TPTP Problem File: SLH0618^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Combinable_Wands/0003_CombinableWands/prob_00654_031969__8463936_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1782 ( 563 unt; 501 typ;   0 def)
%            Number of atoms       : 4566 (1755 equ;   0 cnn)
%            Maximal formula atoms :   23 (   3 avg)
%            Number of connectives : 14934 ( 452   ~;  67   |; 639   &;11952   @)
%                                         (   0 <=>;1824  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   7 avg)
%            Number of types       :   76 (  75 usr)
%            Number of type conns  : 2979 (2979   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  429 ( 426 usr;  36 con; 0-11 aty)
%            Number of variables   : 5117 ( 625   ^;4132   !; 360   ?;5117   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:09:25.985
%------------------------------------------------------------------------------
% Could-be-implicit typings (75)
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
    set_Pr4846723933414538759_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    set_Pr7649655576376376530on_val: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    produc9003781588162623004on_val: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J_J,type,
    set_Pr523392378426063460_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
    produc4357056439811976324_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J_J,type,
    set_Pr401513967981718343_state: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J_J,type,
    set_op8023886964038656367_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
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thf(ty_n_t__PackageLogic__Oaassertion_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
    option2747342629145793465_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr5323702258453113287_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
    set_Pr4231240917283394591_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_Mt__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_J,type,
    produc4324712645871441639_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    produc7272280185086996969_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr5527899070135025255_state: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J_J,type,
    set_op3387789762728285652_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    produc948624807956558599_state: $tType ).

thf(ty_n_t__PackageLogic__Oaassertion_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    aasser3179708585289803787_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr1688445902015331925_state: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    option886743748074664436_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr2391701600042252046_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__PartialHeapSA__Ostate_J_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr5618112136499017889_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    produc3142500478612311029_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    produc3882902684156697774_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__PartialHeapSA__Ostate_J_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    produc597591266584407361_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PackageLogic__Oaassertion_It__PartialHeapSA__Ostate_J_Mt__PartialHeapSA__Ostate_J,type,
    produc2756261550280135502_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J_J,type,
    set_Pr6847263425858009217_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    set_Pr1785066336555260981_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J_J,type,
    set_Pr6039890582625207499_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    produc6251377953716854561_state: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PosRat__Oprat_J_J,type,
    set_Pr3298805672376727137t_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    produc8023240190789890773_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    produc1457211279475724562t_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J,type,
    produc4844402304972151915_state: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PosRat__Oprat_J,type,
    produc2235545006120619777t_prat: $tType ).

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

thf(ty_n_t__PackageLogic__Oaassertion_It__PartialHeapSA__Ostate_J,type,
    aassertion_state: $tType ).

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

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

thf(ty_n_t__Option__Ooption_It__Mask__Oval_J,type,
    option_val: $tType ).

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

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

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

thf(ty_n_t__Mask__Oval,type,
    val: $tType ).

% Explicit typings (426)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    bNF_Ca2815325664443363616_state: set_Pr1688445902015331925_state > ( option_state > set_state ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    bNF_Ca2927435689774128306_state: set_Pr4305388979797279737on_val > ( produc9003781588162623004on_val > set_state ) > $o ).

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

thf(sy_c_CombinableWands_Obinary,type,
    binary: set_state > $o ).

thf(sy_c_CombinableWands_Obinary__mask,type,
    binary_mask: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ).

thf(sy_c_CombinableWands_Ocombinable,type,
    combinable: set_state > $o ).

thf(sy_c_CombinableWands_Ocomp__min__mask,type,
    comp_min_mask: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ).

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__rel,type,
    multiply_rel: produc4844402304972151915_state > produc4844402304972151915_state > $o ).

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

thