA_Ominus__unique,axiom,
    ! [B2: state,A3: state,X3: state,Y2: state] :
      ( ( ( ( some_state @ B2 )
          = ( plus @ A3 @ X3 ) )
        & ( greater @ X3 @ ( core @ B2 ) ) )
     => ( ( ( ( some_state @ B2 )
            = ( plus @ A3 @ Y2 ) )
          & ( greater @ Y2 @ ( core @ B2 ) ) )
       => ( X3 = Y2 ) ) ) ).

% PartialSA.minus_unique
thf(fact_34_PartialSA_Oadd__set__commm,axiom,
    ( add_set
    = ( ^ [A: set_state,B4: set_state] : ( add_set @ B4 @ A ) ) ) ).

% PartialSA.add_set_commm
thf(fact_35_PartialSA_Oempty__set__sum,axiom,
    ! [A4: set_state] :
      ( ( add_set @ bot_bot_set_state @ A4 )
      = bot_bot_set_state ) ).

% PartialSA.empty_set_sum
thf(fact_36_PartialSA_Osmaller__than__core,axiom,
    ! [Y2: state,X3: state,Z: state] :
      ( ( greater @ Y2 @ X3 )
     => ( ( ( some_state @ Z )
          = ( plus @ X3 @ ( core @ Y2 ) ) )
       => ( ( core @ Z )
          = ( core @ Y2 ) ) ) ) ).

% PartialSA.smaller_than_core
thf(fact_37_PartialSA_Osum__then__singleton,axiom,
    ! [A3: state,B2: state,C: state] :
      ( ( ( some_state @ A3 )
        = ( plus @ B2 @ C ) )
      = ( ( insert_state @ A3 @ bot_bot_set_state )
        = ( add_set @ ( insert_state @ B2 @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).

% PartialSA.sum_then_singleton
thf(fact_38_PartialSA_Ox__elem__set__product,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ( member_state @ X3 @ ( add_set @ A4 @ B3 ) )
      = ( ? [A2: state,B: state] :
            ( ( member_state @ A2 @ A4 )
            & ( member_state @ B @ B3 )
            & ( ( some_state @ X3 )
              = ( plus @ A2 @ B ) ) ) ) ) ).

% PartialSA.x_elem_set_product
thf(fact_39_PartialSA_Obigger__core__sum__defined,axiom,
    ! [A3: state,B2: state] :
      ( ( greater @ ( core @ A3 ) @ B2 )
     => ( ( some_state @ A3 )
        = ( plus @ A3 @ B2 ) ) ) ).

% PartialSA.bigger_core_sum_defined
thf(fact_40_PartialSA_Ogreater__than__sum__exists,axiom,
    ! [A3: state,B2: state,B1: state,B22: state] :
      ( ( greater @ A3 @ B2 )
     => ( ( ( some_state @ B2 )
          = ( plus @ B1 @ B22 ) )
       => ? [R: state] :
            ( ( ( some_state @ A3 )
              = ( plus @ R @ B22 ) )
            & ( greater @ ( core @ R ) @ ( core @ A3 ) )
            & ( greater @ R @ B1 ) ) ) ) ).

% PartialSA.greater_than_sum_exists
thf(fact_41_PartialSA_Ominus__equiv__def__any__elem,axiom,
    ! [X3: state,A3: state,B2: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ A3 @ B2 ) )
     => ( ( some_state @ ( minus @ X3 @ A3 ) )
        = ( plus @ B2 @ ( core @ X3 ) ) ) ) ).

% PartialSA.minus_equiv_def_any_elem
thf(fact_42_asso1,axiom,
    ! [A3: state,B2: state,Ab: state,C: state,Bc: state] :
      ( ( ( ( plus @ A3 @ B2 )
          = ( some_state @ Ab ) )
        & ( ( plus @ B2 @ C )
          = ( some_state @ Bc ) ) )
     => ( ( plus @ Ab @ C )
        = ( plus @ A3 @ Bc ) ) ) ).

% asso1
thf(fact_43_core__max,axiom,
    ! [X3: state,C: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ X3 @ C ) )
     => ? [R: state] :
          ( ( some_state @ ( core @ X3 ) )
          = ( plus @ C @ R ) ) ) ).

% core_max
thf(fact_44_core__sum,axiom,
    ! [C: state,A3: state,B2: state] :
      ( ( ( some_state @ C )
        = ( plus @ A3 @ B2 ) )
     => ( ( some_state @ ( core @ C ) )
        = ( plus @ ( core @ A3 ) @ ( core @ B2 ) ) ) ) ).

% core_sum
thf(fact_45_mem__Collect__eq,axiom,
    ! [A3: option_state,P: option_state > $o] :
      ( ( member_option_state @ A3 @ ( collect_option_state @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A3: set_state,P: set_state > $o] :
      ( ( member_set_state @ A3 @ ( collect_set_state @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( member_nat @ A3 @ ( collect_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A3: set_set_state,P: set_set_state > $o] :
      ( ( member_set_set_state @ A3 @ ( collec1217220441665645051_state @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A3: state,P: state > $o] :
      ( ( member_state @ A3 @ ( collect_state @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A4: set_option_state] :
      ( ( collect_option_state
        @ ^ [X2: option_state] : ( member_option_state @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A4: set_set_state] :
      ( ( collect_set_state
        @ ^ [X2: set_state] : ( member_set_state @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A4: set_set_set_state] :
      ( ( collec1217220441665645051_state
        @ ^ [X2: set_set_state] : ( member_set_set_state @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A4: set_state] :
      ( ( collect_state
        @ ^ [X2: state] : ( member_state @ X2 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_55_Collect__cong,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ! [X4: state] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_state @ P )
        = ( collect_state @ Q ) ) ) ).

% Collect_cong
thf(fact_56_positivity,axiom,
    ! [A3: state,B2: state,C: state] :
      ( ( ( plus @ A3 @ B2 )
        = ( some_state @ C ) )
     => ( ( ( some_state @ C )
          = ( plus @ C @ C ) )
       => ( ( some_state @ A3 )
          = ( plus @ A3 @ A3 ) ) ) ) ).

% positivity
thf(fact_57_stable__sum,axiom,
    ! [A3: state,B2: state,X3: state] :
      ( ( stable @ A3 )
     => ( ( stable @ B2 )
       => ( ( ( some_state @ X3 )
            = ( plus @ A3 @ B2 ) )
         => ( stable @ X3 ) ) ) ) ).

% stable_sum
thf(fact_58_commutative,axiom,
    ( plus
    = ( ^ [A2: state,B: state] : ( plus @ B @ A2 ) ) ) ).

% commutative
thf(fact_59_cancellative,axiom,
    ! [A3: state,B2: state,X3: state,Y2: state] :
      ( ( ( some_state @ A3 )
        = ( plus @ B2 @ X3 ) )
     => ( ( ( some_state @ A3 )
          = ( plus @ B2 @ Y2 ) )
       => ( ( ( core @ X3 )
            = ( core @ Y2 ) )
         => ( X3 = Y2 ) ) ) ) ).

% cancellative
thf(fact_60_core__is__pure,axiom,
    ! [X3: state] :
      ( ( some_state @ ( core @ X3 ) )
      = ( plus @ ( core @ X3 ) @ ( core @ X3 ) ) ) ).

% core_is_pure
thf(fact_61_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X2: state] : ( plus @ X2 @ ( core @ X2 ) ) ) ) ).

% core_is_smaller
thf(fact_62_PartialSA_Obigger__sum__smaller,axiom,
    ! [C: state,A3: state,B2: state,A6: state] :
      ( ( ( some_state @ C )
        = ( plus @ A3 @ B2 ) )
     => ( ( greater @ A3 @ A6 )
       => ? [B5: state] :
            ( ( greater @ B5 @ B2 )
            & ( ( some_state @ C )
              = ( plus @ A6 @ B5 ) ) ) ) ) ).

% PartialSA.bigger_sum_smaller
thf(fact_63_PartialSA_Oaddition__bigger,axiom,
    ! [A6: state,A3: state,X5: state,B2: state,X3: state] :
      ( ( greater @ A6 @ A3 )
     => ( ( ( some_state @ X5 )
          = ( plus @ A6 @ B2 ) )
       => ( ( ( some_state @ X3 )
            = ( plus @ A3 @ B2 ) )
         => ( greater @ X5 @ X3 ) ) ) ) ).

% PartialSA.addition_bigger
thf(fact_64_PartialSA_Ogreater__equiv,axiom,
    ( greater
    = ( ^ [A2: state,B: state] :
        ? [C3: state] :
          ( ( some_state @ A2 )
          = ( plus @ C3 @ B ) ) ) ) ).

% PartialSA.greater_equiv
thf(fact_65_PartialSA_Ogreater__def,axiom,
    ( greater
    = ( ^ [A2: state,B: state] :
        ? [C3: state] :
          ( ( some_state @ A2 )
          = ( plus @ B @ C3 ) ) ) ) ).

% PartialSA.greater_def
thf(fact_66_PartialSA_Obigger__sum,axiom,
    ! [Phi: state,A3: state,B2: state,Phi4: state] :
      ( ( ( some_state @ Phi )
        = ( plus @ A3 @ B2 ) )
     => ( ( greater @ Phi4 @ Phi )
       => ? [B5: state] :
            ( ( greater @ B5 @ B2 )
            & ( ( some_state @ Phi4 )
              = ( plus @ A3 @ B5 ) ) ) ) ) ).

% PartialSA.bigger_sum
thf(fact_67_unit__neutral,axiom,
    ( some_state
    = ( ^ [A2: state] : ( plus @ A2 @ unit ) ) ) ).

% unit_neutral
thf(fact_68_PartialSA_Odefined__sum__move,axiom,
    ! [A3: state,B2: state,X3: state,Y2: state,A6: state] :
      ( ( defined @ A3 @ B2 )
     => ( ( ( some_state @ B2 )
          = ( plus @ X3 @ Y2 ) )
       => ( ( ( some_state @ A6 )
            = ( plus @ A3 @ X3 ) )
         => ( defined @ A6 @ Y2 ) ) ) ) ).

% PartialSA.defined_sum_move
thf(fact_69_PartialSA_Ominus__equiv__def,axiom,
    ! [B2: state,A3: state] :
      ( ( greater @ B2 @ A3 )
     => ( ( ( some_state @ B2 )
          = ( plus @ A3 @ ( minus @ B2 @ A3 ) ) )
        & ( greater @ ( minus @ B2 @ A3 ) @ ( core @ B2 ) ) ) ) ).

% PartialSA.minus_equiv_def
thf(fact_70_PartialSA_OminusI,axiom,
    ! [B2: state,A3: state,X3: state] :
      ( ( ( some_state @ B2 )
        = ( plus @ A3 @ X3 ) )
     => ( ( greater @ X3 @ ( core @ B2 ) )
       => ( X3
          = ( minus @ B2 @ A3 ) ) ) ) ).

% PartialSA.minusI
thf(fact_71_PartialSA_Oprove__last__completeness,axiom,
    ! [A6: state,A3: state,Nf1: state,F2: state] :
      ( ( greater @ A6 @ A3 )
     => ( ( ( some_state @ A3 )
          = ( plus @ Nf1 @ F2 ) )
       => ( greater @ ( minus @ A6 @ Nf1 ) @ F2 ) ) ) ).

% PartialSA.prove_last_completeness
thf(fact_72_PartialSA_Ominus__and__plus,axiom,
    ! [Omega: state,Omega2: state,R2: state,A3: state] :
      ( ( ( some_state @ Omega )
        = ( plus @ Omega2 @ R2 ) )
     => ( ( greater @ Omega2 @ A3 )
       => ( ( some_state @ ( minus @ Omega @ A3 ) )
          = ( plus @ ( minus @ Omega2 @ A3 ) @ R2 ) ) ) ) ).

% PartialSA.minus_and_plus
thf(fact_73_PartialSA_Ominus__bigger,axiom,
    ! [X3: state,A3: state,B2: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ A3 @ B2 ) )
     => ( greater @ ( minus @ X3 @ A3 ) @ B2 ) ) ).

% PartialSA.minus_bigger
thf(fact_74_PartialSA_Ominus__some,axiom,
    ! [A3: state,B2: state] :
      ( ( greater @ A3 @ B2 )
     => ( ( some_state @ A3 )
        = ( plus @ B2 @ ( minus @ A3 @ B2 ) ) ) ) ).

% PartialSA.minus_some
thf(fact_75_singleton__insert__inj__eq_H,axiom,
    ! [A3: option_state,A4: set_option_state,B2: option_state] :
      ( ( ( insert_option_state @ A3 @ A4 )
        = ( insert_option_state @ B2 @ bot_bo710180891245420500_state ) )
      = ( ( A3 = B2 )
        & ( ord_le7116032884704190368_state @ A4 @ ( insert_option_state @ B2 @ bot_bo710180891245420500_state ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_76_singleton__insert__inj__eq_H,axiom,
    ! [A3: set_state,A4: set_set_state,B2: set_state] :
      ( ( ( insert_set_state @ A3 @ A4 )
        = ( insert_set_state @ B2 @ bot_bo2271482359692755898_state ) )
      = ( ( A3 = B2 )
        & ( ord_le5175021213330142598_state @ A4 @ ( insert_set_state @ B2 @ bot_bo2271482359692755898_state ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_77_singleton__insert__inj__eq_H,axiom,
    ! [A3: nat,A4: set_nat,B2: nat] :
      ( ( ( insert_nat @ A3 @ A4 )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A3 = B2 )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_78_singleton__insert__inj__eq_H,axiom,
    ! [A3: state,A4: set_state,B2: state] :
      ( ( ( insert_state @ A3 @ A4 )
        = ( insert_state @ B2 @ bot_bot_set_state ) )
      = ( ( A3 = B2 )
        & ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ B2 @ bot_bot_set_state ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_79_singleton__insert__inj__eq,axiom,
    ! [B2: option_state,A3: option_state,A4: set_option_state] :
      ( ( ( insert_option_state @ B2 @ bot_bo710180891245420500_state )
        = ( insert_option_state @ A3 @ A4 ) )
      = ( ( A3 = B2 )
        & ( ord_le7116032884704190368_state @ A4 @ ( insert_option_state @ B2 @ bot_bo710180891245420500_state ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_80_singleton__insert__inj__eq,axiom,
    ! [B2: set_state,A3: set_state,A4: set_set_state] :
      ( ( ( insert_set_state @ B2 @ bot_bo2271482359692755898_state )
        = ( insert_set_state @ A3 @ A4 ) )
      = ( ( A3 = B2 )
        & ( ord_le5175021213330142598_state @ A4 @ ( insert_set_state @ B2 @ bot_bo2271482359692755898_state ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_81_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A3: nat,A4: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A3 @ A4 ) )
      = ( ( A3 = B2 )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_82_singleton__insert__inj__eq,axiom,
    ! [B2: state,A3: state,A4: set_state] :
      ( ( ( insert_state @ B2 @ bot_bot_set_state )
        = ( insert_state @ A3 @ A4 ) )
      = ( ( A3 = B2 )
        & ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ B2 @ bot_bot_set_state ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_83_insert__subset,axiom,
    ! [X3: option_state,A4: set_option_state,B3: set_option_state] :
      ( ( ord_le7116032884704190368_state @ ( insert_option_state @ X3 @ A4 ) @ B3 )
      = ( ( member_option_state @ X3 @ B3 )
        & ( ord_le7116032884704190368_state @ A4 @ B3 ) ) ) ).

% insert_subset
thf(fact_84_insert__subset,axiom,
    ! [X3: set_set_state,A4: set_set_set_state,B3: set_set_set_state] :
      ( ( ord_le3154822624800881980_state @ ( insert_set_set_state @ X3 @ A4 ) @ B3 )
      = ( ( member_set_set_state @ X3 @ B3 )
        & ( ord_le3154822624800881980_state @ A4 @ B3 ) ) ) ).

% insert_subset
thf(fact_85_insert__subset,axiom,
    ! [X3: set_state,A4: set_set_state,B3: set_set_state] :
      ( ( ord_le5175021213330142598_state @ ( insert_set_state @ X3 @ A4 ) @ B3 )
      = ( ( member_set_state @ X3 @ B3 )
        & ( ord_le5175021213330142598_state @ A4 @ B3 ) ) ) ).

% insert_subset
thf(fact_86_insert__subset,axiom,
    ! [X3: nat,A4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A4 ) @ B3 )
      = ( ( member_nat @ X3 @ B3 )
        & ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ).

% insert_subset
thf(fact_87_insert__subset,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( insert_state @ X3 @ A4 ) @ B3 )
      = ( ( member_state @ X3 @ B3 )
        & ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ).

% insert_subset
thf(fact_88_singletonI,axiom,
    ! [A3: set_set_state] : ( member_set_set_state @ A3 @ ( insert_set_set_state @ A3 @ bot_bo7389043141884672880_state ) ) ).

% singletonI
thf(fact_89_singletonI,axiom,
    ! [A3: option_state] : ( member_option_state @ A3 @ ( insert_option_state @ A3 @ bot_bo710180891245420500_state ) ) ).

% singletonI
thf(fact_90_singletonI,axiom,
    ! [A3: set_state] : ( member_set_state @ A3 @ ( insert_set_state @ A3 @ bot_bo2271482359692755898_state ) ) ).

% singletonI
thf(fact_91_singletonI,axiom,
    ! [A3: nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_92_singletonI,axiom,
    ! [A3: state] : ( member_state @ A3 @ ( insert_state @ A3 @ bot_bot_set_state ) ) ).

% singletonI
thf(fact_93_empty__subsetI,axiom,
    ! [A4: set_option_state] : ( ord_le7116032884704190368_state @ bot_bo710180891245420500_state @ A4 ) ).

% empty_subsetI
thf(fact_94_empty__subsetI,axiom,
    ! [A4: set_set_state] : ( ord_le5175021213330142598_state @ bot_bo2271482359692755898_state @ A4 ) ).

% empty_subsetI
thf(fact_95_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_96_empty__subsetI,axiom,
    ! [A4: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A4 ) ).

% empty_subsetI
thf(fact_97_subset__empty,axiom,
    ! [A4: set_option_state] :
      ( ( ord_le7116032884704190368_state @ A4 @ bot_bo710180891245420500_state )
      = ( A4 = bot_bo710180891245420500_state ) ) ).

% subset_empty
thf(fact_98_subset__empty,axiom,
    ! [A4: set_set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ bot_bo2271482359692755898_state )
      = ( A4 = bot_bo2271482359692755898_state ) ) ).

% subset_empty
thf(fact_99_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_100_subset__empty,axiom,
    ! [A4: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ bot_bot_set_state )
      = ( A4 = bot_bot_set_state ) ) ).

% subset_empty
thf(fact_101_in__cwand,axiom,
    ! [A4: set_state,W: state,B3: set_state] :
      ( ! [A5: state,X4: state] :
          ( ( ( member_state @ A5 @ A4 )
            & ( ( some_state @ X4 )
              = ( plus @ ( r @ A5 @ W ) @ A5 ) ) )
         => ( member_state @ X4 @ B3 ) )
     => ( member_state @ W @ ( cwand @ A4 @ B3 ) ) ) ).

% in_cwand
thf(fact_102_subset__singleton__iff,axiom,
    ! [X6: set_option_state,A3: option_state] :
      ( ( ord_le7116032884704190368_state @ X6 @ ( insert_option_state @ A3 @ bot_bo710180891245420500_state ) )
      = ( ( X6 = bot_bo710180891245420500_state )
        | ( X6
          = ( insert_option_state @ A3 @ bot_bo710180891245420500_state ) ) ) ) ).

% subset_singleton_iff
thf(fact_103_subset__singleton__iff,axiom,
    ! [X6: set_set_state,A3: set_state] :
      ( ( ord_le5175021213330142598_state @ X6 @ ( insert_set_state @ A3 @ bot_bo2271482359692755898_state ) )
      = ( ( X6 = bot_bo2271482359692755898_state )
        | ( X6
          = ( insert_set_state @ A3 @ bot_bo2271482359692755898_state ) ) ) ) ).

% subset_singleton_iff
thf(fact_104_subset__singleton__iff,axiom,
    ! [X6: set_nat,A3: nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
      = ( ( X6 = bot_bot_set_nat )
        | ( X6
          = ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_105_subset__singleton__iff,axiom,
    ! [X6: set_state,A3: state] :
      ( ( ord_le2494988322063910608_state @ X6 @ ( insert_state @ A3 @ bot_bot_set_state ) )
      = ( ( X6 = bot_bot_set_state )
        | ( X6
          = ( insert_state @ A3 @ bot_bot_set_state ) ) ) ) ).

% subset_singleton_iff
thf(fact_106_subset__singletonD,axiom,
    ! [A4: set_option_state,X3: option_state] :
      ( ( ord_le7116032884704190368_state @ A4 @ ( insert_option_state @ X3 @ bot_bo710180891245420500_state ) )
     => ( ( A4 = bot_bo710180891245420500_state )
        | ( A4
          = ( insert_option_state @ X3 @ bot_bo710180891245420500_state ) ) ) ) ).

% subset_singletonD
thf(fact_107_subset__singletonD,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ ( insert_set_state @ X3 @ bot_bo2271482359692755898_state ) )
     => ( ( A4 = bot_bo2271482359692755898_state )
        | ( A4
          = ( insert_set_state @ X3 @ bot_bo2271482359692755898_state ) ) ) ) ).

% subset_singletonD
thf(fact_108_subset__singletonD,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
     => ( ( A4 = bot_bot_set_nat )
        | ( A4
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_109_subset__singletonD,axiom,
    ! [A4: set_state,X3: state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) )
     => ( ( A4 = bot_bot_set_state )
        | ( A4
          = ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ).

% subset_singletonD
thf(fact_110_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_111_option_Oinject,axiom,
    ! [X22: state,Y22: state] :
      ( ( ( some_state @ X22 )
        = ( some_state @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_112_insert__absorb2,axiom,
    ! [X3: option_state,A4: set_option_state] :
      ( ( insert_option_state @ X3 @ ( insert_option_state @ X3 @ A4 ) )
      = ( insert_option_state @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_113_insert__absorb2,axiom,
    ! [X3: set_state,A4: set_set_state] :
      ( ( insert_set_state @ X3 @ ( insert_set_state @ X3 @ A4 ) )
      = ( insert_set_state @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_114_insert__absorb2,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A4 ) )
      = ( insert_nat @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_115_insert__absorb2,axiom,
    ! [X3: state,A4: set_state] :
      ( ( insert_state @ X3 @ ( insert_state @ X3 @ A4 ) )
      = ( insert_state @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_116_empty__iff,axiom,
    ! [C: set_set_state] :
      ~ ( member_set_set_state @ C @ bot_bo7389043141884672880_state ) ).

% empty_iff
thf(fact_117_empty__iff,axiom,
    ! [C: option_state] :
      ~ ( member_option_state @ C @ bot_bo710180891245420500_state ) ).

% empty_iff
thf(fact_118_empty__iff,axiom,
    ! [C: set_state] :
      ~ ( member_set_state @ C @ bot_bo2271482359692755898_state ) ).

% empty_iff
thf(fact_119_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_120_empty__iff,axiom,
    ! [C: state] :
      ~ ( member_state @ C @ bot_bot_set_state ) ).

% empty_iff
thf(fact_121_all__not__in__conv,axiom,
    ! [A4: set_set_set_state] :
      ( ( ! [X2: set_set_state] :
            ~ ( member_set_set_state @ X2 @ A4 ) )
      = ( A4 = bot_bo7389043141884672880_state ) ) ).

% all_not_in_conv
thf(fact_122_all__not__in__conv,axiom,
    ! [A4: set_option_state] :
      ( ( ! [X2: option_state] :
            ~ ( member_option_state @ X2 @ A4 ) )
      = ( A4 = bot_bo710180891245420500_state ) ) ).

% all_not_in_conv
thf(fact_123_all__not__in__conv,axiom,
    ! [A4: set_set_state] :
      ( ( ! [X2: set_state] :
            ~ ( member_set_state @ X2 @ A4 ) )
      = ( A4 = bot_bo2271482359692755898_state ) ) ).

% all_not_in_conv
thf(fact_124_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat @ X2 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_125_all__not__in__conv,axiom,
    ! [A4: set_state] :
      ( ( ! [X2: state] :
            ~ ( member_state @ X2 @ A4 ) )
      = ( A4 = bot_bot_set_state ) ) ).

% all_not_in_conv
thf(fact_126_Collect__empty__eq,axiom,
    ! [P: option_state > $o] :
      ( ( ( collect_option_state @ P )
        = bot_bo710180891245420500_state )
      = ( ! [X2: option_state] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_127_Collect__empty__eq,axiom,
    ! [P: set_state > $o] :
      ( ( ( collect_set_state @ P )
        = bot_bo2271482359692755898_state )
      = ( ! [X2: set_state] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_128_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_129_Collect__empty__eq,axiom,
    ! [P: state > $o] :
      ( ( ( collect_state @ P )
        = bot_bot_set_state )
      = ( ! [X2: state] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_130_empty__Collect__eq,axiom,
    ! [P: option_state > $o] :
      ( ( bot_bo710180891245420500_state
        = ( collect_option_state @ P ) )
      = ( ! [X2: option_state] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_131_empty__Collect__eq,axiom,
    ! [P: set_state > $o] :
      ( ( bot_bo2271482359692755898_state
        = ( collect_set_state @ P ) )
      = ( ! [X2: set_state] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_132_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_133_empty__Collect__eq,axiom,
    ! [P: state > $o] :
      ( ( bot_bot_set_state
        = ( collect_state @ P ) )
      = ( ! [X2: state] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_134_subsetI,axiom,
    ! [A4: set_option_state,B3: set_option_state] :
      ( ! [X4: option_state] :
          ( ( member_option_state @ X4 @ A4 )
         => ( member_option_state @ X4 @ B3 ) )
     => ( ord_le7116032884704190368_state @ A4 @ B3 ) ) ).

% subsetI
thf(fact_135_subsetI,axiom,
    ! [A4: set_set_set_state,B3: set_set_set_state] :
      ( ! [X4: set_set_state] :
          ( ( member_set_set_state @ X4 @ A4 )
         => ( member_set_set_state @ X4 @ B3 ) )
     => ( ord_le3154822624800881980_state @ A4 @ B3 ) ) ).

% subsetI
thf(fact_136_subsetI,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ! [X4: set_state] :
          ( ( member_set_state @ X4 @ A4 )
         => ( member_set_state @ X4 @ B3 ) )
     => ( ord_le5175021213330142598_state @ A4 @ B3 ) ) ).

% subsetI
thf(fact_137_subsetI,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( member_nat @ X4 @ B3 ) )
     => ( ord_less_eq_set_nat @ A4 @ B3 ) ) ).

% subsetI
thf(fact_138_subsetI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A4 )
         => ( member_state @ X4 @ B3 ) )
     => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% subsetI
thf(fact_139_insertCI,axiom,
    ! [A3: option_state,B3: set_option_state,B2: option_state] :
      ( ( ~ ( member_option_state @ A3 @ B3 )
       => ( A3 = B2 ) )
     => ( member_option_state @ A3 @ ( insert_option_state @ B2 @ B3 ) ) ) ).

% insertCI
thf(fact_140_insertCI,axiom,
    ! [A3: set_state,B3: set_set_state,B2: set_state] :
      ( ( ~ ( member_set_state @ A3 @ B3 )
       => ( A3 = B2 ) )
     => ( member_set_state @ A3 @ ( insert_set_state @ B2 @ B3 ) ) ) ).

% insertCI
thf(fact_141_insertCI,axiom,
    ! [A3: nat,B3: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A3 @ B3 )
       => ( A3 = B2 ) )
     => ( member_nat @ A3 @ ( insert_nat @ B2 @ B3 ) ) ) ).

% insertCI
thf(fact_142_insertCI,axiom,
    ! [A3: set_set_state,B3: set_set_set_state,B2: set_set_state] :
      ( ( ~ ( member_set_set_state @ A3 @ B3 )
       => ( A3 = B2 ) )
     => ( member_set_set_state @ A3 @ ( insert_set_set_state @ B2 @ B3 ) ) ) ).

% insertCI
thf(fact_143_insertCI,axiom,
    ! [A3: state,B3: set_state,B2: state] :
      ( ( ~ ( member_state @ A3 @ B3 )
       => ( A3 = B2 ) )
     => ( member_state @ A3 @ ( insert_state @ B2 @ B3 ) ) ) ).

% insertCI
thf(fact_144_insert__iff,axiom,
    ! [A3: option_state,B2: option_state,A4: set_option_state] :
      ( ( member_option_state @ A3 @ ( insert_option_state @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_option_state @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_145_insert__iff,axiom,
    ! [A3: set_state,B2: set_state,A4: set_set_state] :
      ( ( member_set_state @ A3 @ ( insert_set_state @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_set_state @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_146_insert__iff,axiom,
    ! [A3: nat,B2: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ ( insert_nat @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_nat @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_147_insert__iff,axiom,
    ! [A3: set_set_state,B2: set_set_state,A4: set_set_set_state] :
      ( ( member_set_set_state @ A3 @ ( insert_set_set_state @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_set_set_state @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_148_insert__iff,axiom,
    ! [A3: state,B2: state,A4: set_state] :
      ( ( member_state @ A3 @ ( insert_state @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_state @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_149_emptyE,axiom,
    ! [A3: set_set_state] :
      ~ ( member_set_set_state @ A3 @ bot_bo7389043141884672880_state ) ).

% emptyE
thf(fact_150_emptyE,axiom,
    ! [A3: option_state] :
      ~ ( member_option_state @ A3 @ bot_bo710180891245420500_state ) ).

% emptyE
thf(fact_151_emptyE,axiom,
    ! [A3: set_state] :
      ~ ( member_set_state @ A3 @ bot_bo2271482359692755898_state ) ).

% emptyE
thf(fact_152_emptyE,axiom,
    ! [A3: nat] :
      ~ ( member_nat @ A3 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_153_emptyE,axiom,
    ! [A3: state] :
      ~ ( member_state @ A3 @ bot_bot_set_state ) ).

% emptyE
thf(fact_154_equals0D,axiom,
    ! [A4: set_set_set_state,A3: set_set_state] :
      ( ( A4 = bot_bo7389043141884672880_state )
     => ~ ( member_set_set_state @ A3 @ A4 ) ) ).

% equals0D
thf(fact_155_equals0D,axiom,
    ! [A4: set_option_state,A3: option_state] :
      ( ( A4 = bot_bo710180891245420500_state )
     => ~ ( member_option_state @ A3 @ A4 ) ) ).

% equals0D
thf(fact_156_equals0D,axiom,
    ! [A4: set_set_state,A3: set_state] :
      ( ( A4 = bot_bo2271482359692755898_state )
     => ~ ( member_set_state @ A3 @ A4 ) ) ).

% equals0D
thf(fact_157_equals0D,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A3 @ A4 ) ) ).

% equals0D
thf(fact_158_equals0D,axiom,
    ! [A4: set_state,A3: state] :
      ( ( A4 = bot_bot_set_state )
     => ~ ( member_state @ A3 @ A4 ) ) ).

% equals0D
thf(fact_159_equals0I,axiom,
    ! [A4: set_set_set_state] :
      ( ! [Y3: set_set_state] :
          ~ ( member_set_set_state @ Y3 @ A4 )
     => ( A4 = bot_bo7389043141884672880_state ) ) ).

% equals0I
thf(fact_160_equals0I,axiom,
    ! [A4: set_option_state] :
      ( ! [Y3: option_state] :
          ~ ( member_option_state @ Y3 @ A4 )
     => ( A4 = bot_bo710180891245420500_state ) ) ).

% equals0I
thf(fact_161_equals0I,axiom,
    ! [A4: set_set_state] :
      ( ! [Y3: set_state] :
          ~ ( member_set_state @ Y3 @ A4 )
     => ( A4 = bot_bo2271482359692755898_state ) ) ).

% equals0I
thf(fact_162_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_163_equals0I,axiom,
    ! [A4: set_state] :
      ( ! [Y3: state] :
          ~ ( member_state @ Y3 @ A4 )
     => ( A4 = bot_bot_set_state ) ) ).

% equals0I
thf(fact_164_ex__in__conv,axiom,
    ! [A4: set_set_set_state] :
      ( ( ? [X2: set_set_state] : ( member_set_set_state @ X2 @ A4 ) )
      = ( A4 != bot_bo7389043141884672880_state ) ) ).

% ex_in_conv
thf(fact_165_ex__in__conv,axiom,
    ! [A4: set_option_state] :
      ( ( ? [X2: option_state] : ( member_option_state @ X2 @ A4 ) )
      = ( A4 != bot_bo710180891245420500_state ) ) ).

% ex_in_conv
thf(fact_166_ex__in__conv,axiom,
    ! [A4: set_set_state] :
      ( ( ? [X2: set_state] : ( member_set_state @ X2 @ A4 ) )
      = ( A4 != bot_bo2271482359692755898_state ) ) ).

% ex_in_conv
thf(fact_167_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X2: nat] : ( member_nat @ X2 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_168_ex__in__conv,axiom,
    ! [A4: set_state] :
      ( ( ? [X2: state] : ( member_state @ X2 @ A4 ) )
      = ( A4 != bot_bot_set_state ) ) ).

% ex_in_conv
thf(fact_169_in__mono,axiom,
    ! [A4: set_option_state,B3: set_option_state,X3: option_state] :
      ( ( ord_le7116032884704190368_state @ A4 @ B3 )
     => ( ( member_option_state @ X3 @ A4 )
       => ( member_option_state @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_170_in__mono,axiom,
    ! [A4: set_set_set_state,B3: set_set_set_state,X3: set_set_state] :
      ( ( ord_le3154822624800881980_state @ A4 @ B3 )
     => ( ( member_set_set_state @ X3 @ A4 )
       => ( member_set_set_state @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_171_in__mono,axiom,
    ! [A4: set_set_state,B3: set_set_state,X3: set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ B3 )
     => ( ( member_set_state @ X3 @ A4 )
       => ( member_set_state @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_172_in__mono,axiom,
    ! [A4: set_nat,B3: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( member_nat @ X3 @ A4 )
       => ( member_nat @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_173_in__mono,axiom,
    ! [A4: set_state,B3: set_state,X3: state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( member_state @ X3 @ A4 )
       => ( member_state @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_174_subsetD,axiom,
    ! [A4: set_option_state,B3: set_option_state,C: option_state] :
      ( ( ord_le7116032884704190368_state @ A4 @ B3 )
     => ( ( member_option_state @ C @ A4 )
       => ( member_option_state @ C @ B3 ) ) ) ).

% subsetD
thf(fact_175_subsetD,axiom,
    ! [A4: set_set_set_state,B3: set_set_set_state,C: set_set_state] :
      ( ( ord_le3154822624800881980_state @ A4 @ B3 )
     => ( ( member_set_set_state @ C @ A4 )
       => ( member_set_set_state @ C @ B3 ) ) ) ).

% subsetD
thf(fact_176_subsetD,axiom,
    ! [A4: set_set_state,B3: set_set_state,C: set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ B3 )
     => ( ( member_set_state @ C @ A4 )
       => ( member_set_state @ C @ B3 ) ) ) ).

% subsetD
thf(fact_177_subsetD,axiom,
    ! [A4: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_178_subsetD,axiom,
    ! [A4: set_state,B3: set_state,C: state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( member_state @ C @ A4 )
       => ( member_state @ C @ B3 ) ) ) ).

% subsetD
thf(fact_179_equalityE,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ( A4 = B3 )
     => ~ ( ( ord_le5175021213330142598_state @ A4 @ B3 )
         => ~ ( ord_le5175021213330142598_state @ B3 @ A4 ) ) ) ).

% equalityE
thf(fact_180_equalityE,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( A4 = B3 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B3 )
         => ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ).

% equalityE
thf(fact_181_equalityE,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( A4 = B3 )
     => ~ ( ( ord_le2494988322063910608_state @ A4 @ B3 )
         => ~ ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ) ).

% equalityE
thf(fact_182_subset__eq,axiom,
    ( ord_le7116032884704190368_state
    = ( ^ [A: set_option_state,B4: set_option_state] :
        ! [X2: option_state] :
          ( ( member_option_state @ X2 @ A )
         => ( member_option_state @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_183_subset__eq,axiom,
    ( ord_le3154822624800881980_state
    = ( ^ [A: set_set_set_state,B4: set_set_set_state] :
        ! [X2: set_set_state] :
          ( ( member_set_set_state @ X2 @ A )
         => ( member_set_set_state @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_184_subset__eq,axiom,
    ( ord_le5175021213330142598_state
    = ( ^ [A: set_set_state,B4: set_set_state] :
        ! [X2: set_state] :
          ( ( member_set_state @ X2 @ A )
         => ( member_set_state @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_185_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A: set_nat,B4: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A )
         => ( member_nat @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_186_subset__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A: set_state,B4: set_state] :
        ! [X2: state] :
          ( ( member_state @ X2 @ A )
         => ( member_state @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_187_equalityD1,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ( A4 = B3 )
     => ( ord_le5175021213330142598_state @ A4 @ B3 ) ) ).

% equalityD1
thf(fact_188_equalityD1,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( A4 = B3 )
     => ( ord_less_eq_set_nat @ A4 @ B3 ) ) ).

% equalityD1
thf(fact_189_equalityD1,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( A4 = B3 )
     => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% equalityD1
thf(fact_190_equalityD2,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ( A4 = B3 )
     => ( ord_le5175021213330142598_state @ B3 @ A4 ) ) ).

% equalityD2
thf(fact_191_equalityD2,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( A4 = B3 )
     => ( ord_less_eq_set_nat @ B3 @ A4 ) ) ).

% equalityD2
thf(fact_192_equalityD2,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( A4 = B3 )
     => ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ).

% equalityD2
thf(fact_193_subset__iff,axiom,
    ( ord_le7116032884704190368_state
    = ( ^ [A: set_option_state,B4: set_option_state] :
        ! [T2: option_state] :
          ( ( member_option_state @ T2 @ A )
         => ( member_option_state @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_194_subset__iff,axiom,
    ( ord_le3154822624800881980_state
    = ( ^ [A: set_set_set_state,B4: set_set_set_state] :
        ! [T2: set_set_state] :
          ( ( member_set_set_state @ T2 @ A )
         => ( member_set_set_state @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_195_subset__iff,axiom,
    ( ord_le5175021213330142598_state
    = ( ^ [A: set_set_state,B4: set_set_state] :
        ! [T2: set_state] :
          ( ( member_set_state @ T2 @ A )
         => ( member_set_state @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_196_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A: set_nat,B4: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A )
         => ( member_nat @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_197_subset__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A: set_state,B4: set_state] :
        ! [T2: state] :
          ( ( member_state @ T2 @ A )
         => ( member_state @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_198_subset__refl,axiom,
    ! [A4: set_set_state] : ( ord_le5175021213330142598_state @ A4 @ A4 ) ).

% subset_refl
thf(fact_199_subset__refl,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).

% subset_refl
thf(fact_200_subset__refl,axiom,
    ! [A4: set_state] : ( ord_le2494988322063910608_state @ A4 @ A4 ) ).

% subset_refl
thf(fact_201_Collect__mono,axiom,
    ! [P: set_state > $o,Q: set_state > $o] :
      ( ! [X4: set_state] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le5175021213330142598_state @ ( collect_set_state @ P ) @ ( collect_set_state @ Q ) ) ) ).

% Collect_mono
thf(fact_202_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_203_Collect__mono,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ! [X4: state] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) ) ) ).

% Collect_mono
thf(fact_204_subset__trans,axiom,
    ! [A4: set_set_state,B3: set_set_state,C2: set_set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ B3 )
     => ( ( ord_le5175021213330142598_state @ B3 @ C2 )
       => ( ord_le5175021213330142598_state @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_205_subset__trans,axiom,
    ! [A4: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_206_subset__trans,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ord_le2494988322063910608_state @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_207_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_state,Z2: set_set_state] : ( Y4 = Z2 ) )
    = ( ^ [A: set_set_state,B4: set_set_state] :
          ( ( ord_le5175021213330142598_state @ A @ B4 )
          & ( ord_le5175021213330142598_state @ B4 @ A ) ) ) ) ).

% set_eq_subset
thf(fact_208_set__eq__subset,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
    = ( ^ [A: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).

% set_eq_subset
thf(fact_209_set__eq__subset,axiom,
    ( ( ^ [Y4: set_state,Z2: set_state] : ( Y4 = Z2 ) )
    = ( ^ [A: set_state,B4: set_state] :
          ( ( ord_le2494988322063910608_state @ A @ B4 )
          & ( ord_le2494988322063910608_state @ B4 @ A ) ) ) ) ).

% set_eq_subset
thf(fact_210_Collect__mono__iff,axiom,
    ! [P: set_state > $o,Q: set_state > $o] :
      ( ( ord_le5175021213330142598_state @ ( collect_set_state @ P ) @ ( collect_set_state @ Q ) )
      = ( ! [X2: set_state] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_211_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_212_Collect__mono__iff,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) )
      = ( ! [X2: state] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_213_insertE,axiom,
    ! [A3: option_state,B2: option_state,A4: set_option_state] :
      ( ( member_option_state @ A3 @ ( insert_option_state @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_option_state @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_214_insertE,axiom,
    ! [A3: set_state,B2: set_state,A4: set_set_state] :
      ( ( member_set_state @ A3 @ ( insert_set_state @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_set_state @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_215_insertE,axiom,
    ! [A3: nat,B2: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ ( insert_nat @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_nat @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_216_insertE,axiom,
    ! [A3: set_set_state,B2: set_set_state,A4: set_set_set_state] :
      ( ( member_set_set_state @ A3 @ ( insert_set_set_state @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_set_set_state @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_217_insertE,axiom,
    ! [A3: state,B2: state,A4: set_state] :
      ( ( member_state @ A3 @ ( insert_state @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_state @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_218_insertI1,axiom,
    ! [A3: option_state,B3: set_option_state] : ( member_option_state @ A3 @ ( insert_option_state @ A3 @ B3 ) ) ).

% insertI1
thf(fact_219_insertI1,axiom,
    ! [A3: set_state,B3: set_set_state] : ( member_set_state @ A3 @ ( insert_set_state @ A3 @ B3 ) ) ).

% insertI1
thf(fact_220_insertI1,axiom,
    ! [A3: nat,B3: set_nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ B3 ) ) ).

% insertI1
thf(fact_221_insertI1,axiom,
    ! [A3: set_set_state,B3: set_set_set_state] : ( member_set_set_state @ A3 @ ( insert_set_set_state @ A3 @ B3 ) ) ).

% insertI1
thf(fact_222_insertI1,axiom,
    ! [A3: state,B3: set_state] : ( member_state @ A3 @ ( insert_state @ A3 @ B3 ) ) ).

% insertI1
thf(fact_223_insertI2,axiom,
    ! [A3: option_state,B3: set_option_state,B2: option_state] :
      ( ( member_option_state @ A3 @ B3 )
     => ( member_option_state @ A3 @ ( insert_option_state @ B2 @ B3 ) ) ) ).

% insertI2
thf(fact_224_insertI2,axiom,
    ! [A3: set_state,B3: set_set_state,B2: set_state] :
      ( ( member_set_state @ A3 @ B3 )
     => ( member_set_state @ A3 @ ( insert_set_state @ B2 @ B3 ) ) ) ).

% insertI2
thf(fact_225_insertI2,axiom,
    ! [A3: nat,B3: set_nat,B2: nat] :
      ( ( member_nat @ A3 @ B3 )
     => ( member_nat @ A3 @ ( insert_nat @ B2 @ B3 ) ) ) ).

% insertI2
thf(fact_226_insertI2,axiom,
    ! [A3: set_set_state,B3: set_set_set_state,B2: set_set_state] :
      ( ( member_set_set_state @ A3 @ B3 )
     => ( member_set_set_state @ A3 @ ( insert_set_set_state @ B2 @ B3 ) ) ) ).

% insertI2
thf(fact_227_insertI2,axiom,
    ! [A3: state,B3: set_state,B2: state] :
      ( ( member_state @ A3 @ B3 )
     => ( member_state @ A3 @ ( insert_state @ B2 @ B3 ) ) ) ).

% insertI2
thf(fact_228_Set_Oset__insert,axiom,
    ! [X3: option_state,A4: set_option_state] :
      ( ( member_option_state @ X3 @ A4 )
     => ~ ! [B6: set_option_state] :
            ( ( A4
              = ( insert_option_state @ X3 @ B6 ) )
           => ( member_option_state @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_229_Set_Oset__insert,axiom,
    ! [X3: set_state,A4: set_set_state] :
      ( ( member_set_state @ X3 @ A4 )
     => ~ ! [B6: set_set_state] :
            ( ( A4
              = ( insert_set_state @ X3 @ B6 ) )
           => ( member_set_state @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_230_Set_Oset__insert,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( member_nat @ X3 @ A4 )
     => ~ ! [B6: set_nat] :
            ( ( A4
              = ( insert_nat @ X3 @ B6 ) )
           => ( member_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_231_Set_Oset__insert,axiom,
    ! [X3: set_set_state,A4: set_set_set_state] :
      ( ( member_set_set_state @ X3 @ A4 )
     => ~ ! [B6: set_set_set_state] :
            ( ( A4
              = ( insert_set_set_state @ X3 @ B6 ) )
           => ( member_set_set_state @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_232_Set_Oset__insert,axiom,
    ! [X3: state,A4: set_state] :
      ( ( member_state @ X3 @ A4 )
     => ~ ! [B6: set_state] :
            ( ( A4
              = ( insert_state @ X3 @ B6 ) )
           => ( member_state @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_233_insert__ident,axiom,
    ! [X3: option_state,A4: set_option_state,B3: set_option_state] :
      ( ~ ( member_option_state @ X3 @ A4 )
     => ( ~ ( member_option_state @ X3 @ B3 )
       => ( ( ( insert_option_state @ X3 @ A4 )
            = ( insert_option_state @ X3 @ B3 ) )
          = ( A4 = B3 ) ) ) ) ).

% insert_ident
thf(fact_234_insert__ident,axiom,
    ! [X3: set_state,A4: set_set_state,B3: set_set_state] :
      ( ~ ( member_set_state @ X3 @ A4 )
     => ( ~ ( member_set_state @ X3 @ B3 )
       => ( ( ( insert_set_state @ X3 @ A4 )
            = ( insert_set_state @ X3 @ B3 ) )
          = ( A4 = B3 ) ) ) ) ).

% insert_ident
thf(fact_235_insert__ident,axiom,
    ! [X3: nat,A4: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ~ ( member_nat @ X3 @ B3 )
       => ( ( ( insert_nat @ X3 @ A4 )
            = ( insert_nat @ X3 @ B3 ) )
          = ( A4 = B3 ) ) ) ) ).

% insert_ident
thf(fact_236_insert__ident,axiom,
    ! [X3: set_set_state,A4: set_set_set_state,B3: set_set_set_state] :
      ( ~ ( member_set_set_state @ X3 @ A4 )
     => ( ~ ( member_set_set_state @ X3 @ B3 )
       => ( ( ( insert_set_set_state @ X3 @ A4 )
            = ( insert_set_set_state @ X3 @ B3 ) )
          = ( A4 = B3 ) ) ) ) ).

% insert_ident
thf(fact_237_insert__ident,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ~ ( member_state @ X3 @ A4 )
     => ( ~ ( member_state @ X3 @ B3 )
       => ( ( ( insert_state @ X3 @ A4 )
            = ( insert_state @ X3 @ B3 ) )
          = ( A4 = B3 ) ) ) ) ).

% insert_ident
thf(fact_238_insert__absorb,axiom,
    ! [A3: option_state,A4: set_option_state] :
      ( ( member_option_state @ A3 @ A4 )
     => ( ( insert_option_state @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_239_insert__absorb,axiom,
    ! [A3: set_state,A4: set_set_state] :
      ( ( member_set_state @ A3 @ A4 )
     => ( ( insert_set_state @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_240_insert__absorb,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ( ( insert_nat @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_241_insert__absorb,axiom,
    ! [A3: set_set_state,A4: set_set_set_state] :
      ( ( member_set_set_state @ A3 @ A4 )
     => ( ( insert_set_set_state @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_242_insert__absorb,axiom,
    ! [A3: state,A4: set_state] :
      ( ( member_state @ A3 @ A4 )
     => ( ( insert_state @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_243_insert__eq__iff,axiom,
    ! [A3: option_state,A4: set_option_state,B2: option_state,B3: set_option_state] :
      ( ~ ( member_option_state @ A3 @ A4 )
     => ( ~ ( member_option_state @ B2 @ B3 )
       => ( ( ( insert_option_state @ A3 @ A4 )
            = ( insert_option_state @ B2 @ B3 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B3 ) )
            & ( ( A3 != B2 )
             => ? [C4: set_option_state] :
                  ( ( A4
                    = ( insert_option_state @ B2 @ C4 ) )
                  & ~ ( member_option_state @ B2 @ C4 )
                  & ( B3
                    = ( insert_option_state @ A3 @ C4 ) )
                  & ~ ( member_option_state @ A3 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_244_insert__eq__iff,axiom,
    ! [A3: set_state,A4: set_set_state,B2: set_state,B3: set_set_state] :
      ( ~ ( member_set_state @ A3 @ A4 )
     => ( ~ ( member_set_state @ B2 @ B3 )
       => ( ( ( insert_set_state @ A3 @ A4 )
            = ( insert_set_state @ B2 @ B3 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B3 ) )
            & ( ( A3 != B2 )
             => ? [C4: set_set_state] :
                  ( ( A4
                    = ( insert_set_state @ B2 @ C4 ) )
                  & ~ ( member_set_state @ B2 @ C4 )
                  & ( B3
                    = ( insert_set_state @ A3 @ C4 ) )
                  & ~ ( member_set_state @ A3 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_245_insert__eq__iff,axiom,
    ! [A3: nat,A4: set_nat,B2: nat,B3: set_nat] :
      ( ~ ( member_nat @ A3 @ A4 )
     => ( ~ ( member_nat @ B2 @ B3 )
       => ( ( ( insert_nat @ A3 @ A4 )
            = ( insert_nat @ B2 @ B3 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B3 ) )
            & ( ( A3 != B2 )
             => ? [C4: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B2 @ C4 ) )
                  & ~ ( member_nat @ B2 @ C4 )
                  & ( B3
                    = ( insert_nat @ A3 @ C4 ) )
                  & ~ ( member_nat @ A3 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_246_insert__eq__iff,axiom,
    ! [A3: set_set_state,A4: set_set_set_state,B2: set_set_state,B3: set_set_set_state] :
      ( ~ ( member_set_set_state @ A3 @ A4 )
     => ( ~ ( member_set_set_state @ B2 @ B3 )
       => ( ( ( insert_set_set_state @ A3 @ A4 )
            = ( insert_set_set_state @ B2 @ B3 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B3 ) )
            & ( ( A3 != B2 )
             => ? [C4: set_set_set_state] :
                  ( ( A4
                    = ( insert_set_set_state @ B2 @ C4 ) )
                  & ~ ( member_set_set_state @ B2 @ C4 )
                  & ( B3
                    = ( insert_set_set_state @ A3 @ C4 ) )
                  & ~ ( member_set_set_state @ A3 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_247_insert__eq__iff,axiom,
    ! [A3: state,A4: set_state,B2: state,B3: set_state] :
      ( ~ ( member_state @ A3 @ A4 )
     => ( ~ ( member_state @ B2 @ B3 )
       => ( ( ( insert_state @ A3 @ A4 )
            = ( insert_state @ B2 @ B3 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B3 ) )
            & ( ( A3 != B2 )
             => ? [C4: set_state] :
                  ( ( A4
                    = ( insert_state @ B2 @ C4 ) )
                  & ~ ( member_state @ B2 @ C4 )
                  & ( B3
                    = ( insert_state @ A3 @ C4 ) )
                  & ~ ( member_state @ A3 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_248_insert__commute,axiom,
    ! [X3: option_state,Y2: option_state,A4: set_option_state] :
      ( ( insert_option_state @ X3 @ ( insert_option_state @ Y2 @ A4 ) )
      = ( insert_option_state @ Y2 @ ( insert_option_state @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_249_insert__commute,axiom,
    ! [X3: set_state,Y2: set_state,A4: set_set_state] :
      ( ( insert_set_state @ X3 @ ( insert_set_state @ Y2 @ A4 ) )
      = ( insert_set_state @ Y2 @ ( insert_set_state @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_250_insert__commute,axiom,
    ! [X3: nat,Y2: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y2 @ A4 ) )
      = ( insert_nat @ Y2 @ ( insert_nat @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_251_insert__commute,axiom,
    ! [X3: state,Y2: state,A4: set_state] :
      ( ( insert_state @ X3 @ ( insert_state @ Y2 @ A4 ) )
      = ( insert_state @ Y2 @ ( insert_state @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_252_mk__disjoint__insert,axiom,
    ! [A3: option_state,A4: set_option_state] :
      ( ( member_option_state @ A3 @ A4 )
     => ? [B6: set_option_state] :
          ( ( A4
            = ( insert_option_state @ A3 @ B6 ) )
          & ~ ( member_option_state @ A3 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_253_mk__disjoint__insert,axiom,
    ! [A3: set_state,A4: set_set_state] :
      ( ( member_set_state @ A3 @ A4 )
     => ? [B6: set_set_state] :
          ( ( A4
            = ( insert_set_state @ A3 @ B6 ) )
          & ~ ( member_set_state @ A3 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_254_mk__disjoint__insert,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ? [B6: set_nat] :
          ( ( A4
            = ( insert_nat @ A3 @ B6 ) )
          & ~ ( member_nat @ A3 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_255_mk__disjoint__insert,axiom,
    ! [A3: set_set_state,A4: set_set_set_state] :
      ( ( member_set_set_state @ A3 @ A4 )
     => ? [B6: set_set_set_state] :
          ( ( A4
            = ( insert_set_set_state @ A3 @ B6 ) )
          & ~ ( member_set_set_state @ A3 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_256_mk__disjoint__insert,axiom,
    ! [A3: state,A4: set_state] :
      ( ( member_state @ A3 @ A4 )
     => ? [B6: set_state] :
          ( ( A4
            = ( insert_state @ A3 @ B6 ) )
          & ~ ( member_state @ A3 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_257_singletonD,axiom,
    ! [B2: set_set_state,A3: set_set_state] :
      ( ( member_set_set_state @ B2 @ ( insert_set_set_state @ A3 @ bot_bo7389043141884672880_state ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_258_singletonD,axiom,
    ! [B2: option_state,A3: option_state] :
      ( ( member_option_state @ B2 @ ( insert_option_state @ A3 @ bot_bo710180891245420500_state ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_259_singletonD,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( member_set_state @ B2 @ ( insert_set_state @ A3 @ bot_bo2271482359692755898_state ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_260_singletonD,axiom,
    ! [B2: nat,A3: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_261_singletonD,axiom,
    ! [B2: state,A3: state] :
      ( ( member_state @ B2 @ ( insert_state @ A3 @ bot_bot_set_state ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_262_singleton__iff,axiom,
    ! [B2: set_set_state,A3: set_set_state] :
      ( ( member_set_set_state @ B2 @ ( insert_set_set_state @ A3 @ bot_bo7389043141884672880_state ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_263_singleton__iff,axiom,
    ! [B2: option_state,A3: option_state] :
      ( ( member_option_state @ B2 @ ( insert_option_state @ A3 @ bot_bo710180891245420500_state ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_264_singleton__iff,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( member_set_state @ B2 @ ( insert_set_state @ A3 @ bot_bo2271482359692755898_state ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_265_singleton__iff,axiom,
    ! [B2: nat,A3: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_266_singleton__iff,axiom,
    ! [B2: state,A3: state] :
      ( ( member_state @ B2 @ ( insert_state @ A3 @ bot_bot_set_state ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_267_doubleton__eq__iff,axiom,
    ! [A3: option_state,B2: option_state,C: option_state,D: option_state] :
      ( ( ( insert_option_state @ A3 @ ( insert_option_state @ B2 @ bot_bo710180891245420500_state ) )
        = ( insert_option_state @ C @ ( insert_option_state @ D @ bot_bo710180891245420500_state ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_268_doubleton__eq__iff,axiom,
    ! [A3: set_state,B2: set_state,C: set_state,D: set_state] :
      ( ( ( insert_set_state @ A3 @ ( insert_set_state @ B2 @ bot_bo2271482359692755898_state ) )
        = ( insert_set_state @ C @ ( insert_set_state @ D @ bot_bo2271482359692755898_state ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_269_doubleton__eq__iff,axiom,
    ! [A3: nat,B2: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A3 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_270_doubleton__eq__iff,axiom,
    ! [A3: state,B2: state,C: state,D: state] :
      ( ( ( insert_state @ A3 @ ( insert_state @ B2 @ bot_bot_set_state ) )
        = ( insert_state @ C @ ( insert_state @ D @ bot_bot_set_state ) ) )
      = ( ( ( A3 = C )
          & ( B2 = D ) )
        | ( ( A3 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_271_insert__not__empty,axiom,
    ! [A3: option_state,A4: set_option_state] :
      ( ( insert_option_state @ A3 @ A4 )
     != bot_bo710180891245420500_state ) ).

% insert_not_empty
thf(fact_272_insert__not__empty,axiom,
    ! [A3: set_state,A4: set_set_state] :
      ( ( insert_set_state @ A3 @ A4 )
     != bot_bo2271482359692755898_state ) ).

% insert_not_empty
thf(fact_273_insert__not__empty,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( insert_nat @ A3 @ A4 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_274_insert__not__empty,axiom,
    ! [A3: state,A4: set_state] :
      ( ( insert_state @ A3 @ A4 )
     != bot_bot_set_state ) ).

% insert_not_empty
thf(fact_275_singleton__inject,axiom,
    ! [A3: state,B2: state] :
      ( ( ( insert_state @ A3 @ bot_bot_set_state )
        = ( insert_state @ B2 @ bot_bot_set_state ) )
     => ( A3 = B2 ) ) ).

% singleton_inject
thf(fact_276_insert__mono,axiom,
    ! [C2: set_state,D2: set_state,A3: state] :
      ( ( ord_le2494988322063910608_state @ C2 @ D2 )
     => ( ord_le2494988322063910608_state @ ( insert_state @ A3 @ C2 ) @ ( insert_state @ A3 @ D2 ) ) ) ).

% insert_mono
thf(fact_277_subset__insert,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ~ ( member_state @ X3 @ A4 )
     => ( ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ X3 @ B3 ) )
        = ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ).

% subset_insert
thf(fact_278_subset__insertI,axiom,
    ! [B3: set_state,A3: state] : ( ord_le2494988322063910608_state @ B3 @ ( insert_state @ A3 @ B3 ) ) ).

% subset_insertI
thf(fact_279_subset__insertI2,axiom,
    ! [A4: set_state,B3: set_state,B2: state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ B2 @ B3 ) ) ) ).

% subset_insertI2
thf(fact_280_double__inclusion,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ A4 )
       => ( A4 = B3 ) ) ) ).

% double_inclusion
thf(fact_281_the__elem__eq,axiom,
    ! [X3: state] :
      ( ( the_elem_state @ ( insert_state @ X3 @ bot_bot_set_state ) )
      = X3 ) ).

% the_elem_eq
thf(fact_282_order__refl,axiom,
    ! [X3: set_state] : ( ord_le2494988322063910608_state @ X3 @ X3 ) ).

% order_refl
thf(fact_283_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_284_dual__order_Orefl,axiom,
    ! [A3: set_state] : ( ord_le2494988322063910608_state @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_285_dual__order_Orefl,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_286_is__singletonI,axiom,
    ! [X3: state] : ( is_singleton_state @ ( insert_state @ X3 @ bot_bot_set_state ) ) ).

% is_singletonI
thf(fact_287_wandI,axiom,
    ! [A4: set_state,W: state,B3: set_state] :
      ( ! [A5: state,X4: state] :
          ( ( ( member_state @ A5 @ A4 )
            & ( ( some_state @ X4 )
              = ( plus @ A5 @ W ) ) )
         => ( member_state @ X4 @ B3 ) )
     => ( member_state @ W @ ( wand @ A4 @ B3 ) ) ) ).

% wandI
thf(fact_288_insert__subsetI,axiom,
    ! [X3: state,A4: set_state,X6: set_state] :
      ( ( member_state @ X3 @ A4 )
     => ( ( ord_le2494988322063910608_state @ X6 @ A4 )
       => ( ord_le2494988322063910608_state @ ( insert_state @ X3 @ X6 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_289_subset__emptyI,axiom,
    ! [A4: set_state] :
      ( ! [X4: state] :
          ~ ( member_state @ X4 @ A4 )
     => ( ord_le2494988322063910608_state @ A4 @ bot_bot_set_state ) ) ).

% subset_emptyI
thf(fact_290_bot_Oextremum,axiom,
    ! [A3: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A3 ) ).

% bot.extremum
thf(fact_291_bot_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).

% bot.extremum
thf(fact_292_bot__set__def,axiom,
    ( bot_bot_set_state
    = ( collect_state @ bot_bot_state_o ) ) ).

% bot_set_def
thf(fact_293_is__singleton__the__elem,axiom,
    ( is_singleton_state
    = ( ^ [A: set_state] :
          ( A
          = ( insert_state @ ( the_elem_state @ A ) @ bot_bot_set_state ) ) ) ) ).

% is_singleton_the_elem
thf(fact_294_is__singletonI_H,axiom,
    ! [A4: set_state] :
      ( ( A4 != bot_bot_set_state )
     => ( ! [X4: state,Y3: state] :
            ( ( member_state @ X4 @ A4 )
           => ( ( member_state @ Y3 @ A4 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_state @ A4 ) ) ) ).

% is_singletonI'
thf(fact_295_order__antisym__conv,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
     => ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_296_order__antisym__conv,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_297_linorder__le__cases,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_le_cases
thf(fact_298_ord__le__eq__subst,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_299_ord__le__eq__subst,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > nat,C: nat] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_300_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F: nat > set_state,C: set_state] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_301_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_302_ord__eq__le__subst,axiom,
    ! [A3: set_state,F: set_state > set_state,B2: set_state,C: set_state] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_303_ord__eq__le__subst,axiom,
    ! [A3: nat,F: set_state > nat,B2: set_state,C: set_state] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_304_ord__eq__le__subst,axiom,
    ! [A3: set_state,F: nat > set_state,B2: nat,C: nat] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_305_ord__eq__le__subst,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_306_linorder__linear,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_linear
thf(fact_307_order__eq__refl,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( X3 = Y2 )
     => ( ord_le2494988322063910608_state @ X3 @ Y2 ) ) ).

% order_eq_refl
thf(fact_308_order__eq__refl,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( X3 = Y2 )
     => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).

% order_eq_refl
thf(fact_309_order__subst2,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ ( F @ B2 ) @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_310_order__subst2,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > nat,C: nat] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_311_order__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > set_state,C: set_state] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_312_order__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_313_order__subst1,axiom,
    ! [A3: set_state,F: set_state > set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_314_order__subst1,axiom,
    ! [A3: set_state,F: nat > set_state,B2: nat,C: nat] :
      ( ( ord_le2494988322063910608_state @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_315_order__subst1,axiom,
    ! [A3: nat,F: set_state > nat,B2: set_state,C: set_state] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_316_order__subst1,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_317_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_state,Z2: set_state] : ( Y4 = Z2 ) )
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ A2 @ B )
          & ( ord_le2494988322063910608_state @ B @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_318_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A2: nat,B: nat] :
          ( ( ord_less_eq_nat @ A2 @ B )
          & ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_319_antisym,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_320_antisym,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_321_dual__order_Otrans,axiom,
    ! [B2: set_state,A3: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( ord_le2494988322063910608_state @ C @ B2 )
       => ( ord_le2494988322063910608_state @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_322_dual__order_Otrans,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_323_dual__order_Oantisym,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( ord_le2494988322063910608_state @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_324_dual__order_Oantisym,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_325_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_state,Z2: set_state] : ( Y4 = Z2 ) )
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ B @ A2 )
          & ( ord_le2494988322063910608_state @ A2 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_326_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A2: nat,B: nat] :
          ( ( ord_less_eq_nat @ B @ A2 )
          & ( ord_less_eq_nat @ A2 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_327_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A3: nat,B2: nat] :
      ( ! [A5: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A5 @ B7 )
         => ( P @ A5 @ B7 ) )
     => ( ! [A5: nat,B7: nat] :
            ( ( P @ B7 @ A5 )
           => ( P @ A5 @ B7 ) )
       => ( P @ A3 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_328_order__trans,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ord_le2494988322063910608_state @ Y2 @ Z )
       => ( ord_le2494988322063910608_state @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_329_order__trans,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_330_order_Otrans,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ord_le2494988322063910608_state @ A3 @ C ) ) ) ).

% order.trans
thf(fact_331_order_Otrans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% order.trans
thf(fact_332_order__antisym,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
       => ( X3 = Y2 ) ) ) ).

% order_antisym
thf(fact_333_order__antisym,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X3 )
       => ( X3 = Y2 ) ) ) ).

% order_antisym
thf(fact_334_ord__le__eq__trans,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_le2494988322063910608_state @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_335_ord__le__eq__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_336_ord__eq__le__trans,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( A3 = B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ord_le2494988322063910608_state @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_337_ord__eq__le__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( A3 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_338_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_state,Z2: set_state] : ( Y4 = Z2 ) )
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( ord_le2494988322063910608_state @ X2 @ Y )
          & ( ord_le2494988322063910608_state @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_339_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X2: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y )
          & ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_340_le__cases3,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_341_nle__le,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A3 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A3 )
        & ( B2 != A3 ) ) ) ).

% nle_le
thf(fact_342_is__singleton__def,axiom,
    ( is_singleton_state
    = ( ^ [A: set_state] :
        ? [X2: state] :
          ( A
          = ( insert_state @ X2 @ bot_bot_set_state ) ) ) ) ).

% is_singleton_def
thf(fact_343_is__singletonE,axiom,
    ! [A4: set_state] :
      ( ( is_singleton_state @ A4 )
     => ~ ! [X4: state] :
            ( A4
           != ( insert_state @ X4 @ bot_bot_set_state ) ) ) ).

% is_singletonE
thf(fact_344_bot_Oextremum__uniqueI,axiom,
    ! [A3: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ bot_bot_set_state )
     => ( A3 = bot_bot_set_state ) ) ).

% bot.extremum_uniqueI
thf(fact_345_bot_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
     => ( A3 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_346_bot_Oextremum__unique,axiom,
    ! [A3: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ bot_bot_set_state )
      = ( A3 = bot_bot_set_state ) ) ).

% bot.extremum_unique
thf(fact_347_bot_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
      = ( A3 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_348_PartialSA_Osetify__sum,axiom,
    ! [P: state > $o,A4: set_state,B3: set_state] :
      ( ( sep_setify_state @ P @ ( add_set @ A4 @ B3 ) )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ( sep_setify_state @ P @ ( add_set @ ( insert_state @ X2 @ bot_bot_set_state ) @ B3 ) ) ) ) ) ).

% PartialSA.setify_sum
thf(fact_349_these__insert__Some,axiom,
    ! [X3: state,A4: set_option_state] :
      ( ( these_state @ ( insert_option_state @ ( some_state @ X3 ) @ A4 ) )
      = ( insert_state @ X3 @ ( these_state @ A4 ) ) ) ).

% these_insert_Some
thf(fact_350_PartialSA_Oup__closed__bigger__subset,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( sep_up_closed_state @ plus @ B3 )
     => ( ( greater_set @ A4 @ B3 )
       => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ).

% PartialSA.up_closed_bigger_subset
thf(fact_351_Set_Ois__empty__def,axiom,
    ( is_empty_state
    = ( ^ [A: set_state] : ( A = bot_bot_set_state ) ) ) ).

% Set.is_empty_def
thf(fact_352_package__logic_Omono__transformer_Ocong,axiom,
    packag4817583915586052229_state = packag4817583915586052229_state ).

% package_logic.mono_transformer.cong
thf(fact_353_PartialSA_Obigger__the,axiom,
    ! [A3: state,X5: state,Y2: state,X3: state] :
      ( ( ( some_state @ A3 )
        = ( plus @ X5 @ Y2 ) )
     => ( ( greater @ X5 @ X3 )
       => ( greater @ ( the_state @ ( plus @ ( core @ A3 ) @ X5 ) ) @ ( the_state @ ( plus @ ( core @ A3 ) @ X3 ) ) ) ) ) ).

% PartialSA.bigger_the
thf(fact_354_PartialSA_Opackage__logic__axioms,axiom,
    package_logic_state @ plus @ core @ unit @ stable ).

% PartialSA.package_logic_axioms
thf(fact_355_PartialSA_Ox__elem__set__product__splus,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ( member_state @ X3 @ ( add_set @ A4 @ B3 ) )
      = ( ? [A2: state,B: state] :
            ( ( member_state @ A2 @ A4 )
            & ( member_state @ B @ B3 )
            & ( ( some_state @ X3 )
              = ( sep_splus_state @ plus @ ( some_state @ A2 ) @ ( some_state @ B ) ) ) ) ) ) ).

% PartialSA.x_elem_set_product_splus
thf(fact_356_PartialSA_Osmaller__pure__sum__smaller,axiom,
    ! [Y2: state,A3: state,B2: state,X3: state] :
      ( ( greater @ Y2 @ A3 )
     => ( ( greater @ Y2 @ B2 )
       => ( ( ( some_state @ X3 )
            = ( plus @ A3 @ B2 ) )
         => ( ( sep_pure_state @ plus @ B2 )
           => ( greater @ Y2 @ X3 ) ) ) ) ) ).

% PartialSA.smaller_pure_sum_smaller
thf(fact_357_these__empty,axiom,
    ( ( these_state @ bot_bo710180891245420500_state )
    = bot_bot_set_state ) ).

% these_empty
thf(fact_358_package__logic_Ounit__neutral,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( some_state @ A3 )
        = ( Plus @ A3 @ Unit ) ) ) ).

% package_logic.unit_neutral
thf(fact_359_package__logic_Ostable__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: state,B2: state,X3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A3 )
       => ( ( Stable @ B2 )
         => ( ( ( some_state @ X3 )
              = ( Plus @ A3 @ B2 ) )
           => ( Stable @ X3 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_360_package__logic_Ounit__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( Core @ Unit )
        = Unit ) ) ).

% package_logic.unit_core
thf(fact_361_package__logic_Ostable__unit,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( Stable @ Unit ) ) ).

% package_logic.stable_unit
thf(fact_362_PartialSA_Osetify__def,axiom,
    ( sep_setify_state
    = ( ^ [P2: state > $o,A: set_state] :
        ! [X2: state] :
          ( ( member_state @ X2 @ A )
         => ( P2 @ X2 ) ) ) ) ).

% PartialSA.setify_def
thf(fact_363_option_Osel,axiom,
    ! [X22: state] :
      ( ( the_state @ ( some_state @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_364_PartialSA_Osplus__comm,axiom,
    ! [A3: option_state,B2: option_state] :
      ( ( sep_splus_state @ plus @ A3 @ B2 )
      = ( sep_splus_state @ plus @ B2 @ A3 ) ) ).

% PartialSA.splus_comm
thf(fact_365_PartialSA_Osplus__asso,axiom,
    ! [A3: option_state,B2: option_state,C: option_state] :
      ( ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ A3 @ B2 ) @ C )
      = ( sep_splus_state @ plus @ A3 @ ( sep_splus_state @ plus @ B2 @ C ) ) ) ).

% PartialSA.splus_asso
thf(fact_366_in__these__eq,axiom,
    ! [X3: state,A4: set_option_state] :
      ( ( member_state @ X3 @ ( these_state @ A4 ) )
      = ( member_option_state @ ( some_state @ X3 ) @ A4 ) ) ).

% in_these_eq
thf(fact_367_PartialSA_Osplus__develop,axiom,
    ! [A3: state,B2: state,C: state,D: state] :
      ( ( ( some_state @ A3 )
        = ( plus @ B2 @ C ) )
     => ( ( plus @ A3 @ D )
        = ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ ( some_state @ B2 ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ).

% PartialSA.splus_develop
thf(fact_368_PartialSA_Osplus_Osimps_I3_J,axiom,
    ! [A3: state,B2: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ A3 ) @ ( some_state @ B2 ) )
      = ( plus @ A3 @ B2 ) ) ).

% PartialSA.splus.simps(3)
thf(fact_369_PartialSA_Opure__stable,axiom,
    ! [A3: state,B2: state,C: state] :
      ( ( sep_pure_state @ plus @ A3 )
     => ( ( sep_pure_state @ plus @ B2 )
       => ( ( ( some_state @ C )
            = ( plus @ A3 @ B2 ) )
         => ( sep_pure_state @ plus @ C ) ) ) ) ).

% PartialSA.pure_stable
thf(fact_370_PartialSA_Opure__def,axiom,
    ! [A3: state] :
      ( ( sep_pure_state @ plus @ A3 )
      = ( ( some_state @ A3 )
        = ( plus @ A3 @ A3 ) ) ) ).

% PartialSA.pure_def
thf(fact_371_PartialSA_Opure__smaller,axiom,
    ! [A3: state,B2: state] :
      ( ( sep_pure_state @ plus @ A3 )
     => ( ( greater @ A3 @ B2 )
       => ( sep_pure_state @ plus @ B2 ) ) ) ).

% PartialSA.pure_smaller
thf(fact_372_PartialSA_Oup__closedI,axiom,
    ! [A4: set_state] :
      ( ! [Phi5: state,Phi6: state] :
          ( ( ( greater @ Phi5 @ Phi6 )
            & ( member_state @ Phi6 @ A4 ) )
         => ( member_state @ Phi5 @ A4 ) )
     => ( sep_up_closed_state @ plus @ A4 ) ) ).

% PartialSA.up_closedI
thf(fact_373_PartialSA_Oup__closed__def,axiom,
    ! [A4: set_state] :
      ( ( sep_up_closed_state @ plus @ A4 )
      = ( ! [Phi3: state] :
            ( ? [X2: state] :
                ( ( member_state @ X2 @ A4 )
                & ( greater @ Phi3 @ X2 ) )
           => ( member_state @ Phi3 @ A4 ) ) ) ) ).

% PartialSA.up_closed_def
thf(fact_374_PartialSA_Oup__closed__sum,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( sep_up_closed_state @ plus @ A4 )
     => ( sep_up_closed_state @ plus @ ( add_set @ A4 @ B3 ) ) ) ).

% PartialSA.up_closed_sum
thf(fact_375_bot__empty__eq,axiom,
    ( bot_bot_state_o
    = ( ^ [X2: state] : ( member_state @ X2 @ bot_bot_set_state ) ) ) ).

% bot_empty_eq
thf(fact_376_Collect__empty__eq__bot,axiom,
    ! [P: state > $o] :
      ( ( ( collect_state @ P )
        = bot_bot_set_state )
      = ( P = bot_bot_state_o ) ) ).

% Collect_empty_eq_bot
thf(fact_377_PartialSA_Omono__prop__set,axiom,
    ! [A4: set_state,B3: set_state,P: state > $o] :
      ( ( greater_set @ A4 @ B3 )
     => ( ( sep_setify_state @ P @ B3 )
       => ( ( sep_mono_prop_state @ plus @ P )
         => ( sep_setify_state @ P @ A4 ) ) ) ) ).

% PartialSA.mono_prop_set
thf(fact_378_package__logic_Omono__pruner__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,P3: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2456304381420842418_state @ Plus @ P3 )
        = ( ! [Phi3: state,Phi2: state,R3: state] :
              ( ( ( sep_pure_state @ Plus @ R3 )
                & ( P3 @ Phi2 )
                & ( ( some_state @ Phi3 )
                  = ( Plus @ Phi2 @ R3 ) ) )
             => ( P3 @ Phi3 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_379_PartialSA_Oequiv__up__closed__subset,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( sep_up_closed_state @ plus @ A4 )
     => ( ( sep_equiv_state @ plus @ B3 @ C2 )
       => ( ( ord_le2494988322063910608_state @ B3 @ A4 )
          = ( ord_le2494988322063910608_state @ C2 @ A4 ) ) ) ) ).

% PartialSA.equiv_up_closed_subset
thf(fact_380_PartialSA_Omono__pure__cond__def,axiom,
    ! [B2: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ B2 )
      = ( ! [Phi2: state] :
            ( ( B2 @ Phi2 )
            = ( B2 @ ( core @ Phi2 ) ) )
        & ! [Phi3: state,Phi2: state,R3: state] :
            ( ( ( sep_pure_state @ plus @ R3 )
              & ( ( some_state @ Phi3 )
                = ( plus @ Phi2 @ R3 ) )
              & ~ ( B2 @ Phi2 ) )
           => ~ ( B2 @ Phi3 ) ) ) ) ).

% PartialSA.mono_pure_cond_def
thf(fact_381_PartialSA_Omono__pure__condI,axiom,
    ! [B2: state > $o] :
      ( ! [Phi6: state] :
          ( ( B2 @ Phi6 )
          = ( B2 @ ( core @ Phi6 ) ) )
     => ( ! [Phi6: state,Phi5: state,R: state] :
            ( ( ( sep_pure_state @ plus @ R )
              & ( ( some_state @ Phi5 )
                = ( plus @ Phi6 @ R ) )
              & ~ ( B2 @ Phi6 ) )
           => ~ ( B2 @ Phi5 ) )
       => ( packag6595153354952283300_state @ plus @ core @ B2 ) ) ) ).

% PartialSA.mono_pure_condI
thf(fact_382_PartialSA_Osetify__sum__image,axiom,
    ! [P: state > $o,F: state > state,A4: set_state,B3: set_state] :
      ( ( sep_setify_state @ P @ ( add_set @ ( image_state_state @ F @ A4 ) @ B3 ) )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ( sep_setify_state @ P @ ( add_set @ ( insert_state @ ( F @ X2 ) @ bot_bot_set_state ) @ B3 ) ) ) ) ) ).

% PartialSA.setify_sum_image
thf(fact_383_image__eqI,axiom,
    ! [B2: state,F: state > state,X3: state,A4: set_state] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_state @ X3 @ A4 )
       => ( member_state @ B2 @ ( image_state_state @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_384_image__is__empty,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( ( image_state_state @ F @ A4 )
        = bot_bot_set_state )
      = ( A4 = bot_bot_set_state ) ) ).

% image_is_empty
thf(fact_385_empty__is__image,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( bot_bot_set_state
        = ( image_state_state @ F @ A4 ) )
      = ( A4 = bot_bot_set_state ) ) ).

% empty_is_image
thf(fact_386_image__empty,axiom,
    ! [F: state > state] :
      ( ( image_state_state @ F @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% image_empty
thf(fact_387_insert__image,axiom,
    ! [X3: state,A4: set_state,F: state > state] :
      ( ( member_state @ X3 @ A4 )
     => ( ( insert_state @ ( F @ X3 ) @ ( image_state_state @ F @ A4 ) )
        = ( image_state_state @ F @ A4 ) ) ) ).

% insert_image
thf(fact_388_image__insert,axiom,
    ! [F: state > state,A3: state,B3: set_state] :
      ( ( image_state_state @ F @ ( insert_state @ A3 @ B3 ) )
      = ( insert_state @ ( F @ A3 ) @ ( image_state_state @ F @ B3 ) ) ) ).

% image_insert
thf(fact_389_package__logic_Omono__pruner_Ocong,axiom,
    packag2456304381420842418_state = packag2456304381420842418_state ).

% package_logic.mono_pruner.cong
thf(fact_390_package__logic_Omono__pure__cond_Ocong,axiom,
    packag6595153354952283300_state = packag6595153354952283300_state ).

% package_logic.mono_pure_cond.cong
thf(fact_391_imageI,axiom,
    ! [X3: state,A4: set_state,F: state > state] :
      ( ( member_state @ X3 @ A4 )
     => ( member_state @ ( F @ X3 ) @ ( image_state_state @ F @ A4 ) ) ) ).

% imageI
thf(fact_392_image__iff,axiom,
    ! [Z: state,F: state > state,A4: set_state] :
      ( ( member_state @ Z @ ( image_state_state @ F @ A4 ) )
      = ( ? [X2: state] :
            ( ( member_state @ X2 @ A4 )
            & ( Z
              = ( F @ X2 ) ) ) ) ) ).

% image_iff
thf(fact_393_bex__imageD,axiom,
    ! [F: state > state,A4: set_state,P: state > $o] :
      ( ? [X: state] :
          ( ( member_state @ X @ ( image_state_state @ F @ A4 ) )
          & ( P @ X ) )
     => ? [X4: state] :
          ( ( member_state @ X4 @ A4 )
          & ( P @ ( F @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_394_image__cong,axiom,
    ! [M: set_state,N: set_state,F: state > state,G: state > state] :
      ( ( M = N )
     => ( ! [X4: state] :
            ( ( member_state @ X4 @ N )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( image_state_state @ F @ M )
          = ( image_state_state @ G @ N ) ) ) ) ).

% image_cong
thf(fact_395_ball__imageD,axiom,
    ! [F: state > state,A4: set_state,P: state > $o] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ ( image_state_state @ F @ A4 ) )
         => ( P @ X4 ) )
     => ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ( P @ ( F @ X ) ) ) ) ).

% ball_imageD
thf(fact_396_rev__image__eqI,axiom,
    ! [X3: state,A4: set_state,B2: state,F: state > state] :
      ( ( member_state @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_state @ B2 @ ( image_state_state @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_397_subset__image__iff,axiom,
    ! [B3: set_state,F: state > state,A4: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
      = ( ? [AA: set_state] :
            ( ( ord_le2494988322063910608_state @ AA @ A4 )
            & ( B3
              = ( image_state_state @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_398_image__subset__iff,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ( member_state @ ( F @ X2 ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_399_subset__imageE,axiom,
    ! [B3: set_state,F: state > state,A4: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
     => ~ ! [C5: set_state] :
            ( ( ord_le2494988322063910608_state @ C5 @ A4 )
           => ( B3
             != ( image_state_state @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_400_image__subsetI,axiom,
    ! [A4: set_state,F: state > state,B3: set_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A4 )
         => ( member_state @ ( F @ X4 ) @ B3 ) )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 ) ) ).

% image_subsetI
thf(fact_401_image__mono,axiom,
    ! [A4: set_state,B3: set_state,F: state > state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) ) ).

% image_mono
thf(fact_402_PartialSA_Omono__prop__set__equiv,axiom,
    ! [P: state > $o,A4: set_state,B3: set_state] :
      ( ( sep_mono_prop_state @ plus @ P )
     => ( ( sep_equiv_state @ plus @ A4 @ B3 )
       => ( ( sep_setify_state @ P @ A4 )
          = ( sep_setify_state @ P @ B3 ) ) ) ) ).

% PartialSA.mono_prop_set_equiv
thf(fact_403_the__elem__image__unique,axiom,
    ! [A4: set_state,F: state > state,X3: state] :
      ( ( A4 != bot_bot_set_state )
     => ( ! [Y3: state] :
            ( ( member_state @ Y3 @ A4 )
           => ( ( F @ Y3 )
              = ( F @ X3 ) ) )
       => ( ( the_elem_state @ ( image_state_state @ F @ A4 ) )
          = ( F @ X3 ) ) ) ) ).

% the_elem_image_unique
thf(fact_404_PartialSA_Oequiv__stable__sum,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( sep_equiv_state @ plus @ A4 @ B3 )
     => ( sep_equiv_state @ plus @ ( add_set @ A4 @ C2 ) @ ( add_set @ B3 @ C2 ) ) ) ).

% PartialSA.equiv_stable_sum
thf(fact_405_PartialSA_Oequiv__def,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( sep_equiv_state @ plus @ A4 @ B3 )
      = ( ( greater_set @ A4 @ B3 )
        & ( greater_set @ B3 @ A4 ) ) ) ).

% PartialSA.equiv_def
thf(fact_406_PartialSA_OequivI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( greater_set @ A4 @ B3 )
     => ( ( greater_set @ B3 @ A4 )
       => ( sep_equiv_state @ plus @ A4 @ B3 ) ) ) ).

% PartialSA.equivI
thf(fact_407_PartialSA_Oup__close__equiv,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( sep_up_closed_state @ plus @ A4 )
     => ( ( sep_up_closed_state @ plus @ B3 )
       => ( ( sep_equiv_state @ plus @ A4 @ B3 )
          = ( A4 = B3 ) ) ) ) ).

% PartialSA.up_close_equiv
thf(fact_408_PartialSA_Omono__propI,axiom,
    ! [P: state > $o] :
      ( ! [X4: state,Y3: state] :
          ( ( ( greater @ Y3 @ X4 )
            & ( P @ X4 ) )
         => ( P @ Y3 ) )
     => ( sep_mono_prop_state @ plus @ P ) ) ).

% PartialSA.mono_propI
thf(fact_409_PartialSA_Omono__prop__def,axiom,
    ! [P: state > $o] :
      ( ( sep_mono_prop_state @ plus @ P )
      = ( ! [X2: state,Y: state] :
            ( ( ( greater @ Y @ X2 )
              & ( P @ X2 ) )
           => ( P @ Y ) ) ) ) ).

% PartialSA.mono_prop_def
thf(fact_410_PartialSA_Omono__pruner__def,axiom,
    ! [P3: state > $o] :
      ( ( packag2456304381420842418_state @ plus @ P3 )
      = ( ! [Phi3: state,Phi2: state,R3: state] :
            ( ( ( sep_pure_state @ plus @ R3 )
              & ( P3 @ Phi2 )
              & ( ( some_state @ Phi3 )
                = ( plus @ Phi2 @ R3 ) ) )
           => ( P3 @ Phi3 ) ) ) ) ).

% PartialSA.mono_pruner_def
thf(fact_411_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ B2 )
        = ( ! [Phi2: state] :
              ( ( B2 @ Phi2 )
              = ( B2 @ ( Core @ Phi2 ) ) )
          & ! [Phi3: state,Phi2: state,R3: state] :
              ( ( ( sep_pure_state @ Plus @ R3 )
                & ( ( some_state @ Phi3 )
                  = ( Plus @ Phi2 @ R3 ) )
                & ~ ( B2 @ Phi2 ) )
             => ~ ( B2 @ Phi3 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_412_package__logic_Omono__pure__condI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi6: state] :
            ( ( B2 @ Phi6 )
            = ( B2 @ ( Core @ Phi6 ) ) )
       => ( ! [Phi6: state,Phi5: state,R: state] :
              ( ( ( sep_pure_state @ Plus @ R )
                & ( ( some_state @ Phi5 )
                  = ( Plus @ Phi6 @ R ) )
                & ~ ( B2 @ Phi6 ) )
             => ~ ( B2 @ Phi5 ) )
         => ( packag6595153354952283300_state @ Plus @ Core @ B2 ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_413_multiply__sem__assertion_Oelims,axiom,
    ! [X3: prat,Xa: set_state,Y2: set_state] :
      ( ( ( multip8064567061438756306ertion @ X3 @ Xa )
        = Y2 )
     => ( Y2
        = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X3 ) @ Xa ) ) ) ) ).

% multiply_sem_assertion.elims
thf(fact_414_multiply__sem__assertion_Osimps,axiom,
    ( multip8064567061438756306ertion
    = ( ^ [P4: prat,P2: set_state] : ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ P4 ) @ P2 ) ) ) ) ).

% multiply_sem_assertion.simps
thf(fact_415_all__subset__image,axiom,
    ! [F: state > state,A4: set_state,P: set_state > $o] :
      ( ( ! [B4: set_state] :
            ( ( ord_le2494988322063910608_state @ B4 @ ( image_state_state @ F @ A4 ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_state] :
            ( ( ord_le2494988322063910608_state @ B4 @ A4 )
           => ( P @ ( image_state_state @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_416_these__empty__eq,axiom,
    ! [B3: set_option_state] :
      ( ( ( these_state @ B3 )
        = bot_bot_set_state )
      = ( ( B3 = bot_bo710180891245420500_state )
        | ( B3
          = ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_empty_eq
thf(fact_417_these__not__empty__eq,axiom,
    ! [B3: set_option_state] :
      ( ( ( these_state @ B3 )
       != bot_bot_set_state )
      = ( ( B3 != bot_bo710180891245420500_state )
        & ( B3
         != ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_not_empty_eq
thf(fact_418_PartialSA_Omono__pure__cond__conj,axiom,
    ! [Pc: state > $o,B2: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ Pc )
     => ( ( packag6595153354952283300_state @ plus @ core @ B2 )
       => ( packag6595153354952283300_state @ plus @ core @ ( packag1330488657391168505_state @ Pc @ B2 ) ) ) ) ).

% PartialSA.mono_pure_cond_conj
thf(fact_419_package__logic_Omono__pure__cond__conj,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,B2: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
       => ( ( packag6595153354952283300_state @ Plus @ Core @ B2 )
         => ( packag6595153354952283300_state @ Plus @ Core @ ( packag1330488657391168505_state @ Pc @ B2 ) ) ) ) ) ).

% package_logic.mono_pure_cond_conj
thf(fact_420_PartialSA_Oupper__closure__up__closed,axiom,
    ! [A4: set_state] : ( sep_up_closed_state @ plus @ ( sep_up1246176804924251236_state @ plus @ A4 ) ) ).

% PartialSA.upper_closure_up_closed
thf(fact_421_not__None__eq,axiom,
    ! [X3: option_state] :
      ( ( X3 != none_state )
      = ( ? [Y: state] :
            ( X3
            = ( some_state @ Y ) ) ) ) ).

% not_None_eq
thf(fact_422_not__Some__eq,axiom,
    ! [X3: option_state] :
      ( ( ! [Y: state] :
            ( X3
           != ( some_state @ Y ) ) )
      = ( X3 = none_state ) ) ).

% not_Some_eq
thf(fact_423_these__image__Some__eq,axiom,
    ! [A4: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A4 ) )
      = A4 ) ).

% these_image_Some_eq
thf(fact_424_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_425_these__insert__None,axiom,
    ! [A4: set_option_state] :
      ( ( these_state @ ( insert_option_state @ none_state @ A4 ) )
      = ( these_state @ A4 ) ) ).

% these_insert_None
thf(fact_426_None__notin__image__Some,axiom,
    ! [A4: set_state] :
      ~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A4 ) ) ).

% None_notin_image_Some
thf(fact_427_PartialSA_Obool__conj__def,axiom,
    ( packag1330488657391168505_state
    = ( ^ [A2: state > $o,B: state > $o,X2: state] :
          ( ( A2 @ X2 )
          & ( B @ X2 ) ) ) ) ).

% PartialSA.bool_conj_def
thf(fact_428_option_Odistinct_I1_J,axiom,
    ! [X22: state] :
      ( none_state
     != ( some_state @ X22 ) ) ).

% option.distinct(1)
thf(fact_429_option_OdiscI,axiom,
    ! [Option: option_state,X22: state] :
      ( ( Option
        = ( some_state @ X22 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_430_option_Oexhaust,axiom,
    ! [Y2: option_state] :
      ( ( Y2 != none_state )
     => ~ ! [X23: state] :
            ( Y2
           != ( some_state @ X23 ) ) ) ).

% option.exhaust
thf(fact_431_split__option__ex,axiom,
    ( ( ^ [P5: option_state > $o] :
        ? [X7: option_state] : ( P5 @ X7 ) )
    = ( ^ [P2: option_state > $o] :
          ( ( P2 @ none_state )
          | ? [X2: state] : ( P2 @ ( some_state @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_432_split__option__all,axiom,
    ( ( ^ [P5: option_state > $o] :
        ! [X7: option_state] : ( P5 @ X7 ) )
    = ( ^ [P2: option_state > $o] :
          ( ( P2 @ none_state )
          & ! [X2: state] : ( P2 @ ( some_state @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_433_combine__options__cases,axiom,
    ! [X3: option_state,P: option_state > option_state > $o,Y2: option_state] :
      ( ( ( X3 = none_state )
       => ( P @ X3 @ Y2 ) )
     => ( ( ( Y2 = none_state )
         => ( P @ X3 @ Y2 ) )
       => ( ! [A5: state,B7: state] :
              ( ( X3
                = ( some_state @ A5 ) )
             => ( ( Y2
                  = ( some_state @ B7 ) )
               => ( P @ X3 @ Y2 ) ) )
         => ( P @ X3 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_434_option_Oexpand,axiom,
    ! [Option: option_state,Option2: option_state] :
      ( ( ( Option = none_state )
        = ( Option2 = none_state ) )
     => ( ( ( Option != none_state )
         => ( ( Option2 != none_state )
           => ( ( the_state @ Option )
              = ( the_state @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_435_asso2,axiom,
    ! [A3: state,B2: state,Ab: state,C: state] :
      ( ( ( ( plus @ A3 @ B2 )
          = ( some_state @ Ab ) )
        & ( ( plus @ B2 @ C )
          = none_state ) )
     => ( ( plus @ Ab @ C )
        = none_state ) ) ).

% asso2
thf(fact_436_PartialSA_Oasso3,axiom,
    ! [A3: state,B2: state,C: state,Bc: state] :
      ( ( ( plus @ A3 @ B2 )
        = none_state )
     => ( ( ( plus @ B2 @ C )
          = ( some_state @ Bc ) )
       => ( ( plus @ A3 @ Bc )
          = none_state ) ) ) ).

% PartialSA.asso3
thf(fact_437_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_438_PartialSA_Odefined__def,axiom,
    ( defined
    = ( ^ [A2: state,B: state] :
          ( ( plus @ A2 @ B )
         != none_state ) ) ) ).

% PartialSA.defined_def
thf(fact_439_package__logic_Obool__conj__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A3: state > $o,B2: state > $o,X3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1330488657391168505_state @ A3 @ B2 @ X3 )
        = ( ( A3 @ X3 )
          & ( B2 @ X3 ) ) ) ) ).

% package_logic.bool_conj_def
thf(fact_440_PartialSA_Osplus_Osimps_I1_J,axiom,
    ! [Uu: option_state] :
      ( ( sep_splus_state @ plus @ none_state @ Uu )
      = none_state ) ).

% PartialSA.splus.simps(1)
thf(fact_441_PartialSA_Osplus_Oelims,axiom,
    ! [X3: option_state,Xa: option_state,Y2: option_state] :
      ( ( ( sep_splus_state @ plus @ X3 @ Xa )
        = Y2 )
     => ( ( ( X3 = none_state )
         => ( Y2 != none_state ) )
       => ( ( ? [V: state] :
                ( X3
                = ( some_state @ V ) )
           => ( ( Xa = none_state )
             => ( Y2 != none_state ) ) )
         => ~ ! [A5: state] :
                ( ( X3
                  = ( some_state @ A5 ) )
               => ! [B7: state] :
                    ( ( Xa
                      = ( some_state @ B7 ) )
                   => ( Y2
                     != ( plus @ A5 @ B7 ) ) ) ) ) ) ) ).

% PartialSA.splus.elims
thf(fact_442_PartialSA_Osplus_Osimps_I2_J,axiom,
    ! [V2: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ V2 ) @ none_state )
      = none_state ) ).

% PartialSA.splus.simps(2)
thf(fact_443_PartialSA_Omax__projection__prop__pure__core,axiom,
    sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).

% PartialSA.max_projection_prop_pure_core
thf(fact_444_sep__algebra_Oup__closed_Ocong,axiom,
    sep_up_closed_state = sep_up_closed_state ).

% sep_algebra.up_closed.cong
thf(fact_445_sep__algebra_Opure_Ocong,axiom,
    sep_pure_state = sep_pure_state ).

% sep_algebra.pure.cong
thf(fact_446_sep__algebra_Osplus_Ocong,axiom,
    sep_splus_state = sep_splus_state ).

% sep_algebra.splus.cong
thf(fact_447_PartialSA_Ompp__invo,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( F @ ( F @ X3 ) )
        = ( F @ X3 ) ) ) ).

% PartialSA.mpp_invo
thf(fact_448_PartialSA_Ompp__prop,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( P @ ( F @ X3 ) ) ) ).

% PartialSA.mpp_prop
thf(fact_449_PartialSA_Omax__projection__prop__def,axiom,
    ! [P: state > $o,F: state > state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
      = ( ! [X2: state] :
            ( ( greater @ X2 @ ( F @ X2 ) )
            & ( P @ ( F @ X2 ) )
            & ! [P4: state] :
                ( ( ( P @ P4 )
                  & ( greater @ X2 @ P4 ) )
               => ( greater @ ( F @ X2 ) @ P4 ) ) ) ) ) ).

% PartialSA.max_projection_prop_def
thf(fact_450_PartialSA_Omax__projection__propI,axiom,
    ! [F: state > state,P: state > $o] :
      ( ! [X4: state] : ( greater @ X4 @ ( F @ X4 ) )
     => ( ! [X4: state] : ( P @ ( F @ X4 ) )
       => ( ! [X4: state,P6: state] :
              ( ( ( P @ P6 )
                & ( greater @ X4 @ P6 ) )
             => ( greater @ ( F @ X4 ) @ P6 ) )
         => ( sep_ma8214210560313151521_state @ plus @ P @ F ) ) ) ) ).

% PartialSA.max_projection_propI
thf(fact_451_PartialSA_Ompp__smaller,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( greater @ X3 @ ( F @ X3 ) ) ) ).

% PartialSA.mpp_smaller
thf(fact_452_PartialSA_Ompp__mono,axiom,
    ! [P: state > $o,F: state > state,A3: state,B2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A3 @ B2 )
       => ( greater @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ).

% PartialSA.mpp_mono
thf(fact_453_PartialSA_OmppI,axiom,
    ! [P: state > $o,F: state > state,A3: state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A3 @ X3 )
       => ( ( P @ X3 )
         => ( ( greater @ X3 @ ( F @ A3 ) )
           => ( X3
              = ( F @ A3 ) ) ) ) ) ) ).

% PartialSA.mppI
thf(fact_454_PartialSA_Omax__projection__propE_I3_J,axiom,
    ! [P: state > $o,F: state > state,P3: state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( ( P @ P3 )
          & ( greater @ X3 @ P3 ) )
       => ( greater @ ( F @ X3 ) @ P3 ) ) ) ).

% PartialSA.max_projection_propE(3)
thf(fact_455_PartialSA_Ompp__compatible,axiom,
    ! [P: state > $o,F: state > state,A3: state,B2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( defined @ A3 @ B2 )
       => ( defined @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ).

% PartialSA.mpp_compatible
thf(fact_456_compatible__options_Oelims_I1_J,axiom,
    ! [X3: option_state,Xa: option_state,Y2: $o] :
      ( ( ( compat2278460363914054422_state @ X3 @ Xa )
        = Y2 )
     => ( ! [A5: state] :
            ( ( X3
              = ( some_state @ A5 ) )
           => ! [B7: state] :
                ( ( Xa
                  = ( some_state @ B7 ) )
               => ( Y2
                  = ( A5 != B7 ) ) ) )
       => ( ( ( X3 = none_state )
           => ~ Y2 )
         => ~ ( ( Xa = none_state )
             => ~ Y2 ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_457_compatible__options_Oelims_I2_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ( compat2278460363914054422_state @ X3 @ Xa )
     => ( ! [A5: state] :
            ( ( X3
              = ( some_state @ A5 ) )
           => ! [B7: state] :
                ( ( Xa
                  = ( some_state @ B7 ) )
               => ( A5 != B7 ) ) )
       => ( ( X3 != none_state )
         => ( Xa = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_458_PartialSA_Oup__closed__plus__UNIV,axiom,
    ! [A4: set_state] : ( sep_up_closed_state @ plus @ ( add_set @ A4 @ top_top_set_state ) ) ).

% PartialSA.up_closed_plus_UNIV
thf(fact_459_PartialSA_Ointuitionistic__def,axiom,
    ! [A4: state > $o] :
      ( ( packag8361946002163212404_state @ plus @ A4 )
      = ( ! [Phi3: state,Phi2: state] :
            ( ( ( greater @ Phi3 @ Phi2 )
              & ( A4 @ Phi2 ) )
           => ( A4 @ Phi3 ) ) ) ) ).

% PartialSA.intuitionistic_def
thf(fact_460_image__Fpow__mono,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
     => ( ord_le5175021213330142598_state @ ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( finite_Fpow_state @ A4 ) ) @ ( finite_Fpow_state @ B3 ) ) ) ).

% image_Fpow_mono
thf(fact_461_option_Osimps_I15_J,axiom,
    ! [X22: state] :
      ( ( set_option_state2 @ ( some_state @ X22 ) )
      = ( insert_state @ X22 @ bot_bot_set_state ) ) ).

% option.simps(15)
thf(fact_462_subset__Compl__singleton,axiom,
    ! [A4: set_state,B2: state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( uminus472742206872269241_state @ ( insert_state @ B2 @ bot_bot_set_state ) ) )
      = ( ~ ( member_state @ B2 @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_463_UNIV__I,axiom,
    ! [X3: state] : ( member_state @ X3 @ top_top_set_state ) ).

% UNIV_I
thf(fact_464_Compl__iff,axiom,
    ! [C: state,A4: set_state] :
      ( ( member_state @ C @ ( uminus472742206872269241_state @ A4 ) )
      = ( ~ ( member_state @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_465_ComplI,axiom,
    ! [C: state,A4: set_state] :
      ( ~ ( member_state @ C @ A4 )
     => ( member_state @ C @ ( uminus472742206872269241_state @ A4 ) ) ) ).

% ComplI
thf(fact_466_Compl__anti__mono,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ B3 ) @ ( uminus472742206872269241_state @ A4 ) ) ) ).

% Compl_anti_mono
thf(fact_467_Compl__subset__Compl__iff,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ A4 ) @ ( uminus472742206872269241_state @ B3 ) )
      = ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ).

% Compl_subset_Compl_iff
thf(fact_468_elem__set,axiom,
    ! [X3: state,Xo: option_state] :
      ( ( member_state @ X3 @ ( set_option_state2 @ Xo ) )
      = ( Xo
        = ( some_state @ X3 ) ) ) ).

% elem_set
thf(fact_469_set__empty__eq,axiom,
    ! [Xo: option_state] :
      ( ( ( set_option_state2 @ Xo )
        = bot_bot_set_state )
      = ( Xo = none_state ) ) ).

% set_empty_eq
thf(fact_470_Compl__UNIV__eq,axiom,
    ( ( uminus472742206872269241_state @ top_top_set_state )
    = bot_bot_set_state ) ).

% Compl_UNIV_eq
thf(fact_471_Compl__empty__eq,axiom,
    ( ( uminus472742206872269241_state @ bot_bot_set_state )
    = top_top_set_state ) ).

% Compl_empty_eq
thf(fact_472_UNIV__witness,axiom,
    ? [X4: state] : ( member_state @ X4 @ top_top_set_state ) ).

% UNIV_witness
thf(fact_473_UNIV__eq__I,axiom,
    ! [A4: set_state] :
      ( ! [X4: state] : ( member_state @ X4 @ A4 )
     => ( top_top_set_state = A4 ) ) ).

% UNIV_eq_I
thf(fact_474_ComplD,axiom,
    ! [C: state,A4: set_state] :
      ( ( member_state @ C @ ( uminus472742206872269241_state @ A4 ) )
     => ~ ( member_state @ C @ A4 ) ) ).

% ComplD
thf(fact_475_package__logic_Ointuitionistic_Ocong,axiom,
    packag8361946002163212404_state = packag8361946002163212404_state ).

% package_logic.intuitionistic.cong
thf(fact_476_top__greatest,axiom,
    ! [A3: set_state] : ( ord_le2494988322063910608_state @ A3 @ top_top_set_state ) ).

% top_greatest
thf(fact_477_top_Oextremum__unique,axiom,
    ! [A3: set_state] :
      ( ( ord_le2494988322063910608_state @ top_top_set_state @ A3 )
      = ( A3 = top_top_set_state ) ) ).

% top.extremum_unique
thf(fact_478_top_Oextremum__uniqueI,axiom,
    ! [A3: set_state] :
      ( ( ord_le2494988322063910608_state @ top_top_set_state @ A3 )
     => ( A3 = top_top_set_state ) ) ).

% top.extremum_uniqueI
thf(fact_479_range__eqI,axiom,
    ! [B2: state,F: state > state,X3: state] :
      ( ( B2
        = ( F @ X3 ) )
     => ( member_state @ B2 @ ( image_state_state @ F @ top_top_set_state ) ) ) ).

% range_eqI
thf(fact_480_rangeI,axiom,
    ! [F: state > state,X3: state] : ( member_state @ ( F @ X3 ) @ ( image_state_state @ F @ top_top_set_state ) ) ).

% rangeI
thf(fact_481_empty__not__UNIV,axiom,
    bot_bot_set_state != top_top_set_state ).

% empty_not_UNIV
thf(fact_482_subset__UNIV,axiom,
    ! [A4: set_state] : ( ord_le2494988322063910608_state @ A4 @ top_top_set_state ) ).

% subset_UNIV
thf(fact_483_insert__UNIV,axiom,
    ! [X3: state] :
      ( ( insert_state @ X3 @ top_top_set_state )
      = top_top_set_state ) ).

% insert_UNIV
thf(fact_484_option_Oset__cases,axiom,
    ! [E: state,A3: option_state] :
      ( ( member_state @ E @ ( set_option_state2 @ A3 ) )
     => ( A3
        = ( some_state @ E ) ) ) ).

% option.set_cases
thf(fact_485_option_Oset__intros,axiom,
    ! [X22: state] : ( member_state @ X22 @ ( set_option_state2 @ ( some_state @ X22 ) ) ) ).

% option.set_intros
thf(fact_486_ospec,axiom,
    ! [A4: option_state,P: state > $o,X3: state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ ( set_option_state2 @ A4 ) )
         => ( P @ X4 ) )
     => ( ( A4
          = ( some_state @ X3 ) )
       => ( P @ X3 ) ) ) ).

% ospec
thf(fact_487_subset__Compl__self__eq,axiom,
    ! [A4: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( uminus472742206872269241_state @ A4 ) )
      = ( A4 = bot_bot_set_state ) ) ).

% subset_Compl_self_eq
thf(fact_488_empty__in__Fpow,axiom,
    ! [A4: set_state] : ( member_set_state @ bot_bot_set_state @ ( finite_Fpow_state @ A4 ) ) ).

% empty_in_Fpow
thf(fact_489_Fpow__mono,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le5175021213330142598_state @ ( finite_Fpow_state @ A4 ) @ ( finite_Fpow_state @ B3 ) ) ) ).

% Fpow_mono
thf(fact_490_range__subsetD,axiom,
    ! [F: state > state,B3: set_state,I: state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ top_top_set_state ) @ B3 )
     => ( member_state @ ( F @ I ) @ B3 ) ) ).

% range_subsetD
thf(fact_491_compatible__options_Osimps_I1_J,axiom,
    ! [A3: state,B2: state] :
      ( ( compat2278460363914054422_state @ ( some_state @ A3 ) @ ( some_state @ B2 ) )
      = ( A3 = B2 ) ) ).

% compatible_options.simps(1)
thf(fact_492_compatible__options_Oelims_I3_J,axiom,
    ! [X3: option_state,Xa: option_state] :
      ( ~ ( compat2278460363914054422_state @ X3 @ Xa )
     => ~ ! [A5: state] :
            ( ( X3
              = ( some_state @ A5 ) )
           => ! [B7: state] :
                ( ( Xa
                  = ( some_state @ B7 ) )
               => ( A5 = B7 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_493_compatible__options_Osimps_I2_J,axiom,
    ! [Uv: option_state] : ( compat2278460363914054422_state @ none_state @ Uv ) ).

% compatible_options.simps(2)
thf(fact_494_compatible__options_Osimps_I3_J,axiom,
    ! [Uu: option_state] : ( compat2278460363914054422_state @ Uu @ none_state ) ).

% compatible_options.simps(3)
thf(fact_495_UNIV__option__conv,axiom,
    ( top_to7666338855062656496_state
    = ( insert_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) ) ).

% UNIV_option_conv
thf(fact_496_option_Osimps_I14_J,axiom,
    ( ( set_option_state2 @ none_state )
    = bot_bot_set_state ) ).

% option.simps(14)
thf(fact_497_option_Oset__sel,axiom,
    ! [A3: option_state] :
      ( ( A3 != none_state )
     => ( member_state @ ( the_state @ A3 ) @ ( set_option_state2 @ A3 ) ) ) ).

% option.set_sel
thf(fact_498_range__eq__singletonD,axiom,
    ! [F: state > state,A3: state,X3: state] :
      ( ( ( image_state_state @ F @ top_top_set_state )
        = ( insert_state @ A3 @ bot_bot_set_state ) )
     => ( ( F @ X3 )
        = A3 ) ) ).

% range_eq_singletonD
thf(fact_499_notin__range__Some,axiom,
    ! [X3: option_state] :
      ( ( ~ ( member_option_state @ X3 @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) )
      = ( X3 = none_state ) ) ).

% notin_range_Some
thf(fact_500_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus472742206872269241_state @ bot_bot_set_state )
    = top_top_set_state ) ).

% boolean_algebra.compl_zero
thf(fact_501_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus472742206872269241_state @ top_top_set_state )
    = bot_bot_set_state ) ).

% boolean_algebra.compl_one
thf(fact_502_compl__le__compl__iff,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ X3 ) @ ( uminus472742206872269241_state @ Y2 ) )
      = ( ord_le2494988322063910608_state @ Y2 @ X3 ) ) ).

% compl_le_compl_iff
thf(fact_503_surj__Compl__image__subset,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( ( image_state_state @ F @ top_top_set_state )
        = top_top_set_state )
     => ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ ( image_state_state @ F @ A4 ) ) @ ( image_state_state @ F @ ( uminus472742206872269241_state @ A4 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_504_top__set__def,axiom,
    ( top_top_set_state
    = ( collect_state @ top_top_state_o ) ) ).

% top_set_def
thf(fact_505_compl__le__swap2,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ Y2 ) @ X3 )
     => ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ X3 ) @ Y2 ) ) ).

% compl_le_swap2
thf(fact_506_compl__le__swap1,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ ( uminus472742206872269241_state @ X3 ) )
     => ( ord_le2494988322063910608_state @ X3 @ ( uminus472742206872269241_state @ Y2 ) ) ) ).

% compl_le_swap1
thf(fact_507_compl__mono,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ Y2 ) @ ( uminus472742206872269241_state @ X3 ) ) ) ).

% compl_mono
thf(fact_508_image__Pow__mono,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
     => ( ord_le5175021213330142598_state @ ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( pow_state @ A4 ) ) @ ( pow_state @ B3 ) ) ) ).

% image_Pow_mono
thf(fact_509_inj__image__Compl__subset,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( inj_on_state_state @ F @ top_top_set_state )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( uminus472742206872269241_state @ A4 ) ) @ ( uminus472742206872269241_state @ ( image_state_state @ F @ A4 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_510_minus__def,axiom,
    ( minus
    = ( sep_minus_state @ plus @ core ) ) ).

% minus_def
thf(fact_511_Greatest__equality,axiom,
    ! [P: set_state > $o,X3: set_state] :
      ( ( P @ X3 )
     => ( ! [Y3: set_state] :
            ( ( P @ Y3 )
           => ( ord_le2494988322063910608_state @ Y3 @ X3 ) )
       => ( ( order_2642746146112740183_state @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_512_Greatest__equality,axiom,
    ! [P: nat > $o,X3: nat] :
      ( ( P @ X3 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X3 ) )
       => ( ( order_Greatest_nat @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_513_GreatestI2__order,axiom,
    ! [P: set_state > $o,X3: set_state,Q: set_state > $o] :
      ( ( P @ X3 )
     => ( ! [Y3: set_state] :
            ( ( P @ Y3 )
           => ( ord_le2494988322063910608_state @ Y3 @ X3 ) )
       => ( ! [X4: set_state] :
              ( ( P @ X4 )
             => ( ! [Y5: set_state] :
                    ( ( P @ Y5 )
                   => ( ord_le2494988322063910608_state @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_2642746146112740183_state @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_514_GreatestI2__order,axiom,
    ! [P: nat > $o,X3: nat,Q: nat > $o] :
      ( ( P @ X3 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X3 ) )
       => ( ! [X4: nat] :
              ( ( P @ X4 )
             => ( ! [Y5: nat] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_nat @ Y5 @ X4 ) )
               => ( Q @ X4 ) ) )
         => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_515_sep__algebra_Omono__prop__set__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_mono_prop_state @ Plus @ P )
       => ( ( sep_equiv_state @ Plus @ A4 @ B3 )
         => ( ( sep_setify_state @ P @ A4 )
            = ( sep_setify_state @ P @ B3 ) ) ) ) ) ).

% sep_algebra.mono_prop_set_equiv
thf(fact_516_Pow__UNIV,axiom,
    ( ( pow_state @ top_top_set_state )
    = top_to5262587396890829782_state ) ).

% Pow_UNIV
thf(fact_517_PowI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( member_set_state @ A4 @ ( pow_state @ B3 ) ) ) ).

% PowI
thf(fact_518_Pow__iff,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( member_set_state @ A4 @ ( pow_state @ B3 ) )
      = ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% Pow_iff
thf(fact_519_Pow__singleton__iff,axiom,
    ! [X6: set_state,Y6: set_state] :
      ( ( ( pow_state @ X6 )
        = ( insert_set_state @ Y6 @ bot_bo2271482359692755898_state ) )
      = ( ( X6 = bot_bot_set_state )
        & ( Y6 = bot_bot_set_state ) ) ) ).

% Pow_singleton_iff
thf(fact_520_Pow__empty,axiom,
    ( ( pow_state @ bot_bot_set_state )
    = ( insert_set_state @ bot_bot_set_state @ bot_bo2271482359692755898_state ) ) ).

% Pow_empty
thf(fact_521_sep__algebra_Oasso1,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,Ab: state,C: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A3 @ B2 )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B2 @ C )
            = ( some_state @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A3 @ Bc ) ) ) ) ).

% sep_algebra.asso1
thf(fact_522_sep__algebra_Ocore__max,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X3 )
          = ( Plus @ X3 @ C ) )
       => ? [R: state] :
            ( ( some_state @ ( Core @ X3 ) )
            = ( Plus @ C @ R ) ) ) ) ).

% sep_algebra.core_max
thf(fact_523_sep__algebra_Ocore__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C: state,A3: state,B2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C )
          = ( Plus @ A3 @ B2 ) )
       => ( ( some_state @ ( Core @ C ) )
          = ( Plus @ ( Core @ A3 ) @ ( Core @ B2 ) ) ) ) ) ).

% sep_algebra.core_sum
thf(fact_524_sep__algebra_Opositivity,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A3 @ B2 )
          = ( some_state @ C ) )
       => ( ( ( some_state @ C )
            = ( Plus @ C @ C ) )
         => ( ( some_state @ A3 )
            = ( Plus @ A3 @ A3 ) ) ) ) ) ).

% sep_algebra.positivity
thf(fact_525_sep__algebra_Ocancellative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,X3: state,Y2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A3 )
          = ( Plus @ B2 @ X3 ) )
       => ( ( ( some_state @ A3 )
            = ( Plus @ B2 @ Y2 ) )
         => ( ( ( Core @ X3 )
              = ( Core @ Y2 ) )
           => ( X3 = Y2 ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_526_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ ( Core @ X3 ) )
        = ( Plus @ ( Core @ X3 ) @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_527_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ X3 )
        = ( Plus @ X3 @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_528_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A3: state,B2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X3 )
          = ( Plus @ A3 @ B2 ) )
       => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ X3 @ A3 ) )
          = ( Plus @ B2 @ ( Core @ X3 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_529_inj__Some,axiom,
    ! [A4: set_state] : ( inj_on3577428053172332983_state @ some_state @ A4 ) ).

% inj_Some
thf(fact_530_package__logic_Oaxioms_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( sep_algebra_state @ Plus @ Core ) ) ).

% package_logic.axioms(1)
thf(fact_531_sep__algebra_Osplus__comm,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: option_state,B2: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ A3 @ B2 )
        = ( sep_splus_state @ Plus @ B2 @ A3 ) ) ) ).

% sep_algebra.splus_comm
thf(fact_532_sep__algebra_Osplus__asso,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: option_state,B2: option_state,C: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ A3 @ B2 ) @ C )
        = ( sep_splus_state @ Plus @ A3 @ ( sep_splus_state @ Plus @ B2 @ C ) ) ) ) ).

% sep_algebra.splus_asso
thf(fact_533_Pow__bottom,axiom,
    ! [B3: set_state] : ( member_set_state @ bot_bot_set_state @ ( pow_state @ B3 ) ) ).

% Pow_bottom
thf(fact_534_PowD,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( member_set_state @ A4 @ ( pow_state @ B3 ) )
     => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% PowD
thf(fact_535_sep__algebra_Osetify__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A4: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ A4 )
        = ( ! [X2: state] :
              ( ( member_state @ X2 @ A4 )
             => ( P @ X2 ) ) ) ) ) ).

% sep_algebra.setify_def
thf(fact_536_sep__algebra_Oasso2,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,Ab: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A3 @ B2 )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B2 @ C )
            = none_state ) )
       => ( ( Plus @ Ab @ C )
          = none_state ) ) ) ).

% sep_algebra.asso2
thf(fact_537_sep__algebra_Oasso3,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,C: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A3 @ B2 )
          = none_state )
       => ( ( ( Plus @ B2 @ C )
            = ( some_state @ Bc ) )
         => ( ( Plus @ A3 @ Bc )
            = none_state ) ) ) ) ).

% sep_algebra.asso3
thf(fact_538_sep__algebra_Ointro,axiom,
    ! [Plus: state > state > option_state,Core: state > state] :
      ( ! [A5: state,B7: state] :
          ( ( Plus @ A5 @ B7 )
          = ( Plus @ B7 @ A5 ) )
     => ( ! [A5: state,B7: state,Ab2: state,C6: state,Bc2: state] :
            ( ( ( ( Plus @ A5 @ B7 )
                = ( some_state @ Ab2 ) )
              & ( ( Plus @ B7 @ C6 )
                = ( some_state @ Bc2 ) ) )
           => ( ( Plus @ Ab2 @ C6 )
              = ( Plus @ A5 @ Bc2 ) ) )
       => ( ! [A5: state,B7: state,Ab2: state,C6: state] :
              ( ( ( ( Plus @ A5 @ B7 )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus @ B7 @ C6 )
                  = none_state ) )
             => ( ( Plus @ Ab2 @ C6 )
                = none_state ) )
         => ( ! [X4: state] :
                ( ( some_state @ X4 )
                = ( Plus @ X4 @ ( Core @ X4 ) ) )
           => ( ! [X4: state] :
                  ( ( some_state @ ( Core @ X4 ) )
                  = ( Plus @ ( Core @ X4 ) @ ( Core @ X4 ) ) )
             => ( ! [X4: state,C6: state] :
                    ( ( ( some_state @ X4 )
                      = ( Plus @ X4 @ C6 ) )
                   => ? [R4: state] :
                        ( ( some_state @ ( Core @ X4 ) )
                        = ( Plus @ C6 @ R4 ) ) )
               => ( ! [C6: state,A5: state,B7: state] :
                      ( ( ( some_state @ C6 )
                        = ( Plus @ A5 @ B7 ) )
                     => ( ( some_state @ ( Core @ C6 ) )
                        = ( Plus @ ( Core @ A5 ) @ ( Core @ B7 ) ) ) )
                 => ( ! [A5: state,B7: state,C6: state] :
                        ( ( ( Plus @ A5 @ B7 )
                          = ( some_state @ C6 ) )
                       => ( ( ( some_state @ C6 )
                            = ( Plus @ C6 @ C6 ) )
                         => ( ( some_state @ A5 )
                            = ( Plus @ A5 @ A5 ) ) ) )
                   => ( ! [A5: state,B7: state,X4: state,Y3: state] :
                          ( ( ( some_state @ A5 )
                            = ( Plus @ B7 @ X4 ) )
                         => ( ( ( some_state @ A5 )
                              = ( Plus @ B7 @ Y3 ) )
                           => ( ( ( Core @ X4 )
                                = ( Core @ Y3 ) )
                             => ( X4 = Y3 ) ) ) )
                     => ( sep_algebra_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_539_sep__algebra__def,axiom,
    ( sep_algebra_state
    = ( ^ [Plus2: state > state > option_state,Core2: state > state] :
          ( ! [A2: state,B: state] :
              ( ( Plus2 @ A2 @ B )
              = ( Plus2 @ B @ A2 ) )
          & ! [A2: state,B: state,Ab3: state,C3: state,Bc3: state] :
              ( ( ( ( Plus2 @ A2 @ B )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus2 @ B @ C3 )
                  = ( some_state @ Bc3 ) ) )
             => ( ( Plus2 @ Ab3 @ C3 )
                = ( Plus2 @ A2 @ Bc3 ) ) )
          & ! [A2: state,B: state,Ab3: state,C3: state] :
              ( ( ( ( Plus2 @ A2 @ B )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus2 @ B @ C3 )
                  = none_state ) )
             => ( ( Plus2 @ Ab3 @ C3 )
                = none_state ) )
          & ! [X2: state] :
              ( ( some_state @ X2 )
              = ( Plus2 @ X2 @ ( Core2 @ X2 ) ) )
          & ! [X2: state] :
              ( ( some_state @ ( Core2 @ X2 ) )
              = ( Plus2 @ ( Core2 @ X2 ) @ ( Core2 @ X2 ) ) )
          & ! [X2: state,C3: state] :
              ( ( ( some_state @ X2 )
                = ( Plus2 @ X2 @ C3 ) )
             => ? [R3: state] :
                  ( ( some_state @ ( Core2 @ X2 ) )
                  = ( Plus2 @ C3 @ R3 ) ) )
          & ! [C3: state,A2: state,B: state] :
              ( ( ( some_state @ C3 )
                = ( Plus2 @ A2 @ B ) )
             => ( ( some_state @ ( Core2 @ C3 ) )
                = ( Plus2 @ ( Core2 @ A2 ) @ ( Core2 @ B ) ) ) )
          & ! [A2: state,B: state,C3: state] :
              ( ( ( Plus2 @ A2 @ B )
                = ( some_state @ C3 ) )
             => ( ( ( some_state @ C3 )
                  = ( Plus2 @ C3 @ C3 ) )
               => ( ( some_state @ A2 )
                  = ( Plus2 @ A2 @ A2 ) ) ) )
          & ! [A2: state,B: state,X2: state,Y: state] :
              ( ( ( some_state @ A2 )
                = ( Plus2 @ B @ X2 ) )
             => ( ( ( some_state @ A2 )
                  = ( Plus2 @ B @ Y ) )
               => ( ( ( Core2 @ X2 )
                    = ( Core2 @ Y ) )
                 => ( X2 = Y ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_540_inj__on__image__eq__iff,axiom,
    ! [F: state > state,C2: set_state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ C2 )
     => ( ( ord_le2494988322063910608_state @ A4 @ C2 )
       => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
         => ( ( ( image_state_state @ F @ A4 )
              = ( image_state_state @ F @ B3 ) )
            = ( A4 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_541_inj__on__image__mem__iff,axiom,
    ! [F: state > state,B3: set_state,A3: state,A4: set_state] :
      ( ( inj_on_state_state @ F @ B3 )
     => ( ( member_state @ A3 @ B3 )
       => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
         => ( ( member_state @ ( F @ A3 ) @ ( image_state_state @ F @ A4 ) )
            = ( member_state @ A3 @ A4 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_542_inj__img__insertE,axiom,
    ! [F: state > state,A4: set_state,X3: state,B3: set_state] :
      ( ( inj_on_state_state @ F @ A4 )
     => ( ~ ( member_state @ X3 @ B3 )
       => ( ( ( insert_state @ X3 @ B3 )
            = ( image_state_state @ F @ A4 ) )
         => ~ ! [X8: state,A8: set_state] :
                ( ~ ( member_state @ X8 @ A8 )
               => ( ( A4
                    = ( insert_state @ X8 @ A8 ) )
                 => ( ( X3
                      = ( F @ X8 ) )
                   => ( B3
                     != ( image_state_state @ F @ A8 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_543_PartialSA_Osep__algebra__axioms,axiom,
    sep_algebra_state @ plus @ core ).

% PartialSA.sep_algebra_axioms
thf(fact_544_sep__algebra_Osplus_Osimps_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ A3 ) @ ( some_state @ B2 ) )
        = ( Plus @ A3 @ B2 ) ) ) ).

% sep_algebra.splus.simps(3)
thf(fact_545_sep__algebra_Osplus__develop,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,C: state,D: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A3 )
          = ( Plus @ B2 @ C ) )
       => ( ( Plus @ A3 @ D )
          = ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ ( some_state @ B2 ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ) ).

% sep_algebra.splus_develop
thf(fact_546_sep__algebra_Opure__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A3 )
        = ( ( some_state @ A3 )
          = ( Plus @ A3 @ A3 ) ) ) ) ).

% sep_algebra.pure_def
thf(fact_547_sep__algebra_Opure__stable,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A3 )
       => ( ( sep_pure_state @ Plus @ B2 )
         => ( ( ( some_state @ C )
              = ( Plus @ A3 @ B2 ) )
           => ( sep_pure_state @ Plus @ C ) ) ) ) ) ).

% sep_algebra.pure_stable
thf(fact_548_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Uu: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ none_state @ Uu )
        = none_state ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_549_sep__algebra_Omax__projection__prop__pure__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_ma8214210560313151521_state @ Plus @ ( sep_pure_state @ Plus ) @ Core ) ) ).

% sep_algebra.max_projection_prop_pure_core
thf(fact_550_sep__algebra_Oup__close__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A4 )
       => ( ( sep_up_closed_state @ Plus @ B3 )
         => ( ( sep_equiv_state @ Plus @ A4 @ B3 )
            = ( A4 = B3 ) ) ) ) ) ).

% sep_algebra.up_close_equiv
thf(fact_551_Pow__mono,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le5175021213330142598_state @ ( pow_state @ A4 ) @ ( pow_state @ B3 ) ) ) ).

% Pow_mono
thf(fact_552_sep__algebra_Oupper__closure__up__closed,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_up_closed_state @ Plus @ ( sep_up1246176804924251236_state @ Plus @ A4 ) ) ) ).

% sep_algebra.upper_closure_up_closed
thf(fact_553_image__Pow__surj,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( ( image_state_state @ F @ A4 )
        = B3 )
     => ( ( image_2476256681063834599_state @ ( image_state_state @ F ) @ ( pow_state @ A4 ) )
        = ( pow_state @ B3 ) ) ) ).

% image_Pow_surj
thf(fact_554_inj__image__subset__iff,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ top_top_set_state )
     => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) )
        = ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ).

% inj_image_subset_iff
thf(fact_555_sep__algebra_Osplus_Osimps_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,V2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ V2 ) @ none_state )
        = none_state ) ) ).

% sep_algebra.splus.simps(2)
thf(fact_556_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: option_state,Xa: option_state,Y2: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X3 @ Xa )
          = Y2 )
       => ( ( ( X3 = none_state )
           => ( Y2 != none_state ) )
         => ( ( ? [V: state] :
                  ( X3
                  = ( some_state @ V ) )
             => ( ( Xa = none_state )
               => ( Y2 != none_state ) ) )
           => ~ ! [A5: state] :
                  ( ( X3
                    = ( some_state @ A5 ) )
                 => ! [B7: state] :
                      ( ( Xa
                        = ( some_state @ B7 ) )
                     => ( Y2
                       != ( Plus @ A5 @ B7 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_557_sep__algebra_Oequiv__up__closed__subset,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state,B3: set_state,C2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A4 )
       => ( ( sep_equiv_state @ Plus @ B3 @ C2 )
         => ( ( ord_le2494988322063910608_state @ B3 @ A4 )
            = ( ord_le2494988322063910608_state @ C2 @ A4 ) ) ) ) ) ).

% sep_algebra.equiv_up_closed_subset
thf(fact_558_inj__on__iff__surj,axiom,
    ! [A4: set_state,A7: set_state] :
      ( ( A4 != bot_bot_set_state )
     => ( ( ? [F3: state > state] :
              ( ( inj_on_state_state @ F3 @ A4 )
              & ( ord_le2494988322063910608_state @ ( image_state_state @ F3 @ A4 ) @ A7 ) ) )
        = ( ? [G2: state > state] :
              ( ( image_state_state @ G2 @ A7 )
              = A4 ) ) ) ) ).

% inj_on_iff_surj
thf(fact_559_subset__image__inj,axiom,
    ! [S: set_state,F: state > state,T: set_state] :
      ( ( ord_le2494988322063910608_state @ S @ ( image_state_state @ F @ T ) )
      = ( ? [U: set_state] :
            ( ( ord_le2494988322063910608_state @ U @ T )
            & ( inj_on_state_state @ F @ U )
            & ( S
              = ( image_state_state @ F @ U ) ) ) ) ) ).

% subset_image_inj
thf(fact_560_sep__algebra_Osetify__sum__image,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( image_state_state @ F @ A4 ) @ B3 ) )
        = ( ! [X2: state] :
              ( ( member_state @ X2 @ A4 )
             => ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( insert_state @ ( F @ X2 ) @ bot_bot_set_state ) @ B3 ) ) ) ) ) ) ).

% sep_algebra.setify_sum_image
thf(fact_561_sep__algebra_Osetify__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ A4 @ B3 ) )
        = ( ! [X2: state] :
              ( ( member_state @ X2 @ A4 )
             => ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( insert_state @ X2 @ bot_bot_set_state ) @ B3 ) ) ) ) ) ) ).

% sep_algebra.setify_sum
thf(fact_562_package__logic__def,axiom,
    ( package_logic_state
    = ( ^ [Plus2: state > state > option_state,Core2: state > state,Unit2: state,Stable2: state > $o] :
          ( ( sep_algebra_state @ Plus2 @ Core2 )
          & ( packag2647621270594721818_state @ Plus2 @ Unit2 @ Stable2 ) ) ) ) ).

% package_logic_def
thf(fact_563_package__logic_Ointro,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( packag2647621270594721818_state @ Plus @ Unit @ Stable )
       => ( package_logic_state @ Plus @ Core @ Unit @ Stable ) ) ) ).

% package_logic.intro
thf(fact_564_sep__algebra_Oempty__set__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ bot_bot_set_state @ A4 )
        = bot_bot_set_state ) ) ).

% sep_algebra.empty_set_sum
thf(fact_565_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Phi: state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ Phi @ ( sep_add_set_state @ Plus @ A4 @ B3 ) )
        = ( ? [A2: state,B: state] :
              ( ( ( some_state @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( member_state @ A2 @ A4 )
              & ( member_state @ B @ B3 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_566_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X3 @ ( sep_add_set_state @ Plus @ A4 @ B3 ) )
        = ( ? [A2: state,B: state] :
              ( ( member_state @ A2 @ A4 )
              & ( member_state @ B @ B3 )
              & ( ( some_state @ X3 )
                = ( Plus @ A2 @ B ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_567_package__logic__axioms__def,axiom,
    ( packag2647621270594721818_state
    = ( ^ [Plus2: state > state > option_state,Unit2: state,Stable2: state > $o] :
          ( ! [A2: state] :
              ( ( some_state @ A2 )
              = ( Plus2 @ A2 @ Unit2 ) )
          & ! [A2: state,B: state,X2: state] :
              ( ( Stable2 @ A2 )
             => ( ( Stable2 @ B )
               => ( ( ( some_state @ X2 )
                    = ( Plus2 @ A2 @ B ) )
                 => ( Stable2 @ X2 ) ) ) )
          & ( Stable2 @ Unit2 ) ) ) ) ).

% package_logic_axioms_def
thf(fact_568_package__logic__axioms_Ointro,axiom,
    ! [Plus: state > state > option_state,Unit: state,Stable: state > $o] :
      ( ! [A5: state] :
          ( ( some_state @ A5 )
          = ( Plus @ A5 @ Unit ) )
     => ( ! [A5: state,B7: state,X4: state] :
            ( ( Stable @ A5 )
           => ( ( Stable @ B7 )
             => ( ( ( some_state @ X4 )
                  = ( Plus @ A5 @ B7 ) )
               => ( Stable @ X4 ) ) ) )
       => ( ( Stable @ Unit )
         => ( packag2647621270594721818_state @ Plus @ Unit @ Stable ) ) ) ) ).

% package_logic_axioms.intro
thf(fact_569_add__set__def,axiom,
    ( add_set
    = ( sep_add_set_state @ plus ) ) ).

% add_set_def
thf(fact_570_package__logic_Oaxioms_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( packag2647621270594721818_state @ Plus @ Unit @ Stable ) ) ).

% package_logic.axioms(2)
thf(fact_571_sep__algebra_Oup__closed__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A4 )
       => ( sep_up_closed_state @ Plus @ ( sep_add_set_state @ Plus @ A4 @ B3 ) ) ) ) ).

% sep_algebra.up_closed_sum
thf(fact_572_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X3 @ ( sep_add_set_state @ Plus @ A4 @ B3 ) )
        = ( ? [A2: state,B: state] :
              ( ( member_state @ A2 @ A4 )
              & ( member_state @ B @ B3 )
              & ( ( some_state @ X3 )
                = ( sep_splus_state @ Plus @ ( some_state @ A2 ) @ ( some_state @ B ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_573_sep__algebra_Oup__closed__plus__UNIV,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_up_closed_state @ Plus @ ( sep_add_set_state @ Plus @ A4 @ top_top_set_state ) ) ) ).

% sep_algebra.up_closed_plus_UNIV
thf(fact_574_sep__algebra_Osum__then__singleton,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A3: state,B2: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A3 )
          = ( Plus @ B2 @ C ) )
        = ( ( insert_state @ A3 @ bot_bot_set_state )
          = ( sep_add_set_state @ Plus @ ( insert_state @ B2 @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ) ).

% sep_algebra.sum_then_singleton
thf(fact_575_sep__algebra_Omono__prop__set,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state,B3: set_state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A4 @ B3 )
       => ( ( sep_setify_state @ P @ B3 )
         => ( ( sep_mono_prop_state @ Plus @ P )
           => ( sep_setify_state @ P @ A4 ) ) ) ) ) ).

% sep_algebra.mono_prop_set
thf(fact_576_inj__on__insert,axiom,
    ! [F: state > state,A3: state,A4: set_state] :
      ( ( inj_on_state_state @ F @ ( insert_state @ A3 @ A4 ) )
      = ( ( inj_on_state_state @ F @ A4 )
        & ~ ( member_state @ ( F @ A3 ) @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ bot_bot_set_state ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_577_sep__algebra_Oup__closed__bigger__subset,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: set_state,A4: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ B3 )
       => ( ( sep_gr7105985528888466643_state @ Plus @ A4 @ B3 )
         => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ).

% sep_algebra.up_closed_bigger_subset
thf(fact_578_the__inv__into__into,axiom,
    ! [F: state > state,A4: set_state,X3: state,B3: set_state] :
      ( ( inj_on_state_state @ F @ A4 )
     => ( ( member_state @ X3 @ ( image_state_state @ F @ A4 ) )
       => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
         => ( member_state @ ( the_in3035302284364921129_state @ A4 @ F @ X3 ) @ B3 ) ) ) ) ).

% the_inv_into_into
thf(fact_579_vimage__subsetI,axiom,
    ! [F: state > state,B3: set_state,A4: set_state] :
      ( ( inj_on_state_state @ F @ top_top_set_state )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
       => ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ B3 ) @ A4 ) ) ) ).

% vimage_subsetI
thf(fact_580_Diff__iff,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
      = ( ( member_state @ C @ A4 )
        & ~ ( member_state @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_581_DiffI,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ A4 )
     => ( ~ ( member_state @ C @ B3 )
       => ( member_state @ C @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ) ).

% DiffI
thf(fact_582_vimage__eq,axiom,
    ! [A3: state,F: state > state,B3: set_state] :
      ( ( member_state @ A3 @ ( vimage_state_state @ F @ B3 ) )
      = ( member_state @ ( F @ A3 ) @ B3 ) ) ).

% vimage_eq
thf(fact_583_vimageI,axiom,
    ! [F: state > state,A3: state,B2: state,B3: set_state] :
      ( ( ( F @ A3 )
        = B2 )
     => ( ( member_state @ B2 @ B3 )
       => ( member_state @ A3 @ ( vimage_state_state @ F @ B3 ) ) ) ) ).

% vimageI
thf(fact_584_Diff__empty,axiom,
    ! [A4: set_state] :
      ( ( minus_3933957440811877961_state @ A4 @ bot_bot_set_state )
      = A4 ) ).

% Diff_empty
thf(fact_585_empty__Diff,axiom,
    ! [A4: set_state] :
      ( ( minus_3933957440811877961_state @ bot_bot_set_state @ A4 )
      = bot_bot_set_state ) ).

% empty_Diff
thf(fact_586_Diff__cancel,axiom,
    ! [A4: set_state] :
      ( ( minus_3933957440811877961_state @ A4 @ A4 )
      = bot_bot_set_state ) ).

% Diff_cancel
thf(fact_587_Diff__insert0,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ~ ( member_state @ X3 @ A4 )
     => ( ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ B3 ) )
        = ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_588_insert__Diff1,axiom,
    ! [X3: state,B3: set_state,A4: set_state] :
      ( ( member_state @ X3 @ B3 )
     => ( ( minus_3933957440811877961_state @ ( insert_state @ X3 @ A4 ) @ B3 )
        = ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_589_vimage__UNIV,axiom,
    ! [F: state > state] :
      ( ( vimage_state_state @ F @ top_top_set_state )
      = top_top_set_state ) ).

% vimage_UNIV
thf(fact_590_vimage__empty,axiom,
    ! [F: state > state] :
      ( ( vimage_state_state @ F @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% vimage_empty
thf(fact_591_Diff__UNIV,axiom,
    ! [A4: set_state] :
      ( ( minus_3933957440811877961_state @ A4 @ top_top_set_state )
      = bot_bot_set_state ) ).

% Diff_UNIV
thf(fact_592_Diff__eq__empty__iff,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( minus_3933957440811877961_state @ A4 @ B3 )
        = bot_bot_set_state )
      = ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_593_insert__Diff__single,axiom,
    ! [A3: state,A4: set_state] :
      ( ( insert_state @ A3 @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ bot_bot_set_state ) ) )
      = ( insert_state @ A3 @ A4 ) ) ).

% insert_Diff_single
thf(fact_594_double__diff,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ( minus_3933957440811877961_state @ B3 @ ( minus_3933957440811877961_state @ C2 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_595_Diff__subset,axiom,
    ! [A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) @ A4 ) ).

% Diff_subset
thf(fact_596_Diff__mono,axiom,
    ! [A4: set_state,C2: set_state,D2: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ C2 )
     => ( ( ord_le2494988322063910608_state @ D2 @ B3 )
       => ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) @ ( minus_3933957440811877961_state @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_597_insert__Diff__if,axiom,
    ! [X3: state,B3: set_state,A4: set_state] :
      ( ( ( member_state @ X3 @ B3 )
       => ( ( minus_3933957440811877961_state @ ( insert_state @ X3 @ A4 ) @ B3 )
          = ( minus_3933957440811877961_state @ A4 @ B3 ) ) )
      & ( ~ ( member_state @ X3 @ B3 )
       => ( ( minus_3933957440811877961_state @ ( insert_state @ X3 @ A4 ) @ B3 )
          = ( insert_state @ X3 @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_598_vimageI2,axiom,
    ! [F: state > state,A3: state,A4: set_state] :
      ( ( member_state @ ( F @ A3 ) @ A4 )
     => ( member_state @ A3 @ ( vimage_state_state @ F @ A4 ) ) ) ).

% vimageI2
thf(fact_599_vimageE,axiom,
    ! [A3: state,F: state > state,B3: set_state] :
      ( ( member_state @ A3 @ ( vimage_state_state @ F @ B3 ) )
     => ( member_state @ ( F @ A3 ) @ B3 ) ) ).

% vimageE
thf(fact_600_vimageD,axiom,
    ! [A3: state,F: state > state,A4: set_state] :
      ( ( member_state @ A3 @ ( vimage_state_state @ F @ A4 ) )
     => ( member_state @ ( F @ A3 ) @ A4 ) ) ).

% vimageD
thf(fact_601_DiffD2,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
     => ~ ( member_state @ C @ B3 ) ) ).

% DiffD2
thf(fact_602_DiffD1,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
     => ( member_state @ C @ A4 ) ) ).

% DiffD1
thf(fact_603_DiffE,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
     => ~ ( ( member_state @ C @ A4 )
         => ( member_state @ C @ B3 ) ) ) ).

% DiffE
thf(fact_604_subset__vimage__iff,axiom,
    ! [A4: set_state,F: state > state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( vimage_state_state @ F @ B3 ) )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ( member_state @ ( F @ X2 ) @ B3 ) ) ) ) ).

% subset_vimage_iff
thf(fact_605_vimage__mono,axiom,
    ! [A4: set_state,B3: set_state,F: state > state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ A4 ) @ ( vimage_state_state @ F @ B3 ) ) ) ).

% vimage_mono
thf(fact_606_image__vimage__subset,axiom,
    ! [F: state > state,A4: set_state] : ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( vimage_state_state @ F @ A4 ) ) @ A4 ) ).

% image_vimage_subset
thf(fact_607_image__subset__iff__subset__vimage,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
      = ( ord_le2494988322063910608_state @ A4 @ ( vimage_state_state @ F @ B3 ) ) ) ).

% image_subset_iff_subset_vimage
thf(fact_608_vimage__singleton__eq,axiom,
    ! [A3: state,F: state > state,B2: state] :
      ( ( member_state @ A3 @ ( vimage_state_state @ F @ ( insert_state @ B2 @ bot_bot_set_state ) ) )
      = ( ( F @ A3 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_609_diff__shunt__var,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ( minus_3933957440811877961_state @ X3 @ Y2 )
        = bot_bot_set_state )
      = ( ord_le2494988322063910608_state @ X3 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_610_image__diff__subset,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ).

% image_diff_subset
thf(fact_611_Diff__insert,axiom,
    ! [A4: set_state,A3: state,B3: set_state] :
      ( ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ B3 ) )
      = ( minus_3933957440811877961_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) @ ( insert_state @ A3 @ bot_bot_set_state ) ) ) ).

% Diff_insert
thf(fact_612_insert__Diff,axiom,
    ! [A3: state,A4: set_state] :
      ( ( member_state @ A3 @ A4 )
     => ( ( insert_state @ A3 @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ bot_bot_set_state ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_613_Diff__insert2,axiom,
    ! [A4: set_state,A3: state,B3: set_state] :
      ( ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ B3 ) )
      = ( minus_3933957440811877961_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ bot_bot_set_state ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_614_Diff__insert__absorb,axiom,
    ! [X3: state,A4: set_state] :
      ( ~ ( member_state @ X3 @ A4 )
     => ( ( minus_3933957440811877961_state @ ( insert_state @ X3 @ A4 ) @ ( insert_state @ X3 @ bot_bot_set_state ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_615_subset__Diff__insert,axiom,
    ! [A4: set_state,B3: set_state,X3: state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( minus_3933957440811877961_state @ B3 @ ( insert_state @ X3 @ C2 ) ) )
      = ( ( ord_le2494988322063910608_state @ A4 @ ( minus_3933957440811877961_state @ B3 @ C2 ) )
        & ~ ( member_state @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_616_Compl__eq__Diff__UNIV,axiom,
    ( uminus472742206872269241_state
    = ( minus_3933957440811877961_state @ top_top_set_state ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_617_surj__vimage__empty,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( ( image_state_state @ F @ top_top_set_state )
        = top_top_set_state )
     => ( ( ( vimage_state_state @ F @ A4 )
          = bot_bot_set_state )
        = ( A4 = bot_bot_set_state ) ) ) ).

% surj_vimage_empty
thf(fact_618_vimage__subsetD,axiom,
    ! [F: state > state,B3: set_state,A4: set_state] :
      ( ( ( image_state_state @ F @ top_top_set_state )
        = top_top_set_state )
     => ( ( ord_le2494988322063910608_state @ ( vimage_state_state @ F @ B3 ) @ A4 )
       => ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) ) ) ) ).

% vimage_subsetD
thf(fact_619_sep__algebra_Osub__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A4: set_state,B3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( sep_gr7105985528888466643_state @ Plus @ A4 @ B3 ) ) ) ).

% sep_algebra.sub_bigger
thf(fact_620_subset__insert__iff,axiom,
    ! [A4: set_state,X3: state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ X3 @ B3 ) )
      = ( ( ( member_state @ X3 @ A4 )
         => ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) @ B3 ) )
        & ( ~ ( member_state @ X3 @ A4 )
         => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_621_Diff__single__insert,axiom,
    ! [A4: set_state,X3: state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) @ B3 )
     => ( ord_le2494988322063910608_state @ A4 @ ( insert_state @ X3 @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_622_inj__on__image__set__diff,axiom,
    ! [F: state > state,C2: set_state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ C2 )
     => ( ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) @ C2 )
       => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
         => ( ( image_state_state @ F @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
            = ( minus_3933957440811877961_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_623_Compl__insert,axiom,
    ! [X3: state,A4: set_state] :
      ( ( uminus472742206872269241_state @ ( insert_state @ X3 @ A4 ) )
      = ( minus_3933957440811877961_state @ ( uminus472742206872269241_state @ A4 ) @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) ).

% Compl_insert
thf(fact_624_greater__set__def,axiom,
    ( greater_set
    = ( sep_gr7105985528888466643_state @ plus ) ) ).

% greater_set_def
thf(fact_625_in__image__insert__iff,axiom,
    ! [B3: set_set_state,X3: state,A4: set_state] :
      ( ! [C5: set_state] :
          ( ( member_set_state @ C5 @ B3 )
         => ~ ( member_state @ X3 @ C5 ) )
     => ( ( member_set_state @ A4 @ ( image_2476256681063834599_state @ ( insert_state @ X3 ) @ B3 ) )
        = ( ( member_state @ X3 @ A4 )
          & ( member_set_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) @ B3 ) ) ) ) ).

% in_image_insert_iff
thf(fact_626_remove__def,axiom,
    ( remove_state
    = ( ^ [X2: state,A: set_state] : ( minus_3933957440811877961_state @ A @ ( insert_state @ X2 @ bot_bot_set_state ) ) ) ) ).

% remove_def
thf(fact_627_fun__upd__image,axiom,
    ! [X3: state,A4: set_state,F: state > state,Y2: state] :
      ( ( ( member_state @ X3 @ A4 )
       => ( ( image_state_state @ ( fun_upd_state_state @ F @ X3 @ Y2 ) @ A4 )
          = ( insert_state @ Y2 @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ) )
      & ( ~ ( member_state @ X3 @ A4 )
       => ( ( image_state_state @ ( fun_upd_state_state @ F @ X3 @ Y2 ) @ A4 )
          = ( image_state_state @ F @ A4 ) ) ) ) ).

% fun_upd_image
thf(fact_628_card__vimage__inj,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( inj_on_state_state @ F @ top_top_set_state )
     => ( ( ord_le2494988322063910608_state @ A4 @ ( image_state_state @ F @ top_top_set_state ) )
       => ( ( finite_card_state @ ( vimage_state_state @ F @ A4 ) )
          = ( finite_card_state @ A4 ) ) ) ) ).

% card_vimage_inj
thf(fact_629_pairwise__alt,axiom,
    ( pairwise_state
    = ( ^ [R5: state > state > $o,S2: set_state] :
        ! [X2: state] :
          ( ( member_state @ X2 @ S2 )
         => ! [Y: state] :
              ( ( member_state @ Y @ ( minus_3933957440811877961_state @ S2 @ ( insert_state @ X2 @ bot_bot_set_state ) ) )
             => ( R5 @ X2 @ Y ) ) ) ) ) ).

% pairwise_alt
thf(fact_630_inf__img__fin__dom,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A4 ) )
            & ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).

% inf_img_fin_dom
thf(fact_631_inf__img__fin__dom,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( finite_finite_state @ ( image_state_state @ F @ A4 ) )
     => ( ~ ( finite_finite_state @ A4 )
       => ? [X4: state] :
            ( ( member_state @ X4 @ ( image_state_state @ F @ A4 ) )
            & ~ ( finite_finite_state @ ( vimage_state_state @ F @ ( insert_state @ X4 @ bot_bot_set_state ) ) ) ) ) ) ).

% inf_img_fin_dom
thf(fact_632_inf__img__fin__dom,axiom,
    ! [F: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ ( image_nat_state @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ? [X4: state] :
            ( ( member_state @ X4 @ ( image_nat_state @ F @ A4 ) )
            & ~ ( finite_finite_nat @ ( vimage_nat_state @ F @ ( insert_state @ X4 @ bot_bot_set_state ) ) ) ) ) ) ).

% inf_img_fin_dom
thf(fact_633_member__remove,axiom,
    ! [X3: state,Y2: state,A4: set_state] :
      ( ( member_state @ X3 @ ( remove_state @ Y2 @ A4 ) )
      = ( ( member_state @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_634_finite__insert,axiom,
    ! [A3: state,A4: set_state] :
      ( ( finite_finite_state @ ( insert_state @ A3 @ A4 ) )
      = ( finite_finite_state @ A4 ) ) ).

% finite_insert
thf(fact_635_finite__insert,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) )
      = ( finite_finite_nat @ A4 ) ) ).

% finite_insert
thf(fact_636_finite__Diff__insert,axiom,
    ! [A4: set_state,A3: state,B3: set_state] :
      ( ( finite_finite_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ B3 ) ) )
      = ( finite_finite_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_637_finite__Diff__insert,axiom,
    ! [A4: set_nat,A3: nat,B3: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ B3 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_638_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_639_finite__option__UNIV,axiom,
    ( ( finite3180955649987104801_state @ top_to7666338855062656496_state )
    = ( finite_finite_state @ top_top_set_state ) ) ).

% finite_option_UNIV
thf(fact_640_card__Diff__subset,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A4 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B3 ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_641_card__Diff__subset,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ A4 )
       => ( ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
          = ( minus_minus_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_642_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_643_diff__card__le__card__Diff,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B3 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_644_finite_OinsertI,axiom,
    ! [A4: set_state,A3: state] :
      ( ( finite_finite_state @ A4 )
     => ( finite_finite_state @ ( insert_state @ A3 @ A4 ) ) ) ).

% finite.insertI
thf(fact_645_finite_OinsertI,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) ) ) ).

% finite.insertI
thf(fact_646_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_nat,C2: nat] :
      ( ! [G3: set_nat] :
          ( ( ord_less_eq_set_nat @ G3 @ F4 )
         => ( ( finite_finite_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
     => ( ( finite_finite_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_647_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_state,C2: nat] :
      ( ! [G3: set_state] :
          ( ( ord_le2494988322063910608_state @ G3 @ F4 )
         => ( ( finite_finite_state @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_state @ G3 ) @ C2 ) ) )
     => ( ( finite_finite_state @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_state @ F4 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_648_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ? [B6: set_nat] :
          ( ( finite_finite_nat @ B6 )
          & ( ( finite_card_nat @ B6 )
            = N2 )
          & ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_649_infinite__arbitrarily__large,axiom,
    ! [A4: set_state,N2: nat] :
      ( ~ ( finite_finite_state @ A4 )
     => ? [B6: set_state] :
          ( ( finite_finite_state @ B6 )
          & ( ( finite_card_state @ B6 )
            = N2 )
          & ( ord_le2494988322063910608_state @ B6 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_650_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
     => ~ ! [T3: set_nat] :
            ( ( ord_less_eq_set_nat @ T3 @ S )
           => ( ( ( finite_card_nat @ T3 )
                = N2 )
             => ~ ( finite_finite_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_651_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_state] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_state @ S ) )
     => ~ ! [T3: set_state] :
            ( ( ord_le2494988322063910608_state @ T3 @ S )
           => ( ( ( finite_card_state @ T3 )
                = N2 )
             => ~ ( finite_finite_state @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_652_exists__subset__between,axiom,
    ! [A4: set_nat,N2: nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
       => ( ( ord_less_eq_set_nat @ A4 @ C2 )
         => ( ( finite_finite_nat @ C2 )
           => ? [B6: set_nat] :
                ( ( ord_less_eq_set_nat @ A4 @ B6 )
                & ( ord_less_eq_set_nat @ B6 @ C2 )
                & ( ( finite_card_nat @ B6 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_653_exists__subset__between,axiom,
    ! [A4: set_state,N2: nat,C2: set_state] :
      ( ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite_card_state @ C2 ) )
       => ( ( ord_le2494988322063910608_state @ A4 @ C2 )
         => ( ( finite_finite_state @ C2 )
           => ? [B6: set_state] :
                ( ( ord_le2494988322063910608_state @ A4 @ B6 )
                & ( ord_le2494988322063910608_state @ B6 @ C2 )
                & ( ( finite_card_state @ B6 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_654_card__subset__eq,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A4 @ B3 )
       => ( ( ( finite_card_nat @ A4 )
            = ( finite_card_nat @ B3 ) )
         => ( A4 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_655_card__subset__eq,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( ( ( finite_card_state @ A4 )
            = ( finite_card_state @ B3 ) )
         => ( A4 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_656_card__seteq,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A4 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A4 ) )
         => ( A4 = B3 ) ) ) ) ).

% card_seteq
thf(fact_657_card__seteq,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_state @ B3 ) @ ( finite_card_state @ A4 ) )
         => ( A4 = B3 ) ) ) ) ).

% card_seteq
thf(fact_658_card__mono,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A4 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_659_card__mono,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ).

% card_mono
thf(fact_660_pairwiseD,axiom,
    ! [R6: state > state > $o,S: set_state,X3: state,Y2: state] :
      ( ( pairwise_state @ R6 @ S )
     => ( ( member_state @ X3 @ S )
       => ( ( member_state @ Y2 @ S )
         => ( ( X3 != Y2 )
           => ( R6 @ X3 @ Y2 ) ) ) ) ) ).

% pairwiseD
thf(fact_661_pairwiseI,axiom,
    ! [S: set_state,R6: state > state > $o] :
      ( ! [X4: state,Y3: state] :
          ( ( member_state @ X4 @ S )
         => ( ( member_state @ Y3 @ S )
           => ( ( X4 != Y3 )
             => ( R6 @ X4 @ Y3 ) ) ) )
     => ( pairwise_state @ R6 @ S ) ) ).

% pairwiseI
thf(fact_662_finite__subset,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_663_finite__subset,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( finite_finite_state @ B3 )
       => ( finite_finite_state @ A4 ) ) ) ).

% finite_subset
thf(fact_664_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_665_infinite__super,axiom,
    ! [S: set_state,T: set_state] :
      ( ( ord_le2494988322063910608_state @ S @ T )
     => ( ~ ( finite_finite_state @ S )
       => ~ ( finite_finite_state @ T ) ) ) ).

% infinite_super
thf(fact_666_rev__finite__subset,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A4 @ B3 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_667_rev__finite__subset,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( finite_finite_state @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_668_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_669_infinite__imp__nonempty,axiom,
    ! [S: set_state] :
      ( ~ ( finite_finite_state @ S )
     => ( S != bot_bot_set_state ) ) ).

% infinite_imp_nonempty
thf(fact_670_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_671_finite_OemptyI,axiom,
    finite_finite_state @ bot_bot_set_state ).

% finite.emptyI
thf(fact_672_finite__has__maximal2,axiom,
    ! [A4: set_set_state,A3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( member_set_state @ A3 @ A4 )
       => ? [X4: set_state] :
            ( ( member_set_state @ X4 @ A4 )
            & ( ord_le2494988322063910608_state @ A3 @ X4 )
            & ! [Xa2: set_state] :
                ( ( member_set_state @ Xa2 @ A4 )
               => ( ( ord_le2494988322063910608_state @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_673_finite__has__maximal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ A3 @ X4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_674_finite__has__minimal2,axiom,
    ! [A4: set_set_state,A3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( member_set_state @ A3 @ A4 )
       => ? [X4: set_state] :
            ( ( member_set_state @ X4 @ A4 )
            & ( ord_le2494988322063910608_state @ X4 @ A3 )
            & ! [Xa2: set_state] :
                ( ( member_set_state @ Xa2 @ A4 )
               => ( ( ord_le2494988322063910608_state @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_675_finite__has__minimal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ X4 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_676_card__insert__le,axiom,
    ! [A4: set_state,X3: state] : ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ ( insert_state @ X3 @ A4 ) ) ) ).

% card_insert_le
thf(fact_677_surj__card__le,axiom,
    ! [A4: set_state,B3: set_state,F: state > state] :
      ( ( finite_finite_state @ A4 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_state @ B3 ) @ ( finite_card_state @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_678_surj__card__le,axiom,
    ! [A4: set_nat,B3: set_state,F: nat > state] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_nat_state @ F @ A4 ) )
       => ( ord_less_eq_nat @ ( finite_card_state @ B3 ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% surj_card_le
thf(fact_679_card__image__le,axiom,
    ! [A4: set_state,F: state > state] :
      ( ( finite_finite_state @ A4 )
     => ( ord_less_eq_nat @ ( finite_card_state @ ( image_state_state @ F @ A4 ) ) @ ( finite_card_state @ A4 ) ) ) ).

% card_image_le
thf(fact_680_infinite__countable__subset,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ? [F5: nat > nat] :
          ( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
          & ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_681_infinite__countable__subset,axiom,
    ! [S: set_state] :
      ( ~ ( finite_finite_state @ S )
     => ? [F5: nat > state] :
          ( ( inj_on_nat_state @ F5 @ top_top_set_nat )
          & ( ord_le2494988322063910608_state @ ( image_nat_state @ F5 @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_682_infinite__iff__countable__subset,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ? [F3: nat > nat] :
            ( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
            & ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_683_infinite__iff__countable__subset,axiom,
    ! [S: set_state] :
      ( ( ~ ( finite_finite_state @ S ) )
      = ( ? [F3: nat > state] :
            ( ( inj_on_nat_state @ F3 @ top_top_set_nat )
            & ( ord_le2494988322063910608_state @ ( image_nat_state @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_684_surjective__iff__injective__gen,axiom,
    ! [S: set_nat,T: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ( finite_card_nat @ S )
            = ( finite_card_nat @ T ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S ) @ T )
           => ( ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ T )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S )
                        & ( ( F @ Y )
                          = X2 ) ) ) )
              = ( inj_on_nat_nat @ F @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_685_surjective__iff__injective__gen,axiom,
    ! [S: set_state,T: set_state,F: state > state] :
      ( ( finite_finite_state @ S )
     => ( ( finite_finite_state @ T )
       => ( ( ( finite_card_state @ S )
            = ( finite_card_state @ T ) )
         => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ S ) @ T )
           => ( ( ! [X2: state] :
                    ( ( member_state @ X2 @ T )
                   => ? [Y: state] :
                        ( ( member_state @ Y @ S )
                        & ( ( F @ Y )
                          = X2 ) ) ) )
              = ( inj_on_state_state @ F @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_686_surjective__iff__injective__gen,axiom,
    ! [S: set_nat,T: set_state,F: nat > state] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_state @ T )
       => ( ( ( finite_card_nat @ S )
            = ( finite_card_state @ T ) )
         => ( ( ord_le2494988322063910608_state @ ( image_nat_state @ F @ S ) @ T )
           => ( ( ! [X2: state] :
                    ( ( member_state @ X2 @ T )
                   => ? [Y: nat] :
                        ( ( member_nat @ Y @ S )
                        & ( ( F @ Y )
                          = X2 ) ) ) )
              = ( inj_on_nat_state @ F @ S ) ) ) ) ) ) ).

% surjective_iff_injective_gen
thf(fact_687_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ? [F3: nat > nat] :
                ( ( inj_on_nat_nat @ F3 @ A4 )
                & ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A4 ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_688_inj__on__iff__card__le,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( finite_finite_state @ A4 )
     => ( ( finite_finite_state @ B3 )
       => ( ( ? [F3: state > state] :
                ( ( inj_on_state_state @ F3 @ A4 )
                & ( ord_le2494988322063910608_state @ ( image_state_state @ F3 @ A4 ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_689_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B3: set_state] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_state @ B3 )
       => ( ( ? [F3: nat > state] :
                ( ( inj_on_nat_state @ F3 @ A4 )
                & ( ord_le2494988322063910608_state @ ( image_nat_state @ F3 @ A4 ) @ B3 ) ) )
          = ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ).

% inj_on_iff_card_le
thf(fact_690_card__inj__on__le,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
       => ( ( finite_finite_state @ B3 )
         => ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ).

% card_inj_on_le
thf(fact_691_card__le__inj,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) )
         => ? [F5: nat > nat] :
              ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B3 )
              & ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).

% card_le_inj
thf(fact_692_card__le__inj,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( finite_finite_state @ A4 )
     => ( ( finite_finite_state @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) )
         => ? [F5: state > state] :
              ( ( ord_le2494988322063910608_state @ ( image_state_state @ F5 @ A4 ) @ B3 )
              & ( inj_on_state_state @ F5 @ A4 ) ) ) ) ) ).

% card_le_inj
thf(fact_693_card__le__inj,axiom,
    ! [A4: set_nat,B3: set_state] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_state @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_state @ B3 ) )
         => ? [F5: nat > state] :
              ( ( ord_le2494988322063910608_state @ ( image_nat_state @ F5 @ A4 ) @ B3 )
              & ( inj_on_nat_state @ F5 @ A4 ) ) ) ) ) ).

% card_le_inj
thf(fact_694_card__bij__eq,axiom,
    ! [F: nat > nat,A4: set_nat,B3: set_nat,G: nat > nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B3 )
       => ( ( inj_on_nat_nat @ G @ B3 )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B3 ) @ A4 )
           => ( ( finite_finite_nat @ A4 )
             => ( ( finite_finite_nat @ B3 )
               => ( ( finite_card_nat @ A4 )
                  = ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_695_card__bij__eq,axiom,
    ! [F: state > nat,A4: set_state,B3: set_nat,G: nat > state] :
      ( ( inj_on_state_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ ( image_state_nat @ F @ A4 ) @ B3 )
       => ( ( inj_on_nat_state @ G @ B3 )
         => ( ( ord_le2494988322063910608_state @ ( image_nat_state @ G @ B3 ) @ A4 )
           => ( ( finite_finite_state @ A4 )
             => ( ( finite_finite_nat @ B3 )
               => ( ( finite_card_state @ A4 )
                  = ( finite_card_nat @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_696_card__bij__eq,axiom,
    ! [F: nat > state,A4: set_nat,B3: set_state,G: state > nat] :
      ( ( inj_on_nat_state @ F @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_nat_state @ F @ A4 ) @ B3 )
       => ( ( inj_on_state_nat @ G @ B3 )
         => ( ( ord_less_eq_set_nat @ ( image_state_nat @ G @ B3 ) @ A4 )
           => ( ( finite_finite_nat @ A4 )
             => ( ( finite_finite_state @ B3 )
               => ( ( finite_card_nat @ A4 )
                  = ( finite_card_state @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_697_card__bij__eq,axiom,
    ! [F: state > state,A4: set_state,B3: set_state,G: state > state] :
      ( ( inj_on_state_state @ F @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
       => ( ( inj_on_state_state @ G @ B3 )
         => ( ( ord_le2494988322063910608_state @ ( image_state_state @ G @ B3 ) @ A4 )
           => ( ( finite_finite_state @ A4 )
             => ( ( finite_finite_state @ B3 )
               => ( ( finite_card_state @ A4 )
                  = ( finite_card_state @ B3 ) ) ) ) ) ) ) ) ).

% card_bij_eq
thf(fact_698_pairwise__imageI,axiom,
    ! [A4: set_state,F: state > state,P: state > state > $o] :
      ( ! [X4: state,Y3: state] :
          ( ( member_state @ X4 @ A4 )
         => ( ( member_state @ Y3 @ A4 )
           => ( ( X4 != Y3 )
             => ( ( ( F @ X4 )
                 != ( F @ Y3 ) )
               => ( P @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) )
     => ( pairwise_state @ P @ ( image_state_state @ F @ A4 ) ) ) ).

% pairwise_imageI
thf(fact_699_pairwise__empty,axiom,
    ! [P: state > state > $o] : ( pairwise_state @ P @ bot_bot_set_state ) ).

% pairwise_empty
thf(fact_700_pairwise__subset,axiom,
    ! [P: state > state > $o,S: set_state,T: set_state] :
      ( ( pairwise_state @ P @ S )
     => ( ( ord_le2494988322063910608_state @ T @ S )
       => ( pairwise_state @ P @ T ) ) ) ).

% pairwise_subset
thf(fact_701_pairwise__mono,axiom,
    ! [P: state > state > $o,A4: set_state,Q: state > state > $o,B3: set_state] :
      ( ( pairwise_state @ P @ A4 )
     => ( ! [X4: state,Y3: state] :
            ( ( P @ X4 @ Y3 )
           => ( Q @ X4 @ Y3 ) )
       => ( ( ord_le2494988322063910608_state @ B3 @ A4 )
         => ( pairwise_state @ Q @ B3 ) ) ) ) ).

% pairwise_mono
thf(fact_702_pairwise__insert,axiom,
    ! [R2: state > state > $o,X3: state,S3: set_state] :
      ( ( pairwise_state @ R2 @ ( insert_state @ X3 @ S3 ) )
      = ( ! [Y: state] :
            ( ( ( member_state @ Y @ S3 )
              & ( Y != X3 ) )
           => ( ( R2 @ X3 @ Y )
              & ( R2 @ Y @ X3 ) ) )
        & ( pairwise_state @ R2 @ S3 ) ) ) ).

% pairwise_insert
thf(fact_703_finite__range__Some,axiom,
    ( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_range_Some
thf(fact_704_finite__range__Some,axiom,
    ( ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) )
    = ( finite_finite_state @ top_top_set_state ) ) ).

% finite_range_Some
thf(fact_705_finite__has__minimal,axiom,
    ! [A4: set_set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ? [X4: set_state] :
            ( ( member_set_state @ X4 @ A4 )
            & ! [Xa2: set_state] :
                ( ( member_set_state @ Xa2 @ A4 )
               => ( ( ord_le2494988322063910608_state @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_706_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_707_finite__has__maximal,axiom,
    ! [A4: set_set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ? [X4: set_state] :
            ( ( member_set_state @ X4 @ A4 )
            & ! [Xa2: set_state] :
                ( ( member_set_state @ Xa2 @ A4 )
               => ( ( ord_le2494988322063910608_state @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_708_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_709_all__finite__subset__image,axiom,
    ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
      ( ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ A4 ) )
           => ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_710_all__finite__subset__image,axiom,
    ! [F: state > nat,A4: set_state,P: set_nat > $o] :
      ( ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ ( image_state_nat @ F @ A4 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_state] :
            ( ( ( finite_finite_state @ B4 )
              & ( ord_le2494988322063910608_state @ B4 @ A4 ) )
           => ( P @ ( image_state_nat @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_711_all__finite__subset__image,axiom,
    ! [F: nat > state,A4: set_nat,P: set_state > $o] :
      ( ( ! [B4: set_state] :
            ( ( ( finite_finite_state @ B4 )
              & ( ord_le2494988322063910608_state @ B4 @ ( image_nat_state @ F @ A4 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_nat] :
            ( ( ( finite_finite_nat @ B4 )
              & ( ord_less_eq_set_nat @ B4 @ A4 ) )
           => ( P @ ( image_nat_state @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_712_all__finite__subset__image,axiom,
    ! [F: state > state,A4: set_state,P: set_state > $o] :
      ( ( ! [B4: set_state] :
            ( ( ( finite_finite_state @ B4 )
              & ( ord_le2494988322063910608_state @ B4 @ ( image_state_state @ F @ A4 ) ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_state] :
            ( ( ( finite_finite_state @ B4 )
              & ( ord_le2494988322063910608_state @ B4 @ A4 ) )
           => ( P @ ( image_state_state @ F @ B4 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_713_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A4: set_nat,P: set_nat > $o] :
      ( ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A4 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ A4 )
            & ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_714_ex__finite__subset__image,axiom,
    ! [F: state > nat,A4: set_state,P: set_nat > $o] :
      ( ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ ( image_state_nat @ F @ A4 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_state] :
            ( ( finite_finite_state @ B4 )
            & ( ord_le2494988322063910608_state @ B4 @ A4 )
            & ( P @ ( image_state_nat @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_715_ex__finite__subset__image,axiom,
    ! [F: nat > state,A4: set_nat,P: set_state > $o] :
      ( ( ? [B4: set_state] :
            ( ( finite_finite_state @ B4 )
            & ( ord_le2494988322063910608_state @ B4 @ ( image_nat_state @ F @ A4 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_nat] :
            ( ( finite_finite_nat @ B4 )
            & ( ord_less_eq_set_nat @ B4 @ A4 )
            & ( P @ ( image_nat_state @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_716_ex__finite__subset__image,axiom,
    ! [F: state > state,A4: set_state,P: set_state > $o] :
      ( ( ? [B4: set_state] :
            ( ( finite_finite_state @ B4 )
            & ( ord_le2494988322063910608_state @ B4 @ ( image_state_state @ F @ A4 ) )
            & ( P @ B4 ) ) )
      = ( ? [B4: set_state] :
            ( ( finite_finite_state @ B4 )
            & ( ord_le2494988322063910608_state @ B4 @ A4 )
            & ( P @ ( image_state_state @ F @ B4 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_717_finite__subset__image,axiom,
    ! [B3: set_nat,F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_nat_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_718_finite__subset__image,axiom,
    ! [B3: set_nat,F: state > nat,A4: set_state] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_state_nat @ F @ A4 ) )
       => ? [C5: set_state] :
            ( ( ord_le2494988322063910608_state @ C5 @ A4 )
            & ( finite_finite_state @ C5 )
            & ( B3
              = ( image_state_nat @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_719_finite__subset__image,axiom,
    ! [B3: set_state,F: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_nat_state @ F @ A4 ) )
       => ? [C5: set_nat] :
            ( ( ord_less_eq_set_nat @ C5 @ A4 )
            & ( finite_finite_nat @ C5 )
            & ( B3
              = ( image_nat_state @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_720_finite__subset__image,axiom,
    ! [B3: set_state,F: state > state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
       => ? [C5: set_state] :
            ( ( ord_le2494988322063910608_state @ C5 @ A4 )
            & ( finite_finite_state @ C5 )
            & ( B3
              = ( image_state_state @ F @ C5 ) ) ) ) ) ).

% finite_subset_image
thf(fact_721_finite__surj,axiom,
    ! [A4: set_nat,B3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A4 ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_722_finite__surj,axiom,
    ! [A4: set_state,B3: set_state,F: state > state] :
      ( ( finite_finite_state @ A4 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_state_state @ F @ A4 ) )
       => ( finite_finite_state @ B3 ) ) ) ).

% finite_surj
thf(fact_723_finite__surj,axiom,
    ! [A4: set_nat,B3: set_state,F: nat > state] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le2494988322063910608_state @ B3 @ ( image_nat_state @ F @ A4 ) )
       => ( finite_finite_state @ B3 ) ) ) ).

% finite_surj
thf(fact_724_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A4: set_nat] :
      ( ! [A9: set_nat] :
          ( ~ ( finite_finite_nat @ A9 )
         => ( P @ A9 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_725_infinite__finite__induct,axiom,
    ! [P: set_state > $o,A4: set_state] :
      ( ! [A9: set_state] :
          ( ~ ( finite_finite_state @ A9 )
         => ( P @ A9 ) )
     => ( ( P @ bot_bot_set_state )
       => ( ! [X4: state,F6: set_state] :
              ( ( finite_finite_state @ F6 )
             => ( ~ ( member_state @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_state @ X4 @ F6 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_726_finite__ne__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( F6 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_727_finite__ne__induct,axiom,
    ! [F4: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ F4 )
     => ( ( F4 != bot_bot_set_state )
       => ( ! [X4: state] : ( P @ ( insert_state @ X4 @ bot_bot_set_state ) )
         => ( ! [X4: state,F6: set_state] :
                ( ( finite_finite_state @ F6 )
               => ( ( F6 != bot_bot_set_state )
                 => ( ~ ( member_state @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_state @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_728_finite__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_729_finite__induct,axiom,
    ! [F4: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ F4 )
     => ( ( P @ bot_bot_set_state )
       => ( ! [X4: state,F6: set_state] :
              ( ( finite_finite_state @ F6 )
             => ( ~ ( member_state @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_state @ X4 @ F6 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_730_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A2: set_nat] :
          ( ( A2 = bot_bot_set_nat )
          | ? [A: set_nat,B: nat] :
              ( ( A2
                = ( insert_nat @ B @ A ) )
              & ( finite_finite_nat @ A ) ) ) ) ) ).

% finite.simps
thf(fact_731_finite_Osimps,axiom,
    ( finite_finite_state
    = ( ^ [A2: set_state] :
          ( ( A2 = bot_bot_set_state )
          | ? [A: set_state,B: state] :
              ( ( A2
                = ( insert_state @ B @ A ) )
              & ( finite_finite_state @ A ) ) ) ) ) ).

% finite.simps
thf(fact_732_finite_Ocases,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ~ ! [A9: set_nat] :
              ( ? [A5: nat] :
                  ( A3
                  = ( insert_nat @ A5 @ A9 ) )
             => ~ ( finite_finite_nat @ A9 ) ) ) ) ).

% finite.cases
thf(fact_733_finite_Ocases,axiom,
    ! [A3: set_state] :
      ( ( finite_finite_state @ A3 )
     => ( ( A3 != bot_bot_set_state )
       => ~ ! [A9: set_state] :
              ( ? [A5: state] :
                  ( A3
                  = ( insert_state @ A5 @ A9 ) )
             => ~ ( finite_finite_state @ A9 ) ) ) ) ).

% finite.cases
thf(fact_734_pairwise__singleton,axiom,
    ! [P: state > state > $o,A4: state] : ( pairwise_state @ P @ ( insert_state @ A4 @ bot_bot_set_state ) ) ).

% pairwise_singleton
thf(fact_735_card__Diff1__le,axiom,
    ! [A4: set_state,X3: state] : ( ord_less_eq_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) @ ( finite_card_state @ A4 ) ) ).

% card_Diff1_le
thf(fact_736_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A5 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F6 @ A4 )
                   => ( ~ ( member_nat @ A5 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_nat @ A5 @ F6 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_737_finite__subset__induct_H,axiom,
    ! [F4: set_state,A4: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ F4 )
     => ( ( ord_le2494988322063910608_state @ F4 @ A4 )
       => ( ( P @ bot_bot_set_state )
         => ( ! [A5: state,F6: set_state] :
                ( ( finite_finite_state @ F6 )
               => ( ( member_state @ A5 @ A4 )
                 => ( ( ord_le2494988322063910608_state @ F6 @ A4 )
                   => ( ~ ( member_state @ A5 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_state @ A5 @ F6 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_738_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A5 @ A4 )
                 => ( ~ ( member_nat @ A5 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ A5 @ F6 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_739_finite__subset__induct,axiom,
    ! [F4: set_state,A4: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ F4 )
     => ( ( ord_le2494988322063910608_state @ F4 @ A4 )
       => ( ( P @ bot_bot_set_state )
         => ( ! [A5: state,F6: set_state] :
                ( ( finite_finite_state @ F6 )
               => ( ( member_state @ A5 @ A4 )
                 => ( ~ ( member_state @ A5 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_state @ A5 @ F6 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_740_finite__surj__inj,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
       => ( inj_on_nat_nat @ F @ A4 ) ) ) ).

% finite_surj_inj
thf(fact_741_finite__surj__inj,axiom,
    ! [A4: set_state,F: state > state] :
      ( ( finite_finite_state @ A4 )
     => ( ( ord_le2494988322063910608_state @ A4 @ ( image_state_state @ F @ A4 ) )
       => ( inj_on_state_state @ F @ A4 ) ) ) ).

% finite_surj_inj
thf(fact_742_inj__on__finite,axiom,
    ! [F: nat > nat,A4: set_nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F @ A4 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B3 )
       => ( ( finite_finite_nat @ B3 )
         => ( finite_finite_nat @ A4 ) ) ) ) ).

% inj_on_finite
thf(fact_743_inj__on__finite,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ B3 )
       => ( ( finite_finite_state @ B3 )
         => ( finite_finite_state @ A4 ) ) ) ) ).

% inj_on_finite
thf(fact_744_inj__on__finite,axiom,
    ! [F: nat > state,A4: set_nat,B3: set_state] :
      ( ( inj_on_nat_state @ F @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_nat_state @ F @ A4 ) @ B3 )
       => ( ( finite_finite_state @ B3 )
         => ( finite_finite_nat @ A4 ) ) ) ) ).

% inj_on_finite
thf(fact_745_endo__inj__surj,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ A4 )
       => ( ( inj_on_nat_nat @ F @ A4 )
         => ( ( image_nat_nat @ F @ A4 )
            = A4 ) ) ) ) ).

% endo_inj_surj
thf(fact_746_endo__inj__surj,axiom,
    ! [A4: set_state,F: state > state] :
      ( ( finite_finite_state @ A4 )
     => ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A4 ) @ A4 )
       => ( ( inj_on_state_state @ F @ A4 )
         => ( ( image_state_state @ F @ A4 )
            = A4 ) ) ) ) ).

% endo_inj_surj
thf(fact_747_infinite__remove,axiom,
    ! [S: set_nat,A3: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_748_infinite__remove,axiom,
    ! [S: set_state,A3: state] :
      ( ~ ( finite_finite_state @ S )
     => ~ ( finite_finite_state @ ( minus_3933957440811877961_state @ S @ ( insert_state @ A3 @ bot_bot_set_state ) ) ) ) ).

% infinite_remove
thf(fact_749_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A4: set_nat] :
      ( ( X6 @ A4 )
     => ( ! [A9: set_nat] :
            ( ( X6 @ A9 )
           => ? [X: nat] :
                ( ( member_nat @ X @ A9 )
                & ( ( X6 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_750_infinite__coinduct,axiom,
    ! [X6: set_state > $o,A4: set_state] :
      ( ( X6 @ A4 )
     => ( ! [A9: set_state] :
            ( ( X6 @ A9 )
           => ? [X: state] :
                ( ( member_state @ X @ A9 )
                & ( ( X6 @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X @ bot_bot_set_state ) ) )
                  | ~ ( finite_finite_state @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X @ bot_bot_set_state ) ) ) ) ) )
       => ~ ( finite_finite_state @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_751_finite__empty__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ A4 )
       => ( ! [A5: nat,A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ( member_nat @ A5 @ A9 )
               => ( ( P @ A9 )
                 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_752_finite__empty__induct,axiom,
    ! [A4: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ A4 )
     => ( ( P @ A4 )
       => ( ! [A5: state,A9: set_state] :
              ( ( finite_finite_state @ A9 )
             => ( ( member_state @ A5 @ A9 )
               => ( ( P @ A9 )
                 => ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ A5 @ bot_bot_set_state ) ) ) ) ) )
         => ( P @ bot_bot_set_state ) ) ) ) ).

% finite_empty_induct
thf(fact_753_finite__remove__induct,axiom,
    ! [B3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ( A9 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A9 @ B3 )
                 => ( ! [X: nat] :
                        ( ( member_nat @ X @ A9 )
                       => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_754_finite__remove__induct,axiom,
    ! [B3: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ B3 )
     => ( ( P @ bot_bot_set_state )
       => ( ! [A9: set_state] :
              ( ( finite_finite_state @ A9 )
             => ( ( A9 != bot_bot_set_state )
               => ( ( ord_le2494988322063910608_state @ A9 @ B3 )
                 => ( ! [X: state] :
                        ( ( member_state @ X @ A9 )
                       => ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X @ bot_bot_set_state ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_755_remove__induct,axiom,
    ! [P: set_nat > $o,B3: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B3 )
         => ( P @ B3 ) )
       => ( ! [A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ( A9 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A9 @ B3 )
                 => ( ! [X: nat] :
                        ( ( member_nat @ X @ A9 )
                       => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_756_remove__induct,axiom,
    ! [P: set_state > $o,B3: set_state] :
      ( ( P @ bot_bot_set_state )
     => ( ( ~ ( finite_finite_state @ B3 )
         => ( P @ B3 ) )
       => ( ! [A9: set_state] :
              ( ( finite_finite_state @ A9 )
             => ( ( A9 != bot_bot_set_state )
               => ( ( ord_le2494988322063910608_state @ A9 @ B3 )
                 => ( ! [X: state] :
                        ( ( member_state @ X @ A9 )
                       => ( P @ ( minus_3933957440811877961_state @ A9 @ ( insert_state @ X @ bot_bot_set_state ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_757_finite__vimageD_H,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A4 ) )
     => ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_vimageD'
thf(fact_758_finite__vimageD_H,axiom,
    ! [F: state > nat,A4: set_nat] :
      ( ( finite_finite_state @ ( vimage_state_nat @ F @ A4 ) )
     => ( ( ord_less_eq_set_nat @ A4 @ ( image_state_nat @ F @ top_top_set_state ) )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_vimageD'
thf(fact_759_finite__vimageD_H,axiom,
    ! [F: nat > state,A4: set_state] :
      ( ( finite_finite_nat @ ( vimage_nat_state @ F @ A4 ) )
     => ( ( ord_le2494988322063910608_state @ A4 @ ( image_nat_state @ F @ top_top_set_nat ) )
       => ( finite_finite_state @ A4 ) ) ) ).

% finite_vimageD'
thf(fact_760_finite__vimageD_H,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( finite_finite_state @ ( vimage_state_state @ F @ A4 ) )
     => ( ( ord_le2494988322063910608_state @ A4 @ ( image_state_state @ F @ top_top_set_state ) )
       => ( finite_finite_state @ A4 ) ) ) ).

% finite_vimageD'
thf(fact_761_inf__img__fin__domE,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ~ ! [Y3: nat] :
              ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A4 ) )
             => ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) ) ) ) ).

% inf_img_fin_domE
thf(fact_762_inf__img__fin__domE,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( finite_finite_state @ ( image_state_state @ F @ A4 ) )
     => ( ~ ( finite_finite_state @ A4 )
       => ~ ! [Y3: state] :
              ( ( member_state @ Y3 @ ( image_state_state @ F @ A4 ) )
             => ( finite_finite_state @ ( vimage_state_state @ F @ ( insert_state @ Y3 @ bot_bot_set_state ) ) ) ) ) ) ).

% inf_img_fin_domE
thf(fact_763_inf__img__fin__domE,axiom,
    ! [F: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ ( image_nat_state @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ~ ! [Y3: state] :
              ( ( member_state @ Y3 @ ( image_nat_state @ F @ A4 ) )
             => ( finite_finite_nat @ ( vimage_nat_state @ F @ ( insert_state @ Y3 @ bot_bot_set_state ) ) ) ) ) ) ).

% inf_img_fin_domE
thf(fact_764_image__map__upd,axiom,
    ! [X3: state,A4: set_state,M2: state > option_state,Y2: state] :
      ( ~ ( member_state @ X3 @ A4 )
     => ( ( image_6076465424260689483_state @ ( fun_up8843634000204221123_state @ M2 @ X3 @ ( some_state @ Y2 ) ) @ A4 )
        = ( image_6076465424260689483_state @ M2 @ A4 ) ) ) ).

% image_map_upd
thf(fact_765_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X4 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_766_finite__ranking__induct,axiom,
    ! [S: set_state,P: set_state > $o,F: state > nat] :
      ( ( finite_finite_state @ S )
     => ( ( P @ bot_bot_set_state )
       => ( ! [X4: state,S4: set_state] :
              ( ( finite_finite_state @ S4 )
             => ( ! [Y5: state] :
                    ( ( member_state @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_state @ X4 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_767_finite__range__updI,axiom,
    ! [F: state > option_state,A3: state,B2: state] :
      ( ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ F @ top_top_set_state ) )
     => ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ ( fun_up8843634000204221123_state @ F @ A3 @ ( some_state @ B2 ) ) @ top_top_set_state ) ) ) ).

% finite_range_updI
thf(fact_768_cofinite__bot,axiom,
    ( ( cofinite_nat = bot_bot_filter_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% cofinite_bot
thf(fact_769_cofinite__bot,axiom,
    ( ( cofinite_state = bot_bot_filter_state )
    = ( finite_finite_state @ top_top_set_state ) ) ).

% cofinite_bot
thf(fact_770_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_771_card__le__if__inj__on__rel,axiom,
    ! [B3: set_state,A4: set_state,R2: state > state > $o] :
      ( ( finite_finite_state @ B3 )
     => ( ! [A5: state] :
            ( ( member_state @ A5 @ A4 )
           => ? [B8: state] :
                ( ( member_state @ B8 @ B3 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A1: state,A22: state,B7: state] :
              ( ( member_state @ A1 @ A4 )
             => ( ( member_state @ A22 @ A4 )
               => ( ( member_state @ B7 @ B3 )
                 => ( ( R2 @ A1 @ B7 )
                   => ( ( R2 @ A22 @ B7 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_772_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A4: set_state,R2: state > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A5: state] :
            ( ( member_state @ A5 @ A4 )
           => ? [B8: nat] :
                ( ( member_nat @ B8 @ B3 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A1: state,A22: state,B7: nat] :
              ( ( member_state @ A1 @ A4 )
             => ( ( member_state @ A22 @ A4 )
               => ( ( member_nat @ B7 @ B3 )
                 => ( ( R2 @ A1 @ B7 )
                   => ( ( R2 @ A22 @ B7 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_state @ A4 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_773_restrict__upd__same,axiom,
    ! [M2: state > option_state,X3: state,Y2: state] :
      ( ( restri2287918369865870758_state @ ( fun_up8843634000204221123_state @ M2 @ X3 @ ( some_state @ Y2 ) ) @ ( uminus472742206872269241_state @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
      = ( restri2287918369865870758_state @ M2 @ ( uminus472742206872269241_state @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ).

% restrict_upd_same
thf(fact_774_fun__upd__None__restrict,axiom,
    ! [X3: state,D2: set_state,M2: state > option_state] :
      ( ( ( member_state @ X3 @ D2 )
       => ( ( fun_up8843634000204221123_state @ ( restri2287918369865870758_state @ M2 @ D2 ) @ X3 @ none_state )
          = ( restri2287918369865870758_state @ M2 @ ( minus_3933957440811877961_state @ D2 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) )
      & ( ~ ( member_state @ X3 @ D2 )
       => ( ( fun_up8843634000204221123_state @ ( restri2287918369865870758_state @ M2 @ D2 ) @ X3 @ none_state )
          = ( restri2287918369865870758_state @ M2 @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_775_restrict__map__to__empty,axiom,
    ! [M2: state > option_state] :
      ( ( restri2287918369865870758_state @ M2 @ bot_bot_set_state )
      = ( ^ [X2: state] : none_state ) ) ).

% restrict_map_to_empty
thf(fact_776_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_777_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_778_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_779_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_780_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_781_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_782_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_783_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_784_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_785_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M: nat] :
      ( ( P @ X3 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X: nat] :
                    ( ( P @ X )
                   => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_786_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N3: set_nat] :
        ? [M4: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N3 )
         => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_787_restrict__complement__singleton__eq,axiom,
    ! [F: state > option_state,X3: state] :
      ( ( restri2287918369865870758_state @ F @ ( uminus472742206872269241_state @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
      = ( fun_up8843634000204221123_state @ F @ X3 @ none_state ) ) ).

% restrict_complement_singleton_eq
thf(fact_788_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_789_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_790_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_791_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_792_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_793_le__diff__iff_H,axiom,
    ! [A3: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% le_diff_iff'
thf(fact_794_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_795_arg__min__least,axiom,
    ! [S: set_nat,Y2: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y2 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).

% arg_min_least
thf(fact_796_arg__min__least,axiom,
    ! [S: set_state,Y2: state,F: state > nat] :
      ( ( finite_finite_state @ S )
     => ( ( S != bot_bot_set_state )
       => ( ( member_state @ Y2 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic8930993470425071781te_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).

% arg_min_least
thf(fact_797_dom__fun__upd,axiom,
    ! [Y2: option_state,F: state > option_state,X3: state] :
      ( ( ( Y2 = none_state )
       => ( ( dom_state_state @ ( fun_up8843634000204221123_state @ F @ X3 @ Y2 ) )
          = ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) )
      & ( ( Y2 != none_state )
       => ( ( dom_state_state @ ( fun_up8843634000204221123_state @ F @ X3 @ Y2 ) )
          = ( insert_state @ X3 @ ( dom_state_state @ F ) ) ) ) ) ).

% dom_fun_upd
thf(fact_798_card__Diff__singleton,axiom,
    ! [X3: state,A4: set_state] :
      ( ( member_state @ X3 @ A4 )
     => ( ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
        = ( minus_minus_nat @ ( finite_card_state @ A4 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_799_dom__eq__empty__conv,axiom,
    ! [F: state > option_state] :
      ( ( ( dom_state_state @ F )
        = bot_bot_set_state )
      = ( F
        = ( ^ [X2: state] : none_state ) ) ) ).

% dom_eq_empty_conv
thf(fact_800_card__Diff__insert,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( member_state @ A3 @ A4 )
     => ( ~ ( member_state @ A3 @ B3 )
       => ( ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ A3 @ B3 ) ) )
          = ( minus_minus_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_801_domD,axiom,
    ! [A3: state,M2: state > option_state] :
      ( ( member_state @ A3 @ ( dom_state_state @ M2 ) )
     => ? [B7: state] :
          ( ( M2 @ A3 )
          = ( some_state @ B7 ) ) ) ).

% domD
thf(fact_802_domI,axiom,
    ! [M2: state > option_state,A3: state,B2: state] :
      ( ( ( M2 @ A3 )
        = ( some_state @ B2 ) )
     => ( member_state @ A3 @ ( dom_state_state @ M2 ) ) ) ).

% domI
thf(fact_803_insert__dom,axiom,
    ! [F: state > option_state,X3: state,Y2: state] :
      ( ( ( F @ X3 )
        = ( some_state @ Y2 ) )
     => ( ( insert_state @ X3 @ ( dom_state_state @ F ) )
        = ( dom_state_state @ F ) ) ) ).

% insert_dom
thf(fact_804_dom__minus,axiom,
    ! [F: state > option_state,X3: state,A4: set_state] :
      ( ( ( F @ X3 )
        = none_state )
     => ( ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ ( insert_state @ X3 @ A4 ) )
        = ( minus_3933957440811877961_state @ ( dom_state_state @ F ) @ A4 ) ) ) ).

% dom_minus
thf(fact_805_card__1__singletonE,axiom,
    ! [A4: set_state] :
      ( ( ( finite_card_state @ A4 )
        = one_one_nat )
     => ~ ! [X4: state] :
            ( A4
           != ( insert_state @ X4 @ bot_bot_set_state ) ) ) ).

% card_1_singletonE
thf(fact_806_card__Diff__singleton__if,axiom,
    ! [X3: state,A4: set_state] :
      ( ( ( member_state @ X3 @ A4 )
       => ( ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
          = ( minus_minus_nat @ ( finite_card_state @ A4 ) @ one_one_nat ) ) )
      & ( ~ ( member_state @ X3 @ A4 )
       => ( ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) )
          = ( finite_card_state @ A4 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_807_UnCI,axiom,
    ! [C: state,B3: set_state,A4: set_state] :
      ( ( ~ ( member_state @ C @ B3 )
       => ( member_state @ C @ A4 ) )
     => ( member_state @ C @ ( sup_sup_set_state @ A4 @ B3 ) ) ) ).

% UnCI
thf(fact_808_Un__iff,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( sup_sup_set_state @ A4 @ B3 ) )
      = ( ( member_state @ C @ A4 )
        | ( member_state @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_809_Un__empty,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( sup_sup_set_state @ A4 @ B3 )
        = bot_bot_set_state )
      = ( ( A4 = bot_bot_set_state )
        & ( B3 = bot_bot_set_state ) ) ) ).

% Un_empty
thf(fact_810_Un__subset__iff,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A4 @ B3 ) @ C2 )
      = ( ( ord_le2494988322063910608_state @ A4 @ C2 )
        & ( ord_le2494988322063910608_state @ B3 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_811_Un__insert__left,axiom,
    ! [A3: state,B3: set_state,C2: set_state] :
      ( ( sup_sup_set_state @ ( insert_state @ A3 @ B3 ) @ C2 )
      = ( insert_state @ A3 @ ( sup_sup_set_state @ B3 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_812_Un__insert__right,axiom,
    ! [A4: set_state,A3: state,B3: set_state] :
      ( ( sup_sup_set_state @ A4 @ ( insert_state @ A3 @ B3 ) )
      = ( insert_state @ A3 @ ( sup_sup_set_state @ A4 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_813_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_state] :
      ( ( sup_sup_set_state @ X3 @ bot_bot_set_state )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_814_image__Un,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( image_state_state @ F @ ( sup_sup_set_state @ A4 @ B3 ) )
      = ( sup_sup_set_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) ) ).

% image_Un
thf(fact_815_Un__UNIV__left,axiom,
    ! [B3: set_state] :
      ( ( sup_sup_set_state @ top_top_set_state @ B3 )
      = top_top_set_state ) ).

% Un_UNIV_left
thf(fact_816_Un__UNIV__right,axiom,
    ! [A4: set_state] :
      ( ( sup_sup_set_state @ A4 @ top_top_set_state )
      = top_top_set_state ) ).

% Un_UNIV_right
thf(fact_817_Un__empty__right,axiom,
    ! [A4: set_state] :
      ( ( sup_sup_set_state @ A4 @ bot_bot_set_state )
      = A4 ) ).

% Un_empty_right
thf(fact_818_Un__empty__left,axiom,
    ! [B3: set_state] :
      ( ( sup_sup_set_state @ bot_bot_set_state @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_819_subset__Un__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A: set_state,B4: set_state] :
          ( ( sup_sup_set_state @ A @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_820_subset__UnE,axiom,
    ! [C2: set_state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ ( sup_sup_set_state @ A4 @ B3 ) )
     => ~ ! [A8: set_state] :
            ( ( ord_le2494988322063910608_state @ A8 @ A4 )
           => ! [B9: set_state] :
                ( ( ord_le2494988322063910608_state @ B9 @ B3 )
               => ( C2
                 != ( sup_sup_set_state @ A8 @ B9 ) ) ) ) ) ).

% subset_UnE
thf(fact_821_Un__absorb2,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A4 )
     => ( ( sup_sup_set_state @ A4 @ B3 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_822_Un__absorb1,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( sup_sup_set_state @ A4 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_823_Un__upper2,axiom,
    ! [B3: set_state,A4: set_state] : ( ord_le2494988322063910608_state @ B3 @ ( sup_sup_set_state @ A4 @ B3 ) ) ).

% Un_upper2
thf(fact_824_Un__upper1,axiom,
    ! [A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ A4 @ ( sup_sup_set_state @ A4 @ B3 ) ) ).

% Un_upper1
thf(fact_825_Un__least,axiom,
    ! [A4: set_state,C2: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ C2 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A4 @ B3 ) @ C2 ) ) ) ).

% Un_least
thf(fact_826_Un__mono,axiom,
    ! [A4: set_state,C2: set_state,B3: set_state,D2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ C2 )
     => ( ( ord_le2494988322063910608_state @ B3 @ D2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A4 @ B3 ) @ ( sup_sup_set_state @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_827_UnE,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( sup_sup_set_state @ A4 @ B3 ) )
     => ( ~ ( member_state @ C @ A4 )
       => ( member_state @ C @ B3 ) ) ) ).

% UnE
thf(fact_828_UnI1,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ A4 )
     => ( member_state @ C @ ( sup_sup_set_state @ A4 @ B3 ) ) ) ).

% UnI1
thf(fact_829_UnI2,axiom,
    ! [C: state,B3: set_state,A4: set_state] :
      ( ( member_state @ C @ B3 )
     => ( member_state @ C @ ( sup_sup_set_state @ A4 @ B3 ) ) ) ).

% UnI2
thf(fact_830_singleton__Un__iff,axiom,
    ! [X3: state,A4: set_state,B3: set_state] :
      ( ( ( insert_state @ X3 @ bot_bot_set_state )
        = ( sup_sup_set_state @ A4 @ B3 ) )
      = ( ( ( A4 = bot_bot_set_state )
          & ( B3
            = ( insert_state @ X3 @ bot_bot_set_state ) ) )
        | ( ( A4
            = ( insert_state @ X3 @ bot_bot_set_state ) )
          & ( B3 = bot_bot_set_state ) )
        | ( ( A4
            = ( insert_state @ X3 @ bot_bot_set_state ) )
          & ( B3
            = ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_831_Un__singleton__iff,axiom,
    ! [A4: set_state,B3: set_state,X3: state] :
      ( ( ( sup_sup_set_state @ A4 @ B3 )
        = ( insert_state @ X3 @ bot_bot_set_state ) )
      = ( ( ( A4 = bot_bot_set_state )
          & ( B3
            = ( insert_state @ X3 @ bot_bot_set_state ) ) )
        | ( ( A4
            = ( insert_state @ X3 @ bot_bot_set_state ) )
          & ( B3 = bot_bot_set_state ) )
        | ( ( A4
            = ( insert_state @ X3 @ bot_bot_set_state ) )
          & ( B3
            = ( insert_state @ X3 @ bot_bot_set_state ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_832_insert__is__Un,axiom,
    ( insert_state
    = ( ^ [A2: state] : ( sup_sup_set_state @ ( insert_state @ A2 @ bot_bot_set_state ) ) ) ) ).

% insert_is_Un
thf(fact_833_Diff__partition,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( sup_sup_set_state @ A4 @ ( minus_3933957440811877961_state @ B3 @ A4 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_834_Diff__subset__conv,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( minus_3933957440811877961_state @ A4 @ B3 ) @ C2 )
      = ( ord_le2494988322063910608_state @ A4 @ ( sup_sup_set_state @ B3 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_835_Compl__partition,axiom,
    ! [A4: set_state] :
      ( ( sup_sup_set_state @ A4 @ ( uminus472742206872269241_state @ A4 ) )
      = top_top_set_state ) ).

% Compl_partition
thf(fact_836_Compl__partition2,axiom,
    ! [A4: set_state] :
      ( ( sup_sup_set_state @ ( uminus472742206872269241_state @ A4 ) @ A4 )
      = top_top_set_state ) ).

% Compl_partition2
thf(fact_837_sup__shunt,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ( sup_sup_set_state @ X3 @ Y2 )
        = top_top_set_state )
      = ( ord_le2494988322063910608_state @ ( uminus472742206872269241_state @ X3 ) @ Y2 ) ) ).

% sup_shunt
thf(fact_838_Pow__insert,axiom,
    ! [A3: state,A4: set_state] :
      ( ( pow_state @ ( insert_state @ A3 @ A4 ) )
      = ( sup_su4188871578264421970_state @ ( pow_state @ A4 ) @ ( image_2476256681063834599_state @ ( insert_state @ A3 ) @ ( pow_state @ A4 ) ) ) ) ).

% Pow_insert
thf(fact_839_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_840_sup__bot_Oright__neutral,axiom,
    ! [A3: set_state] :
      ( ( sup_sup_set_state @ A3 @ bot_bot_set_state )
      = A3 ) ).

% sup_bot.right_neutral
thf(fact_841_sup__bot_Oneutr__eq__iff,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( bot_bot_set_state
        = ( sup_sup_set_state @ A3 @ B2 ) )
      = ( ( A3 = bot_bot_set_state )
        & ( B2 = bot_bot_set_state ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_842_sup__bot_Oleft__neutral,axiom,
    ! [A3: set_state] :
      ( ( sup_sup_set_state @ bot_bot_set_state @ A3 )
      = A3 ) ).

% sup_bot.left_neutral
thf(fact_843_sup_Obounded__iff,axiom,
    ! [B2: set_state,C: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B2 @ C ) @ A3 )
      = ( ( ord_le2494988322063910608_state @ B2 @ A3 )
        & ( ord_le2494988322063910608_state @ C @ A3 ) ) ) ).

% sup.bounded_iff
thf(fact_844_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
      = ( ( ord_less_eq_nat @ B2 @ A3 )
        & ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% sup.bounded_iff
thf(fact_845_le__sup__iff,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ X3 @ Y2 ) @ Z )
      = ( ( ord_le2494988322063910608_state @ X3 @ Z )
        & ( ord_le2494988322063910608_state @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_846_le__sup__iff,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ Z )
      = ( ( ord_less_eq_nat @ X3 @ Z )
        & ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_847_sup__bot__left,axiom,
    ! [X3: set_state] :
      ( ( sup_sup_set_state @ bot_bot_set_state @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_848_sup__bot__right,axiom,
    ! [X3: set_state] :
      ( ( sup_sup_set_state @ X3 @ bot_bot_set_state )
      = X3 ) ).

% sup_bot_right
thf(fact_849_bot__eq__sup__iff,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( bot_bot_set_state
        = ( sup_sup_set_state @ X3 @ Y2 ) )
      = ( ( X3 = bot_bot_set_state )
        & ( Y2 = bot_bot_set_state ) ) ) ).

% bot_eq_sup_iff
thf(fact_850_sup__eq__bot__iff,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ( sup_sup_set_state @ X3 @ Y2 )
        = bot_bot_set_state )
      = ( ( X3 = bot_bot_set_state )
        & ( Y2 = bot_bot_set_state ) ) ) ).

% sup_eq_bot_iff
thf(fact_851_sup__bot_Oeq__neutr__iff,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ( sup_sup_set_state @ A3 @ B2 )
        = bot_bot_set_state )
      = ( ( A3 = bot_bot_set_state )
        & ( B2 = bot_bot_set_state ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_852_inf__sup__ord_I4_J,axiom,
    ! [Y2: set_state,X3: set_state] : ( ord_le2494988322063910608_state @ Y2 @ ( sup_sup_set_state @ X3 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_853_inf__sup__ord_I4_J,axiom,
    ! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_854_inf__sup__ord_I3_J,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ X3 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_855_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_856_le__supE,axiom,
    ! [A3: set_state,B2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ X3 )
     => ~ ( ( ord_le2494988322063910608_state @ A3 @ X3 )
         => ~ ( ord_le2494988322063910608_state @ B2 @ X3 ) ) ) ).

% le_supE
thf(fact_857_le__supE,axiom,
    ! [A3: nat,B2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X3 )
     => ~ ( ( ord_less_eq_nat @ A3 @ X3 )
         => ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).

% le_supE
thf(fact_858_le__supI,axiom,
    ! [A3: set_state,X3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ X3 )
     => ( ( ord_le2494988322063910608_state @ B2 @ X3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ X3 ) ) ) ).

% le_supI
thf(fact_859_le__supI,axiom,
    ! [A3: nat,X3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ X3 )
     => ( ( ord_less_eq_nat @ B2 @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X3 ) ) ) ).

% le_supI
thf(fact_860_sup__ge1,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ X3 @ Y2 ) ) ).

% sup_ge1
thf(fact_861_sup__ge1,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% sup_ge1
thf(fact_862_sup__ge2,axiom,
    ! [Y2: set_state,X3: set_state] : ( ord_le2494988322063910608_state @ Y2 @ ( sup_sup_set_state @ X3 @ Y2 ) ) ).

% sup_ge2
thf(fact_863_sup__ge2,axiom,
    ! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).

% sup_ge2
thf(fact_864_le__supI1,axiom,
    ! [X3: set_state,A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ A3 )
     => ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).

% le_supI1
thf(fact_865_le__supI1,axiom,
    ! [X3: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X3 @ A3 )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% le_supI1
thf(fact_866_le__supI2,axiom,
    ! [X3: set_state,B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ B2 )
     => ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).

% le_supI2
thf(fact_867_le__supI2,axiom,
    ! [X3: nat,B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ X3 @ B2 )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% le_supI2
thf(fact_868_sup_Omono,axiom,
    ! [C: set_state,A3: set_state,D: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ A3 )
     => ( ( ord_le2494988322063910608_state @ D @ B2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ C @ D ) @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_869_sup_Omono,axiom,
    ! [C: nat,A3: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_870_sup__mono,axiom,
    ! [A3: set_state,C: set_state,B2: set_state,D: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ C )
     => ( ( ord_le2494988322063910608_state @ B2 @ D )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A3 @ B2 ) @ ( sup_sup_set_state @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_871_sup__mono,axiom,
    ! [A3: nat,C: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_872_sup__least,axiom,
    ! [Y2: set_state,X3: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
     => ( ( ord_le2494988322063910608_state @ Z @ X3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ Y2 @ Z ) @ X3 ) ) ) ).

% sup_least
thf(fact_873_sup__least,axiom,
    ! [Y2: nat,X3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( ord_less_eq_nat @ Z @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X3 ) ) ) ).

% sup_least
thf(fact_874_le__iff__sup,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( sup_sup_set_state @ X2 @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_875_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y: nat] :
          ( ( sup_sup_nat @ X2 @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_876_sup_OorderE,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( A3
        = ( sup_sup_set_state @ A3 @ B2 ) ) ) ).

% sup.orderE
thf(fact_877_sup_OorderE,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( A3
        = ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.orderE
thf(fact_878_sup_OorderI,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( A3
        = ( sup_sup_set_state @ A3 @ B2 ) )
     => ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ).

% sup.orderI
thf(fact_879_sup_OorderI,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3
        = ( sup_sup_nat @ A3 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A3 ) ) ).

% sup.orderI
thf(fact_880_sup__unique,axiom,
    ! [F: set_state > set_state > set_state,X3: set_state,Y2: set_state] :
      ( ! [X4: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ X4 @ ( F @ X4 @ Y3 ) )
     => ( ! [X4: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ Y3 @ ( F @ X4 @ Y3 ) )
       => ( ! [X4: set_state,Y3: set_state,Z3: set_state] :
              ( ( ord_le2494988322063910608_state @ Y3 @ X4 )
             => ( ( ord_le2494988322063910608_state @ Z3 @ X4 )
               => ( ord_le2494988322063910608_state @ ( F @ Y3 @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_set_state @ X3 @ Y2 )
            = ( F @ X3 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_881_sup__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y2: nat] :
      ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y3 ) )
     => ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X4 @ Y3 ) )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X4 )
             => ( ( ord_less_eq_nat @ Z3 @ X4 )
               => ( ord_less_eq_nat @ ( F @ Y3 @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_nat @ X3 @ Y2 )
            = ( F @ X3 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_882_sup_Oabsorb1,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( sup_sup_set_state @ A3 @ B2 )
        = A3 ) ) ).

% sup.absorb1
thf(fact_883_sup_Oabsorb1,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( sup_sup_nat @ A3 @ B2 )
        = A3 ) ) ).

% sup.absorb1
thf(fact_884_sup_Oabsorb2,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( sup_sup_set_state @ A3 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_885_sup_Oabsorb2,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( sup_sup_nat @ A3 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_886_sup__absorb1,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
     => ( ( sup_sup_set_state @ X3 @ Y2 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_887_sup__absorb1,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( sup_sup_nat @ X3 @ Y2 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_888_sup__absorb2,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( sup_sup_set_state @ X3 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_889_sup__absorb2,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( sup_sup_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_890_sup_OboundedE,axiom,
    ! [B2: set_state,C: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B2 @ C ) @ A3 )
     => ~ ( ( ord_le2494988322063910608_state @ B2 @ A3 )
         => ~ ( ord_le2494988322063910608_state @ C @ A3 ) ) ) ).

% sup.boundedE
thf(fact_891_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A3 )
         => ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% sup.boundedE
thf(fact_892_sup_OboundedI,axiom,
    ! [B2: set_state,A3: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( ord_le2494988322063910608_state @ C @ A3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B2 @ C ) @ A3 ) ) ) ).

% sup.boundedI
thf(fact_893_sup_OboundedI,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ C @ A3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 ) ) ) ).

% sup.boundedI
thf(fact_894_sup_Oorder__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B: set_state,A2: set_state] :
          ( A2
          = ( sup_sup_set_state @ A2 @ B ) ) ) ) ).

% sup.order_iff
thf(fact_895_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B: nat,A2: nat] :
          ( A2
          = ( sup_sup_nat @ A2 @ B ) ) ) ) ).

% sup.order_iff
thf(fact_896_sup_Ocobounded1,axiom,
    ! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ A3 @ ( sup_sup_set_state @ A3 @ B2 ) ) ).

% sup.cobounded1
thf(fact_897_sup_Ocobounded1,axiom,
    ! [A3: nat,B2: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B2 ) ) ).

% sup.cobounded1
thf(fact_898_sup_Ocobounded2,axiom,
    ! [B2: set_state,A3: set_state] : ( ord_le2494988322063910608_state @ B2 @ ( sup_sup_set_state @ A3 @ B2 ) ) ).

% sup.cobounded2
thf(fact_899_sup_Ocobounded2,axiom,
    ! [B2: nat,A3: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A3 @ B2 ) ) ).

% sup.cobounded2
thf(fact_900_sup_Oabsorb__iff1,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B: set_state,A2: set_state] :
          ( ( sup_sup_set_state @ A2 @ B )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_901_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B: nat,A2: nat] :
          ( ( sup_sup_nat @ A2 @ B )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_902_sup_Oabsorb__iff2,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A2: set_state,B: set_state] :
          ( ( sup_sup_set_state @ A2 @ B )
          = B ) ) ) ).

% sup.absorb_iff2
thf(fact_903_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B: nat] :
          ( ( sup_sup_nat @ A2 @ B )
          = B ) ) ) ).

% sup.absorb_iff2
thf(fact_904_sup_OcoboundedI1,axiom,
    ! [C: set_state,A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ A3 )
     => ( ord_le2494988322063910608_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_905_sup_OcoboundedI1,axiom,
    ! [C: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_906_sup_OcoboundedI2,axiom,
    ! [C: set_state,B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ B2 )
     => ( ord_le2494988322063910608_state @ C @ ( sup_sup_set_state @ A3 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_907_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_908_chains__extend,axiom,
    ! [C: set_set_state,S: set_set_state,Z: set_state] :
      ( ( member_set_set_state @ C @ ( chains_state @ S ) )
     => ( ( member_set_state @ Z @ S )
       => ( ! [X4: set_state] :
              ( ( member_set_state @ X4 @ C )
             => ( ord_le2494988322063910608_state @ X4 @ Z ) )
         => ( member_set_set_state @ ( sup_su4188871578264421970_state @ ( insert_set_state @ Z @ bot_bo2271482359692755898_state ) @ C ) @ ( chains_state @ S ) ) ) ) ) ).

% chains_extend
thf(fact_909_Sup__fin_Oremove,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A4 )
              = X3 ) )
          & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A4 )
              = ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_910_Sup__fin_Oinsert__remove,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A4 ) )
            = X3 ) )
        & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A4 ) )
            = ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_911_Sup__fin_Oinsert,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A4 ) )
          = ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_912_Zorn__Lemma2,axiom,
    ! [A4: set_set_state] :
      ( ! [X4: set_set_state] :
          ( ( member_set_set_state @ X4 @ ( chains_state @ A4 ) )
         => ? [Xa2: set_state] :
              ( ( member_set_state @ Xa2 @ A4 )
              & ! [Xb: set_state] :
                  ( ( member_set_state @ Xb @ X4 )
                 => ( ord_le2494988322063910608_state @ Xb @ Xa2 ) ) ) )
     => ? [X4: set_state] :
          ( ( member_set_state @ X4 @ A4 )
          & ! [Xa2: set_state] :
              ( ( member_set_state @ Xa2 @ A4 )
             => ( ( ord_le2494988322063910608_state @ X4 @ Xa2 )
               => ( Xa2 = X4 ) ) ) ) ) ).

% Zorn_Lemma2
thf(fact_913_chainsD,axiom,
    ! [C: set_set_state,S: set_set_state,X3: set_state,Y2: set_state] :
      ( ( member_set_set_state @ C @ ( chains_state @ S ) )
     => ( ( member_set_state @ X3 @ C )
       => ( ( member_set_state @ Y2 @ C )
         => ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
            | ( ord_le2494988322063910608_state @ Y2 @ X3 ) ) ) ) ) ).

% chainsD
thf(fact_914_Sup__fin_OcoboundedI,axiom,
    ! [A4: set_set_state,A3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( member_set_state @ A3 @ A4 )
       => ( ord_le2494988322063910608_state @ A3 @ ( lattic1454283544731368441_state @ A4 ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_915_Sup__fin_OcoboundedI,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ( ord_less_eq_nat @ A3 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_916_Sup__fin_OboundedE,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A4 ) @ X3 )
         => ! [A10: set_state] :
              ( ( member_set_state @ A10 @ A4 )
             => ( ord_le2494988322063910608_state @ A10 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_917_Sup__fin_OboundedE,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X3 )
         => ! [A10: nat] :
              ( ( member_nat @ A10 @ A4 )
             => ( ord_less_eq_nat @ A10 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_918_Sup__fin_OboundedI,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ! [A5: set_state] :
              ( ( member_set_state @ A5 @ A4 )
             => ( ord_le2494988322063910608_state @ A5 @ X3 ) )
         => ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A4 ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_919_Sup__fin_OboundedI,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A4 )
             => ( ord_less_eq_nat @ A5 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_920_Sup__fin_Obounded__iff,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A4 ) @ X3 )
          = ( ! [X2: set_state] :
                ( ( member_set_state @ X2 @ A4 )
               => ( ord_le2494988322063910608_state @ X2 @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_921_Sup__fin_Obounded__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X3 )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A4 )
               => ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_922_Sup__fin_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( lattic1093996805478795353in_nat @ A4 )
        = ( the_nat @ none_nat ) ) ) ).

% Sup_fin.infinite
thf(fact_923_Sup__fin_Osubset__imp,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ B3 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( finite4951987536711252743_state @ B3 )
         => ( ord_le2494988322063910608_state @ ( lattic1454283544731368441_state @ A4 ) @ ( lattic1454283544731368441_state @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_924_Sup__fin_Osubset__imp,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_925_Sup__fin_Ohom__commute,axiom,
    ! [H: nat > nat,N: set_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( H @ ( sup_sup_nat @ X4 @ Y3 ) )
          = ( sup_sup_nat @ ( H @ X4 ) @ ( H @ Y3 ) ) )
     => ( ( finite_finite_nat @ N )
       => ( ( N != bot_bot_set_nat )
         => ( ( H @ ( lattic1093996805478795353in_nat @ N ) )
            = ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_926_Sup__fin_Osubset,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A4 )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B3 ) @ ( lattic1093996805478795353in_nat @ A4 ) )
            = ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_927_Sup__fin_Oclosed,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X4: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ ( insert_nat @ X4 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ A4 ) ) ) ) ).

% Sup_fin.closed
thf(fact_928_Sup__fin_Oinsert__not__elem,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X3 @ A4 )
       => ( ( A4 != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A4 ) )
            = ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_929_Sup__fin_Ounion,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A4 @ B3 ) )
              = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B3 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_930_card__Diff1__less,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_931_card__Diff1__less,axiom,
    ! [A4: set_state,X3: state] :
      ( ( finite_finite_state @ A4 )
     => ( ( member_state @ X3 @ A4 )
       => ( ord_less_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) @ ( finite_card_state @ A4 ) ) ) ) ).

% card_Diff1_less
thf(fact_932_card__Diff2__less,axiom,
    ! [A4: set_nat,X3: nat,Y2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( member_nat @ Y2 @ A4 )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_933_card__Diff2__less,axiom,
    ! [A4: set_state,X3: state,Y2: state] :
      ( ( finite_finite_state @ A4 )
     => ( ( member_state @ X3 @ A4 )
       => ( ( member_state @ Y2 @ A4 )
         => ( ord_less_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) @ ( insert_state @ Y2 @ bot_bot_set_state ) ) ) @ ( finite_card_state @ A4 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_934_card__Diff1__less__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) )
      = ( ( finite_finite_nat @ A4 )
        & ( member_nat @ X3 @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_935_card__Diff1__less__iff,axiom,
    ! [A4: set_state,X3: state] :
      ( ( ord_less_nat @ ( finite_card_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) ) @ ( finite_card_state @ A4 ) )
      = ( ( finite_finite_state @ A4 )
        & ( member_state @ X3 @ A4 ) ) ) ).

% card_Diff1_less_iff
thf(fact_936_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_937_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_938_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_939_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N4: nat] :
          ( ( ord_less_nat @ M4 @ N4 )
          | ( M4 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_940_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_941_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M4 @ N4 )
          & ( M4 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_942_order__less__imp__not__less,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_943_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( Y2 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_944_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( X3 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_945_linorder__less__linear,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
      | ( X3 = Y2 )
      | ( ord_less_nat @ Y2 @ X3 ) ) ).

% linorder_less_linear
thf(fact_946_order__less__imp__triv,axiom,
    ! [X3: nat,Y2: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_947_order__less__not__sym,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).

% order_less_not_sym
thf(fact_948_order__less__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_949_order__less__subst1,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_950_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_951_ord__less__eq__subst,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_952_ord__eq__less__subst,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A3
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_953_order__less__trans,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_954_order__less__asym_H,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A3 ) ) ).

% order_less_asym'
thf(fact_955_linorder__neq__iff,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( X3 != Y2 )
      = ( ( ord_less_nat @ X3 @ Y2 )
        | ( ord_less_nat @ Y2 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_956_order__less__asym,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X3 ) ) ).

% order_less_asym
thf(fact_957_linorder__neqE,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( X3 != Y2 )
     => ( ~ ( ord_less_nat @ X3 @ Y2 )
       => ( ord_less_nat @ Y2 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_958_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_nat @ B2 @ A3 )
     => ( A3 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_959_order_Ostrict__implies__not__eq,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( A3 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_960_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A3 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A3 ) ) ) ).

% dual_order.strict_trans
thf(fact_961_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
      = ( ( ord_less_nat @ Y2 @ X3 )
        | ( X3 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_962_order_Ostrict__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% order.strict_trans
thf(fact_963_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A3: nat,B2: nat] :
      ( ! [A5: nat,B7: nat] :
          ( ( ord_less_nat @ A5 @ B7 )
         => ( P @ A5 @ B7 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B7: nat] :
              ( ( P @ B7 @ A5 )
             => ( P @ A5 @ B7 ) )
         => ( P @ A3 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_964_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X7: nat] : ( P5 @ X7 ) )
    = ( ^ [P2: nat > $o] :
        ? [N4: nat] :
          ( ( P2 @ N4 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N4 )
             => ~ ( P2 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_965_dual__order_Oirrefl,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ A3 ) ).

% dual_order.irrefl
thf(fact_966_dual__order_Oasym,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_nat @ B2 @ A3 )
     => ~ ( ord_less_nat @ A3 @ B2 ) ) ).

% dual_order.asym
thf(fact_967_linorder__cases,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y2 )
     => ( ( X3 != Y2 )
       => ( ord_less_nat @ Y2 @ X3 ) ) ) ).

% linorder_cases
thf(fact_968_antisym__conv3,axiom,
    ! [Y2: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y2 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
        = ( X3 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_969_less__induct,axiom,
    ! [P: nat > $o,A3: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A3 ) ) ).

% less_induct
thf(fact_970_ord__less__eq__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_971_ord__eq__less__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( A3 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_972_order_Oasym,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A3 ) ) ).

% order.asym
thf(fact_973_less__imp__neq,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( X3 != Y2 ) ) ).

% less_imp_neq
thf(fact_974_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X3 @ X_12 ) ).

% gt_ex
thf(fact_975_top_Onot__eq__extremum,axiom,
    ! [A3: set_state] :
      ( ( A3 != top_top_set_state )
      = ( ord_less_set_state @ A3 @ top_top_set_state ) ) ).

% top.not_eq_extremum
thf(fact_976_top_Oextremum__strict,axiom,
    ! [A3: set_state] :
      ~ ( ord_less_set_state @ top_top_set_state @ A3 ) ).

% top.extremum_strict
thf(fact_977_bot_Oextremum__strict,axiom,
    ! [A3: set_state] :
      ~ ( ord_less_set_state @ A3 @ bot_bot_set_state ) ).

% bot.extremum_strict
thf(fact_978_bot_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_979_bot_Onot__eq__extremum,axiom,
    ! [A3: set_state] :
      ( ( A3 != bot_bot_set_state )
      = ( ord_less_set_state @ bot_bot_set_state @ A3 ) ) ).

% bot.not_eq_extremum
thf(fact_980_bot_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).

% bot.not_eq_extremum
thf(fact_981_leD,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
     => ~ ( ord_less_set_state @ X3 @ Y2 ) ) ).

% leD
thf(fact_982_leD,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y2 ) ) ).

% leD
thf(fact_983_leI,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% leI
thf(fact_984_nless__le,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ~ ( ord_less_set_state @ A3 @ B2 ) )
      = ( ~ ( ord_le2494988322063910608_state @ A3 @ B2 )
        | ( A3 = B2 ) ) ) ).

% nless_le
thf(fact_985_nless__le,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A3 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A3 @ B2 )
        | ( A3 = B2 ) ) ) ).

% nless_le
thf(fact_986_antisym__conv1,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ~ ( ord_less_set_state @ X3 @ Y2 )
     => ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_987_antisym__conv1,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_988_antisym__conv2,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ~ ( ord_less_set_state @ X3 @ Y2 ) )
        = ( X3 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_989_antisym__conv2,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
        = ( X3 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_990_less__le__not__le,axiom,
    ( ord_less_set_state
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( ord_le2494988322063910608_state @ X2 @ Y )
          & ~ ( ord_le2494988322063910608_state @ Y @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_991_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_992_not__le__imp__less,axiom,
    ! [Y2: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ord_less_nat @ X3 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_993_order_Oorder__iff__strict,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_less_set_state @ A2 @ B )
          | ( A2 = B ) ) ) ) ).

% order.order_iff_strict
thf(fact_994_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B: nat] :
          ( ( ord_less_nat @ A2 @ B )
          | ( A2 = B ) ) ) ) ).

% order.order_iff_strict
thf(fact_995_order_Ostrict__iff__order,axiom,
    ( ord_less_set_state
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ A2 @ B )
          & ( A2 != B ) ) ) ) ).

% order.strict_iff_order
thf(fact_996_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B: nat] :
          ( ( ord_less_eq_nat @ A2 @ B )
          & ( A2 != B ) ) ) ) ).

% order.strict_iff_order
thf(fact_997_order_Ostrict__trans1,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_less_set_state @ B2 @ C )
       => ( ord_less_set_state @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_998_order_Ostrict__trans1,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_999_order_Ostrict__trans2,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_less_set_state @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ord_less_set_state @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1000_order_Ostrict__trans2,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1001_order_Ostrict__iff__not,axiom,
    ( ord_less_set_state
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ A2 @ B )
          & ~ ( ord_le2494988322063910608_state @ B @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1002_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B: nat] :
          ( ( ord_less_eq_nat @ A2 @ B )
          & ~ ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1003_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B: set_state,A2: set_state] :
          ( ( ord_less_set_state @ B @ A2 )
          | ( A2 = B ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1004_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B: nat,A2: nat] :
          ( ( ord_less_nat @ B @ A2 )
          | ( A2 = B ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1005_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_state
    = ( ^ [B: set_state,A2: set_state] :
          ( ( ord_le2494988322063910608_state @ B @ A2 )
          & ( A2 != B ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1006_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B @ A2 )
          & ( A2 != B ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1007_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_state,A3: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( ord_less_set_state @ C @ B2 )
       => ( ord_less_set_state @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1008_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1009_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_state,A3: set_state,C: set_state] :
      ( ( ord_less_set_state @ B2 @ A3 )
     => ( ( ord_le2494988322063910608_state @ C @ B2 )
       => ( ord_less_set_state @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1010_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1011_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_state
    = ( ^ [B: set_state,A2: set_state] :
          ( ( ord_le2494988322063910608_state @ B @ A2 )
          & ~ ( ord_le2494988322063910608_state @ A2 @ B ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1012_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1013_order_Ostrict__implies__order,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_less_set_state @ A3 @ B2 )
     => ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1014_order_Ostrict__implies__order,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ord_less_eq_nat @ A3 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1015_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( ord_less_set_state @ B2 @ A3 )
     => ( ord_le2494988322063910608_state @ B2 @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1016_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_nat @ B2 @ A3 )
     => ( ord_less_eq_nat @ B2 @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1017_order__le__less,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( ord_less_set_state @ X2 @ Y )
          | ( X2 = Y ) ) ) ) ).

% order_le_less
thf(fact_1018_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y: nat] :
          ( ( ord_less_nat @ X2 @ Y )
          | ( X2 = Y ) ) ) ) ).

% order_le_less
thf(fact_1019_order__less__le,axiom,
    ( ord_less_set_state
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( ord_le2494988322063910608_state @ X2 @ Y )
          & ( X2 != Y ) ) ) ) ).

% order_less_le
thf(fact_1020_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y )
          & ( X2 != Y ) ) ) ) ).

% order_less_le
thf(fact_1021_linorder__not__le,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
      = ( ord_less_nat @ Y2 @ X3 ) ) ).

% linorder_not_le
thf(fact_1022_linorder__not__less,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
      = ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_not_less
thf(fact_1023_order__less__imp__le,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_less_set_state @ X3 @ Y2 )
     => ( ord_le2494988322063910608_state @ X3 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_1024_order__less__imp__le,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_1025_order__le__neq__trans,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( A3 != B2 )
       => ( ord_less_set_state @ A3 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1026_order__le__neq__trans,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( A3 != B2 )
       => ( ord_less_nat @ A3 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1027_order__neq__le__trans,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( A3 != B2 )
     => ( ( ord_le2494988322063910608_state @ A3 @ B2 )
       => ( ord_less_set_state @ A3 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1028_order__neq__le__trans,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != B2 )
     => ( ( ord_less_eq_nat @ A3 @ B2 )
       => ( ord_less_nat @ A3 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1029_order__le__less__trans,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ord_less_set_state @ Y2 @ Z )
       => ( ord_less_set_state @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1030_order__le__less__trans,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1031_order__less__le__trans,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_less_set_state @ X3 @ Y2 )
     => ( ( ord_le2494988322063910608_state @ Y2 @ Z )
       => ( ord_less_set_state @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1032_order__less__le__trans,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1033_order__le__less__subst1,axiom,
    ! [A3: set_state,F: nat > set_state,B2: nat,C: nat] :
      ( ( ord_le2494988322063910608_state @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_set_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1034_order__le__less__subst1,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1035_order__le__less__subst2,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_less_set_state @ ( F @ B2 ) @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1036_order__le__less__subst2,axiom,
    ! [A3: set_state,B2: set_state,F: set_state > nat,C: nat] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1037_order__le__less__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > set_state,C: set_state] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_set_state @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1038_order__le__less__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1039_order__less__le__subst1,axiom,
    ! [A3: set_state,F: set_state > set_state,B2: set_state,C: set_state] :
      ( ( ord_less_set_state @ A3 @ ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ A3 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1040_order__less__le__subst1,axiom,
    ! [A3: nat,F: set_state > nat,B2: set_state,C: set_state] :
      ( ( ord_less_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_le2494988322063910608_state @ B2 @ C )
       => ( ! [X4: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1041_order__less__le__subst1,axiom,
    ! [A3: set_state,F: nat > set_state,B2: nat,C: nat] :
      ( ( ord_less_set_state @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ A3 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1042_order__less__le__subst1,axiom,
    ! [A3: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1043_order__less__le__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > set_state,C: set_state] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_set_state @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_state @ ( F @ A3 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1044_order__less__le__subst2,axiom,
    ! [A3: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1045_linorder__le__less__linear,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
      | ( ord_less_nat @ Y2 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1046_order__le__imp__less__or__eq,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ord_less_set_state @ X3 @ Y2 )
        | ( X3 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1047_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_nat @ X3 @ Y2 )
        | ( X3 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1048_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1049_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X6 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X6 )
                & ( ord_less_nat @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_1050_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M2 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_1051_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ C @ A3 )
       => ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1052_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B7: nat,A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ! [X: nat] :
                    ( ( member_nat @ X @ A9 )
                   => ( ord_less_nat @ B7 @ X ) )
               => ( ( P @ A9 )
                 => ( P @ ( insert_nat @ B7 @ A9 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1053_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B7: nat,A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ! [X: nat] :
                    ( ( member_nat @ X @ A9 )
                   => ( ord_less_nat @ X @ B7 ) )
               => ( ( P @ A9 )
                 => ( P @ ( insert_nat @ B7 @ A9 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1054_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X: nat] :
              ( ( member_nat @ X @ S )
              & ( ord_less_nat @ ( F @ X ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1055_arg__min__if__finite_I2_J,axiom,
    ! [S: set_state,F: state > nat] :
      ( ( finite_finite_state @ S )
     => ( ( S != bot_bot_set_state )
       => ~ ? [X: state] :
              ( ( member_state @ X @ S )
              & ( ord_less_nat @ ( F @ X ) @ ( F @ ( lattic8930993470425071781te_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1056_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K2 ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1057_card__insert__le__m1,axiom,
    ! [N2: nat,Y2: set_state,X3: state] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite_card_state @ Y2 ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_state @ ( insert_state @ X3 @ Y2 ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_1058_minf_I8_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ~ ( ord_less_eq_nat @ T4 @ X ) ) ).

% minf(8)
thf(fact_1059_psubsetI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( A4 != B3 )
       => ( ord_less_set_state @ A4 @ B3 ) ) ) ).

% psubsetI
thf(fact_1060_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1061_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).

% bot_nat_0.extremum
thf(fact_1062_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1063_card_Oempty,axiom,
    ( ( finite_card_state @ bot_bot_set_state )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1064_diff__is__0__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1065_diff__is__0__eq_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1066_card__0__eq,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( finite_card_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_1067_card__0__eq,axiom,
    ! [A4: set_state] :
      ( ( finite_finite_state @ A4 )
     => ( ( ( finite_card_state @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_state ) ) ) ).

% card_0_eq
thf(fact_1068_not__psubset__empty,axiom,
    ! [A4: set_state] :
      ~ ( ord_less_set_state @ A4 @ bot_bot_set_state ) ).

% not_psubset_empty
thf(fact_1069_psubsetE,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_less_set_state @ A4 @ B3 )
     => ~ ( ( ord_le2494988322063910608_state @ A4 @ B3 )
         => ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ) ).

% psubsetE
thf(fact_1070_psubset__eq,axiom,
    ( ord_less_set_state
    = ( ^ [A: set_state,B4: set_state] :
          ( ( ord_le2494988322063910608_state @ A @ B4 )
          & ( A != B4 ) ) ) ) ).

% psubset_eq
thf(fact_1071_psubset__imp__subset,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_less_set_state @ A4 @ B3 )
     => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_1072_psubset__subset__trans,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_less_set_state @ A4 @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ord_less_set_state @ A4 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_1073_subset__not__subset__eq,axiom,
    ( ord_less_set_state
    = ( ^ [A: set_state,B4: set_state] :
          ( ( ord_le2494988322063910608_state @ A @ B4 )
          & ~ ( ord_le2494988322063910608_state @ B4 @ A ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1074_subset__psubset__trans,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( ord_less_set_state @ B3 @ C2 )
       => ( ord_less_set_state @ A4 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_1075_subset__iff__psubset__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A: set_state,B4: set_state] :
          ( ( ord_less_set_state @ A @ B4 )
          | ( A = B4 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1076_psubset__imp__ex__mem,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_less_set_state @ A4 @ B3 )
     => ? [B7: state] : ( member_state @ B7 @ ( minus_3933957440811877961_state @ B3 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1077_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_1078_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1079_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1080_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1081_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1082_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1083_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1084_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1085_card__eq__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_1086_card__eq__0__iff,axiom,
    ! [A4: set_state] :
      ( ( ( finite_card_state @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_state )
        | ~ ( finite_finite_state @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_1087_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T3: set_nat] :
              ( ( ord_less_set_nat @ T3 @ S )
             => ( ( P @ T3 )
               => ? [X: nat] :
                    ( ( member_nat @ X @ ( minus_minus_set_nat @ S @ T3 ) )
                    & ( P @ ( insert_nat @ X @ T3 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_1088_finite__induct__select,axiom,
    ! [S: set_state,P: set_state > $o] :
      ( ( finite_finite_state @ S )
     => ( ( P @ bot_bot_set_state )
       => ( ! [T3: set_state] :
              ( ( ord_less_set_state @ T3 @ S )
             => ( ( P @ T3 )
               => ? [X: state] :
                    ( ( member_state @ X @ ( minus_3933957440811877961_state @ S @ T3 ) )
                    & ( P @ ( insert_state @ X @ T3 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_1089_psubset__insert__iff,axiom,
    ! [A4: set_state,X3: state,B3: set_state] :
      ( ( ord_less_set_state @ A4 @ ( insert_state @ X3 @ B3 ) )
      = ( ( ( member_state @ X3 @ B3 )
         => ( ord_less_set_state @ A4 @ B3 ) )
        & ( ~ ( member_state @ X3 @ B3 )
         => ( ( ( member_state @ X3 @ A4 )
             => ( ord_less_set_state @ ( minus_3933957440811877961_state @ A4 @ ( insert_state @ X3 @ bot_bot_set_state ) ) @ B3 ) )
            & ( ~ ( member_state @ X3 @ A4 )
             => ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1090_card__psubset,axiom,
    ! [B3: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A4 @ B3 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) )
         => ( ord_less_set_nat @ A4 @ B3 ) ) ) ) ).

% card_psubset
thf(fact_1091_card__psubset,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( finite_finite_state @ B3 )
     => ( ( ord_le2494988322063910608_state @ A4 @ B3 )
       => ( ( ord_less_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) )
         => ( ord_less_set_state @ A4 @ B3 ) ) ) ) ).

% card_psubset
thf(fact_1092_card__gt__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_nat )
        & ( finite_finite_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_1093_card__gt__0__iff,axiom,
    ! [A4: set_state] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_state @ A4 ) )
      = ( ( A4 != bot_bot_set_state )
        & ( finite_finite_state @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_1094_pinf_I6_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ~ ( ord_less_eq_nat @ X @ T4 ) ) ).

% pinf(6)
thf(fact_1095_pinf_I8_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ( ord_less_eq_nat @ T4 @ X ) ) ).

% pinf(8)
thf(fact_1096_minf_I6_J,axiom,
    ! [T4: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ( ord_less_eq_nat @ X @ T4 ) ) ).

% minf(6)
thf(fact_1097_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1098_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_1099_psubsetD,axiom,
    ! [A4: set_state,B3: set_state,C: state] :
      ( ( ord_less_set_state @ A4 @ B3 )
     => ( ( member_state @ C @ A4 )
       => ( member_state @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1100_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_1101_Mask_Onull__def,axiom,
    null_nat = zero_zero_nat ).

% Mask.null_def
thf(fact_1102_verit__comp__simplify1_I3_J,axiom,
    ! [B10: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B10 @ A6 ) )
      = ( ord_less_nat @ A6 @ B10 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1103_verit__la__disequality,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = B2 )
      | ~ ( ord_less_eq_nat @ A3 @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ).

% verit_la_disequality
thf(fact_1104_verit__comp__simplify1_I2_J,axiom,
    ! [A3: set_state] : ( ord_le2494988322063910608_state @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_1105_verit__comp__simplify1_I2_J,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_1106_complete__interval,axiom,
    ! [A3: nat,B2: nat,P: nat > $o] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( P @ A3 )
       => ( ~ ( P @ B2 )
         => ? [C6: nat] :
              ( ( ord_less_eq_nat @ A3 @ C6 )
              & ( ord_less_eq_nat @ C6 @ B2 )
              & ! [X: nat] :
                  ( ( ( ord_less_eq_nat @ A3 @ X )
                    & ( ord_less_nat @ X @ C6 ) )
                 => ( P @ X ) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A3 @ X4 )
                        & ( ord_less_nat @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D3 @ C6 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1107_Int__iff,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( inf_inf_set_state @ A4 @ B3 ) )
      = ( ( member_state @ C @ A4 )
        & ( member_state @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_1108_IntI,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ A4 )
     => ( ( member_state @ C @ B3 )
       => ( member_state @ C @ ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% IntI
thf(fact_1109_le__inf__iff,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ ( inf_inf_set_state @ Y2 @ Z ) )
      = ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
        & ( ord_le2494988322063910608_state @ X3 @ Z ) ) ) ).

% le_inf_iff
thf(fact_1110_le__inf__iff,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) )
      = ( ( ord_less_eq_nat @ X3 @ Y2 )
        & ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% le_inf_iff
thf(fact_1111_inf_Obounded__iff,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ ( inf_inf_set_state @ B2 @ C ) )
      = ( ( ord_le2494988322063910608_state @ A3 @ B2 )
        & ( ord_le2494988322063910608_state @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1112_inf_Obounded__iff,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) )
      = ( ( ord_less_eq_nat @ A3 @ B2 )
        & ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_1113_inf__bot__left,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ bot_bot_set_state @ X3 )
      = bot_bot_set_state ) ).

% inf_bot_left
thf(fact_1114_inf__bot__right,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ X3 @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% inf_bot_right
thf(fact_1115_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ X3 @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% boolean_algebra.conj_zero_right
thf(fact_1116_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ bot_bot_set_state @ X3 )
      = bot_bot_set_state ) ).

% boolean_algebra.conj_zero_left
thf(fact_1117_Int__UNIV,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ A4 @ B3 )
        = top_top_set_state )
      = ( ( A4 = top_top_set_state )
        & ( B3 = top_top_set_state ) ) ) ).

% Int_UNIV
thf(fact_1118_Int__subset__iff,axiom,
    ! [C2: set_state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ ( inf_inf_set_state @ A4 @ B3 ) )
      = ( ( ord_le2494988322063910608_state @ C2 @ A4 )
        & ( ord_le2494988322063910608_state @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1119_Int__insert__right__if1,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( member_state @ A3 @ A4 )
     => ( ( inf_inf_set_state @ A4 @ ( insert_state @ A3 @ B3 ) )
        = ( insert_state @ A3 @ ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1120_Int__insert__right__if0,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ~ ( member_state @ A3 @ A4 )
     => ( ( inf_inf_set_state @ A4 @ ( insert_state @ A3 @ B3 ) )
        = ( inf_inf_set_state @ A4 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1121_insert__inter__insert,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( inf_inf_set_state @ ( insert_state @ A3 @ A4 ) @ ( insert_state @ A3 @ B3 ) )
      = ( insert_state @ A3 @ ( inf_inf_set_state @ A4 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_1122_Int__insert__left__if1,axiom,
    ! [A3: state,C2: set_state,B3: set_state] :
      ( ( member_state @ A3 @ C2 )
     => ( ( inf_inf_set_state @ ( insert_state @ A3 @ B3 ) @ C2 )
        = ( insert_state @ A3 @ ( inf_inf_set_state @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1123_Int__insert__left__if0,axiom,
    ! [A3: state,C2: set_state,B3: set_state] :
      ( ~ ( member_state @ A3 @ C2 )
     => ( ( inf_inf_set_state @ ( insert_state @ A3 @ B3 ) @ C2 )
        = ( inf_inf_set_state @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1124_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ X3 @ ( uminus472742206872269241_state @ X3 ) )
      = bot_bot_set_state ) ).

% boolean_algebra.conj_cancel_right
thf(fact_1125_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_state] :
      ( ( inf_inf_set_state @ ( uminus472742206872269241_state @ X3 ) @ X3 )
      = bot_bot_set_state ) ).

% boolean_algebra.conj_cancel_left
thf(fact_1126_inf__compl__bot__right,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( inf_inf_set_state @ X3 @ ( inf_inf_set_state @ Y2 @ ( uminus472742206872269241_state @ X3 ) ) )
      = bot_bot_set_state ) ).

% inf_compl_bot_right
thf(fact_1127_inf__compl__bot__left2,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( inf_inf_set_state @ X3 @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X3 ) @ Y2 ) )
      = bot_bot_set_state ) ).

% inf_compl_bot_left2
thf(fact_1128_inf__compl__bot__left1,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( inf_inf_set_state @ ( uminus472742206872269241_state @ X3 ) @ ( inf_inf_set_state @ X3 @ Y2 ) )
      = bot_bot_set_state ) ).

% inf_compl_bot_left1
thf(fact_1129_insert__disjoint_I1_J,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ ( insert_state @ A3 @ A4 ) @ B3 )
        = bot_bot_set_state )
      = ( ~ ( member_state @ A3 @ B3 )
        & ( ( inf_inf_set_state @ A4 @ B3 )
          = bot_bot_set_state ) ) ) ).

% insert_disjoint(1)
thf(fact_1130_insert__disjoint_I2_J,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( bot_bot_set_state
        = ( inf_inf_set_state @ ( insert_state @ A3 @ A4 ) @ B3 ) )
      = ( ~ ( member_state @ A3 @ B3 )
        & ( bot_bot_set_state
          = ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1131_disjoint__insert_I1_J,axiom,
    ! [B3: set_state,A3: state,A4: set_state] :
      ( ( ( inf_inf_set_state @ B3 @ ( insert_state @ A3 @ A4 ) )
        = bot_bot_set_state )
      = ( ~ ( member_state @ A3 @ B3 )
        & ( ( inf_inf_set_state @ B3 @ A4 )
          = bot_bot_set_state ) ) ) ).

% disjoint_insert(1)
thf(fact_1132_disjoint__insert_I2_J,axiom,
    ! [A4: set_state,B2: state,B3: set_state] :
      ( ( bot_bot_set_state
        = ( inf_inf_set_state @ A4 @ ( insert_state @ B2 @ B3 ) ) )
      = ( ~ ( member_state @ B2 @ A4 )
        & ( bot_bot_set_state
          = ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1133_Diff__disjoint,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( inf_inf_set_state @ A4 @ ( minus_3933957440811877961_state @ B3 @ A4 ) )
      = bot_bot_set_state ) ).

% Diff_disjoint
thf(fact_1134_Compl__disjoint,axiom,
    ! [A4: set_state] :
      ( ( inf_inf_set_state @ A4 @ ( uminus472742206872269241_state @ A4 ) )
      = bot_bot_set_state ) ).

% Compl_disjoint
thf(fact_1135_Compl__disjoint2,axiom,
    ! [A4: set_state] :
      ( ( inf_inf_set_state @ ( uminus472742206872269241_state @ A4 ) @ A4 )
      = bot_bot_set_state ) ).

% Compl_disjoint2
thf(fact_1136_image__vimage__eq,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( image_state_state @ F @ ( vimage_state_state @ F @ A4 ) )
      = ( inf_inf_set_state @ A4 @ ( image_state_state @ F @ top_top_set_state ) ) ) ).

% image_vimage_eq
thf(fact_1137_Un__Int__assoc__eq,axiom,
    ! [A4: set_state,B3: set_state,C2: set_state] :
      ( ( ( sup_sup_set_state @ ( inf_inf_set_state @ A4 @ B3 ) @ C2 )
        = ( inf_inf_set_state @ A4 @ ( sup_sup_set_state @ B3 @ C2 ) ) )
      = ( ord_le2494988322063910608_state @ C2 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_1138_distrib__inf__le,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] : ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ ( inf_inf_set_state @ X3 @ Z ) ) @ ( inf_inf_set_state @ X3 @ ( sup_sup_set_state @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1139_distrib__inf__le,axiom,
    ! [X3: nat,Y2: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( inf_inf_nat @ X3 @ Z ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y2 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1140_distrib__sup__le,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] : ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ X3 @ ( inf_inf_set_state @ Y2 @ Z ) ) @ ( inf_inf_set_state @ ( sup_sup_set_state @ X3 @ Y2 ) @ ( sup_sup_set_state @ X3 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1141_distrib__sup__le,axiom,
    ! [X3: nat,Y2: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( sup_sup_nat @ X3 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1142_Int__insert__right,axiom,
    ! [A3: state,A4: set_state,B3: set_state] :
      ( ( ( member_state @ A3 @ A4 )
       => ( ( inf_inf_set_state @ A4 @ ( insert_state @ A3 @ B3 ) )
          = ( insert_state @ A3 @ ( inf_inf_set_state @ A4 @ B3 ) ) ) )
      & ( ~ ( member_state @ A3 @ A4 )
       => ( ( inf_inf_set_state @ A4 @ ( insert_state @ A3 @ B3 ) )
          = ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_1143_Int__insert__left,axiom,
    ! [A3: state,C2: set_state,B3: set_state] :
      ( ( ( member_state @ A3 @ C2 )
       => ( ( inf_inf_set_state @ ( insert_state @ A3 @ B3 ) @ C2 )
          = ( insert_state @ A3 @ ( inf_inf_set_state @ B3 @ C2 ) ) ) )
      & ( ~ ( member_state @ A3 @ C2 )
       => ( ( inf_inf_set_state @ ( insert_state @ A3 @ B3 ) @ C2 )
          = ( inf_inf_set_state @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1144_IntD2,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( inf_inf_set_state @ A4 @ B3 ) )
     => ( member_state @ C @ B3 ) ) ).

% IntD2
thf(fact_1145_IntD1,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( inf_inf_set_state @ A4 @ B3 ) )
     => ( member_state @ C @ A4 ) ) ).

% IntD1
thf(fact_1146_IntE,axiom,
    ! [C: state,A4: set_state,B3: set_state] :
      ( ( member_state @ C @ ( inf_inf_set_state @ A4 @ B3 ) )
     => ~ ( ( member_state @ C @ A4 )
         => ~ ( member_state @ C @ B3 ) ) ) ).

% IntE
thf(fact_1147_Int__mono,axiom,
    ! [A4: set_state,C2: set_state,B3: set_state,D2: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ C2 )
     => ( ( ord_le2494988322063910608_state @ B3 @ D2 )
       => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A4 @ B3 ) @ ( inf_inf_set_state @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_1148_Int__lower1,axiom,
    ! [A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A4 @ B3 ) @ A4 ) ).

% Int_lower1
thf(fact_1149_Int__lower2,axiom,
    ! [A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A4 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1150_Int__absorb1,axiom,
    ! [B3: set_state,A4: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A4 )
     => ( ( inf_inf_set_state @ A4 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1151_Int__absorb2,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ( inf_inf_set_state @ A4 @ B3 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_1152_Int__greatest,axiom,
    ! [C2: set_state,A4: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ A4 )
     => ( ( ord_le2494988322063910608_state @ C2 @ B3 )
       => ( ord_le2494988322063910608_state @ C2 @ ( inf_inf_set_state @ A4 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1153_Int__Collect__mono,axiom,
    ! [A4: set_state,B3: set_state,P: state > $o,Q: state > $o] :
      ( ( ord_le2494988322063910608_state @ A4 @ B3 )
     => ( ! [X4: state] :
            ( ( member_state @ X4 @ A4 )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A4 @ ( collect_state @ P ) ) @ ( inf_inf_set_state @ B3 @ ( collect_state @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1154_Int__UNIV__right,axiom,
    ! [A4: set_state] :
      ( ( inf_inf_set_state @ A4 @ top_top_set_state )
      = A4 ) ).

% Int_UNIV_right
thf(fact_1155_Int__UNIV__left,axiom,
    ! [B3: set_state] :
      ( ( inf_inf_set_state @ top_top_set_state @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_1156_inf_OcoboundedI2,axiom,
    ! [B2: set_state,C: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ C )
     => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1157_inf_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1158_inf_OcoboundedI1,axiom,
    ! [A3: set_state,C: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ C )
     => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1159_inf_OcoboundedI1,axiom,
    ! [A3: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1160_inf_Oabsorb__iff2,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B: set_state,A2: set_state] :
          ( ( inf_inf_set_state @ A2 @ B )
          = B ) ) ) ).

% inf.absorb_iff2
thf(fact_1161_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B: nat,A2: nat] :
          ( ( inf_inf_nat @ A2 @ B )
          = B ) ) ) ).

% inf.absorb_iff2
thf(fact_1162_inf_Oabsorb__iff1,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A2: set_state,B: set_state] :
          ( ( inf_inf_set_state @ A2 @ B )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1163_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B: nat] :
          ( ( inf_inf_nat @ A2 @ B )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1164_inf_Ocobounded2,axiom,
    ! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1165_inf_Ocobounded2,axiom,
    ! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1166_inf_Ocobounded1,axiom,
    ! [A3: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ A3 ) ).

% inf.cobounded1
thf(fact_1167_inf_Ocobounded1,axiom,
    ! [A3: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ A3 ) ).

% inf.cobounded1
thf(fact_1168_inf_Oorder__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A2: set_state,B: set_state] :
          ( A2
          = ( inf_inf_set_state @ A2 @ B ) ) ) ) ).

% inf.order_iff
thf(fact_1169_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B: nat] :
          ( A2
          = ( inf_inf_nat @ A2 @ B ) ) ) ) ).

% inf.order_iff
thf(fact_1170_inf__greatest,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( ord_le2494988322063910608_state @ X3 @ Z )
       => ( ord_le2494988322063910608_state @ X3 @ ( inf_inf_set_state @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1171_inf__greatest,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ X3 @ Z )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1172_inf_OboundedI,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( ord_le2494988322063910608_state @ A3 @ C )
       => ( ord_le2494988322063910608_state @ A3 @ ( inf_inf_set_state @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1173_inf_OboundedI,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ A3 @ C )
       => ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1174_inf_OboundedE,axiom,
    ! [A3: set_state,B2: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ ( inf_inf_set_state @ B2 @ C ) )
     => ~ ( ( ord_le2494988322063910608_state @ A3 @ B2 )
         => ~ ( ord_le2494988322063910608_state @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_1175_inf_OboundedE,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A3 @ B2 )
         => ~ ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_1176_inf__absorb2,axiom,
    ! [Y2: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y2 @ X3 )
     => ( ( inf_inf_set_state @ X3 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_1177_inf__absorb2,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( inf_inf_nat @ X3 @ Y2 )
        = Y2 ) ) ).

% inf_absorb2
thf(fact_1178_inf__absorb1,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y2 )
     => ( ( inf_inf_set_state @ X3 @ Y2 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1179_inf__absorb1,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( inf_inf_nat @ X3 @ Y2 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_1180_inf_Oabsorb2,axiom,
    ! [B2: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A3 )
     => ( ( inf_inf_set_state @ A3 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1181_inf_Oabsorb2,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( inf_inf_nat @ A3 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1182_inf_Oabsorb1,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( ( inf_inf_set_state @ A3 @ B2 )
        = A3 ) ) ).

% inf.absorb1
thf(fact_1183_inf_Oabsorb1,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( inf_inf_nat @ A3 @ B2 )
        = A3 ) ) ).

% inf.absorb1
thf(fact_1184_le__iff__inf,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [X2: set_state,Y: set_state] :
          ( ( inf_inf_set_state @ X2 @ Y )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1185_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y: nat] :
          ( ( inf_inf_nat @ X2 @ Y )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1186_inf__unique,axiom,
    ! [F: set_state > set_state > set_state,X3: set_state,Y2: set_state] :
      ( ! [X4: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: set_state,Y3: set_state,Z3: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y3 )
             => ( ( ord_le2494988322063910608_state @ X4 @ Z3 )
               => ( ord_le2494988322063910608_state @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_set_state @ X3 @ Y2 )
            = ( F @ X3 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_1187_inf__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y2: nat] :
      ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ( ord_less_eq_nat @ X4 @ Z3 )
               => ( ord_less_eq_nat @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X3 @ Y2 )
            = ( F @ X3 @ Y2 ) ) ) ) ) ).

% inf_unique
thf(fact_1188_inf_OorderI,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( A3
        = ( inf_inf_set_state @ A3 @ B2 ) )
     => ( ord_le2494988322063910608_state @ A3 @ B2 ) ) ).

% inf.orderI
thf(fact_1189_inf_OorderI,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3
        = ( inf_inf_nat @ A3 @ B2 ) )
     => ( ord_less_eq_nat @ A3 @ B2 ) ) ).

% inf.orderI
thf(fact_1190_inf_OorderE,axiom,
    ! [A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ B2 )
     => ( A3
        = ( inf_inf_set_state @ A3 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1191_inf_OorderE,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( A3
        = ( inf_inf_nat @ A3 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1192_le__infI2,axiom,
    ! [B2: set_state,X3: set_state,A3: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ X3 )
     => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ X3 ) ) ).

% le_infI2
thf(fact_1193_le__infI2,axiom,
    ! [B2: nat,X3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X3 ) ) ).

% le_infI2
thf(fact_1194_le__infI1,axiom,
    ! [A3: set_state,X3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ X3 )
     => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ X3 ) ) ).

% le_infI1
thf(fact_1195_le__infI1,axiom,
    ! [A3: nat,X3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ X3 ) ) ).

% le_infI1
thf(fact_1196_inf__mono,axiom,
    ! [A3: set_state,C: set_state,B2: set_state,D: set_state] :
      ( ( ord_le2494988322063910608_state @ A3 @ C )
     => ( ( ord_le2494988322063910608_state @ B2 @ D )
       => ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ A3 @ B2 ) @ ( inf_inf_set_state @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_1197_inf__mono,axiom,
    ! [A3: nat,C: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_1198_le__infI,axiom,
    ! [X3: set_state,A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ A3 )
     => ( ( ord_le2494988322063910608_state @ X3 @ B2 )
       => ( ord_le2494988322063910608_state @ X3 @ ( inf_inf_set_state @ A3 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1199_le__infI,axiom,
    ! [X3: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X3 @ A3 )
     => ( ( ord_less_eq_nat @ X3 @ B2 )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A3 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1200_le__infE,axiom,
    ! [X3: set_state,A3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ ( inf_inf_set_state @ A3 @ B2 ) )
     => ~ ( ( ord_le2494988322063910608_state @ X3 @ A3 )
         => ~ ( ord_le2494988322063910608_state @ X3 @ B2 ) ) ) ).

% le_infE
thf(fact_1201_le__infE,axiom,
    ! [X3: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A3 @ B2 ) )
     => ~ ( ( ord_less_eq_nat @ X3 @ A3 )
         => ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).

% le_infE
thf(fact_1202_inf__le2,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_1203_inf__le2,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_le2
thf(fact_1204_inf__le1,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ X3 ) ).

% inf_le1
thf(fact_1205_inf__le1,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_le1
thf(fact_1206_inf__sup__ord_I1_J,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1207_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_1208_inf__sup__ord_I2_J,axiom,
    ! [X3: set_state,Y2: set_state] : ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_1209_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).

% inf_sup_ord(2)
thf(fact_1210_image__Int__subset,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( inf_inf_set_state @ A4 @ B3 ) ) @ ( inf_inf_set_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_1211_disjoint__iff__not__equal,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ A4 @ B3 )
        = bot_bot_set_state )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ! [Y: state] :
                ( ( member_state @ Y @ B3 )
               => ( X2 != Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1212_Int__empty__right,axiom,
    ! [A4: set_state] :
      ( ( inf_inf_set_state @ A4 @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% Int_empty_right
thf(fact_1213_Int__empty__left,axiom,
    ! [B3: set_state] :
      ( ( inf_inf_set_state @ bot_bot_set_state @ B3 )
      = bot_bot_set_state ) ).

% Int_empty_left
thf(fact_1214_disjoint__iff,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ A4 @ B3 )
        = bot_bot_set_state )
      = ( ! [X2: state] :
            ( ( member_state @ X2 @ A4 )
           => ~ ( member_state @ X2 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1215_Int__emptyI,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A4 )
         => ~ ( member_state @ X4 @ B3 ) )
     => ( ( inf_inf_set_state @ A4 @ B3 )
        = bot_bot_set_state ) ) ).

% Int_emptyI
thf(fact_1216_inf__cancel__left2,axiom,
    ! [X3: set_state,A3: set_state,B2: set_state] :
      ( ( inf_inf_set_state @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X3 ) @ A3 ) @ ( inf_inf_set_state @ X3 @ B2 ) )
      = bot_bot_set_state ) ).

% inf_cancel_left2
thf(fact_1217_inf__cancel__left1,axiom,
    ! [X3: set_state,A3: set_state,B2: set_state] :
      ( ( inf_inf_set_state @ ( inf_inf_set_state @ X3 @ A3 ) @ ( inf_inf_set_state @ ( uminus472742206872269241_state @ X3 ) @ B2 ) )
      = bot_bot_set_state ) ).

% inf_cancel_left1
thf(fact_1218_Diff__triv,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ A4 @ B3 )
        = bot_bot_set_state )
     => ( ( minus_3933957440811877961_state @ A4 @ B3 )
        = A4 ) ) ).

% Diff_triv
thf(fact_1219_Int__Diff__disjoint,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( inf_inf_set_state @ ( inf_inf_set_state @ A4 @ B3 ) @ ( minus_3933957440811877961_state @ A4 @ B3 ) )
      = bot_bot_set_state ) ).

% Int_Diff_disjoint
thf(fact_1220_inf__shunt,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ( inf_inf_set_state @ X3 @ Y2 )
        = bot_bot_set_state )
      = ( ord_le2494988322063910608_state @ X3 @ ( uminus472742206872269241_state @ Y2 ) ) ) ).

% inf_shunt
thf(fact_1221_sup__neg__inf,axiom,
    ! [P3: set_state,Q2: set_state,R2: set_state] :
      ( ( ord_le2494988322063910608_state @ P3 @ ( sup_sup_set_state @ Q2 @ R2 ) )
      = ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ P3 @ ( uminus472742206872269241_state @ Q2 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1222_shunt2,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ ( uminus472742206872269241_state @ Y2 ) ) @ Z )
      = ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ Y2 @ Z ) ) ) ).

% shunt2
thf(fact_1223_shunt1,axiom,
    ! [X3: set_state,Y2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ ( inf_inf_set_state @ X3 @ Y2 ) @ Z )
      = ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ ( uminus472742206872269241_state @ Y2 ) @ Z ) ) ) ).

% shunt1
thf(fact_1224_boolean__algebra_Ocomplement__unique,axiom,
    ! [A3: set_state,X3: set_state,Y2: set_state] :
      ( ( ( inf_inf_set_state @ A3 @ X3 )
        = bot_bot_set_state )
     => ( ( ( sup_sup_set_state @ A3 @ X3 )
          = top_top_set_state )
       => ( ( ( inf_inf_set_state @ A3 @ Y2 )
            = bot_bot_set_state )
         => ( ( ( sup_sup_set_state @ A3 @ Y2 )
              = top_top_set_state )
           => ( X3 = Y2 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1225_inj__on__image__Int,axiom,
    ! [F: state > state,C2: set_state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ C2 )
     => ( ( ord_le2494988322063910608_state @ A4 @ C2 )
       => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
         => ( ( image_state_state @ F @ ( inf_inf_set_state @ A4 @ B3 ) )
            = ( inf_inf_set_state @ ( image_state_state @ F @ A4 ) @ ( image_state_state @ F @ B3 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_1226_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( ( inf_inf_set_state @ A4 @ B3 )
        = bot_bot_set_state )
      = ( ord_le2494988322063910608_state @ A4 @ ( uminus472742206872269241_state @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_1227_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X3: set_state,Y2: set_state] :
      ( ( ( inf_inf_set_state @ X3 @ Y2 )
        = bot_bot_set_state )
     => ( ( ( sup_sup_set_state @ X3 @ Y2 )
          = top_top_set_state )
       => ( ( uminus472742206872269241_state @ X3 )
          = Y2 ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1228_finite__finite__vimage__IntI,axiom,
    ! [F4: set_nat,H: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ F4 )
           => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ A4 ) ) )
       => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H @ F4 ) @ A4 ) ) ) ) ).

% finite_finite_vimage_IntI
thf(fact_1229_finite__finite__vimage__IntI,axiom,
    ! [F4: set_state,H: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ F4 )
     => ( ! [Y3: state] :
            ( ( member_state @ Y3 @ F4 )
           => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_state @ H @ ( insert_state @ Y3 @ bot_bot_set_state ) ) @ A4 ) ) )
       => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_state @ H @ F4 ) @ A4 ) ) ) ) ).

% finite_finite_vimage_IntI
thf(fact_1230_inj__on__Un,axiom,
    ! [F: state > state,A4: set_state,B3: set_state] :
      ( ( inj_on_state_state @ F @ ( sup_sup_set_state @ A4 @ B3 ) )
      = ( ( inj_on_state_state @ F @ A4 )
        & ( inj_on_state_state @ F @ B3 )
        & ( ( inf_inf_set_state @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ A4 @ B3 ) ) @ ( image_state_state @ F @ ( minus_3933957440811877961_state @ B3 @ A4 ) ) )
          = bot_bot_set_state ) ) ) ).

% inj_on_Un
thf(fact_1231_inf__img__fin__dom_H,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A4 ) )
            & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_dom'
thf(fact_1232_inf__img__fin__dom_H,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( finite_finite_state @ ( image_state_state @ F @ A4 ) )
     => ( ~ ( finite_finite_state @ A4 )
       => ? [X4: state] :
            ( ( member_state @ X4 @ ( image_state_state @ F @ A4 ) )
            & ~ ( finite_finite_state @ ( inf_inf_set_state @ ( vimage_state_state @ F @ ( insert_state @ X4 @ bot_bot_set_state ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_dom'
thf(fact_1233_inf__img__fin__dom_H,axiom,
    ! [F: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ ( image_nat_state @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ? [X4: state] :
            ( ( member_state @ X4 @ ( image_nat_state @ F @ A4 ) )
            & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_state @ F @ ( insert_state @ X4 @ bot_bot_set_state ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_dom'
thf(fact_1234_inf__img__fin__domE_H,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ~ ! [Y3: nat] :
              ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A4 ) )
             => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_domE'
thf(fact_1235_inf__img__fin__domE_H,axiom,
    ! [F: state > state,A4: set_state] :
      ( ( finite_finite_state @ ( image_state_state @ F @ A4 ) )
     => ( ~ ( finite_finite_state @ A4 )
       => ~ ! [Y3: state] :
              ( ( member_state @ Y3 @ ( image_state_state @ F @ A4 ) )
             => ( finite_finite_state @ ( inf_inf_set_state @ ( vimage_state_state @ F @ ( insert_state @ Y3 @ bot_bot_set_state ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_domE'
thf(fact_1236_inf__img__fin__domE_H,axiom,
    ! [F: nat > state,A4: set_nat] :
      ( ( finite_finite_state @ ( image_nat_state @ F @ A4 ) )
     => ( ~ ( finite_finite_nat @ A4 )
       => ~ ! [Y3: state] :
              ( ( member_state @ Y3 @ ( image_nat_state @ F @ A4 ) )
             => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_state @ F @ ( insert_state @ Y3 @ bot_bot_set_state ) ) @ A4 ) ) ) ) ) ).

% inf_img_fin_domE'
thf(fact_1237_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila3858001251800985922_state @ inf_inf_set_state @ top_top_set_state @ ord_le2494988322063910608_state @ ord_less_set_state ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_1238_map__add__upd__left,axiom,
    ! [M2: state,E2: state > option_state,E1: state > option_state,U1: state] :
      ( ~ ( member_state @ M2 @ ( dom_state_state @ E2 ) )
     => ( ( map_add_state_state @ ( fun_up8843634000204221123_state @ E1 @ M2 @ ( some_state @ U1 ) ) @ E2 )
        = ( fun_up8843634000204221123_state @ ( map_add_state_state @ E1 @ E2 ) @ M2 @ ( some_state @ U1 ) ) ) ) ).

% map_add_upd_left
thf(fact_1239_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5298875108296682874_state @ inf_inf_set_state @ sup_sup_set_state @ uminus472742206872269241_state @ bot_bot_set_state @ top_top_set_state ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1240_Inf__fin_Oinsert__remove,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A4 ) )
            = X3 ) )
        & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A4 ) )
            = ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_1241_Inf__fin_Oinsert,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A4 ) )
          = ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1242_Inf__fin_OcoboundedI,axiom,
    ! [A4: set_set_state,A3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( member_set_state @ A3 @ A4 )
       => ( ord_le2494988322063910608_state @ ( lattic4879230916095660051_state @ A4 ) @ A3 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_1243_Inf__fin_OcoboundedI,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A3 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_1244_Inf__fin_Obounded__iff,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( ord_le2494988322063910608_state @ X3 @ ( lattic4879230916095660051_state @ A4 ) )
          = ( ! [X2: set_state] :
                ( ( member_set_state @ X2 @ A4 )
               => ( ord_le2494988322063910608_state @ X3 @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_1245_Inf__fin_Obounded__iff,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A4 ) )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A4 )
               => ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_1246_Inf__fin_OboundedI,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ! [A5: set_state] :
              ( ( member_set_state @ A5 @ A4 )
             => ( ord_le2494988322063910608_state @ X3 @ A5 ) )
         => ( ord_le2494988322063910608_state @ X3 @ ( lattic4879230916095660051_state @ A4 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_1247_Inf__fin_OboundedI,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A4 )
             => ( ord_less_eq_nat @ X3 @ A5 ) )
         => ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_1248_Inf__fin_OboundedE,axiom,
    ! [A4: set_set_state,X3: set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( ord_le2494988322063910608_state @ X3 @ ( lattic4879230916095660051_state @ A4 ) )
         => ! [A10: set_state] :
              ( ( member_set_state @ A10 @ A4 )
             => ( ord_le2494988322063910608_state @ X3 @ A10 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_1249_Inf__fin_OboundedE,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A4 ) )
         => ! [A10: nat] :
              ( ( member_nat @ A10 @ A4 )
             => ( ord_less_eq_nat @ X3 @ A10 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_1250_Inf__fin_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( lattic5238388535129920115in_nat @ A4 )
        = ( the_nat @ none_nat ) ) ) ).

% Inf_fin.infinite
thf(fact_1251_Inf__fin_Osubset__imp,axiom,
    ! [A4: set_set_state,B3: set_set_state] :
      ( ( ord_le5175021213330142598_state @ A4 @ B3 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ( finite4951987536711252743_state @ B3 )
         => ( ord_le2494988322063910608_state @ ( lattic4879230916095660051_state @ B3 ) @ ( lattic4879230916095660051_state @ A4 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_1252_Inf__fin_Osubset__imp,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B3 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_1253_Inf__fin_Ohom__commute,axiom,
    ! [H: nat > nat,N: set_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( H @ ( inf_inf_nat @ X4 @ Y3 ) )
          = ( inf_inf_nat @ ( H @ X4 ) @ ( H @ Y3 ) ) )
     => ( ( finite_finite_nat @ N )
       => ( ( N != bot_bot_set_nat )
         => ( ( H @ ( lattic5238388535129920115in_nat @ N ) )
            = ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_1254_Inf__fin_Osubset,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A4 )
         => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A4 ) )
            = ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1255_Inf__fin_Oclosed,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X4: nat,Y3: nat] : ( member_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ ( insert_nat @ X4 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A4 ) ) ) ) ).

% Inf_fin.closed
thf(fact_1256_Inf__fin_Oinsert__not__elem,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X3 @ A4 )
       => ( ( A4 != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A4 ) )
            = ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_1257_Inf__fin_Ounion,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A4 @ B3 ) )
              = ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic5238388535129920115in_nat @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1258_Inf__fin__le__Sup__fin,axiom,
    ! [A4: set_set_state] :
      ( ( finite4951987536711252743_state @ A4 )
     => ( ( A4 != bot_bo2271482359692755898_state )
       => ( ord_le2494988322063910608_state @ ( lattic4879230916095660051_state @ A4 ) @ ( lattic1454283544731368441_state @ A4 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1259_Inf__fin__le__Sup__fin,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1260_Inf__fin_Oremove,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A4 )
              = X3 ) )
          & ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A4 )
              = ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_1261_card__Un__disjoint,axiom,
    ! [A4: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A4 @ B3 )
            = bot_bot_set_nat )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B3 ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_1262_card__Un__disjoint,axiom,
    ! [A4: set_state,B3: set_state] :
      ( ( finite_finite_state @ A4 )
     => ( ( finite_finite_state @ B3 )
       => ( ( ( inf_inf_set_state @ A4 @ B3 )
            = bot_bot_set_state )
         => ( ( finite_card_state @ ( sup_sup_set_state @ A4 @ B3 ) )
            = ( plus_plus_nat @ ( finite_card_state @ A4 ) @ ( finite_card_state @ B3 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_1263_add__le__cancel__left,axiom,
    ! [C: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A3 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_1264_add__le__cancel__right,axiom,
    ! [A3: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A3 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_1265_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1266_le__add__same__cancel2,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1267_le__add__same__cancel1,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1268_add__le__same__cancel2,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1269_add__le__same__cancel1,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1270_le__add__diff__inverse,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1271_le__add__diff__inverse2,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ B2 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1272_image__add__0,axiom,
    ! [S: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
      = S ) ).

% image_add_0
thf(fact_1273_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_1274_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_1275_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

% Conjectures (1)
thf(conj_0,conjecture,
    ? [X: state] :
      ( ( member_state @ X @ a )
      & ( greater @ x @ ( multiply @ p @ X ) ) ) ).

%------------------------------------------------------------------------------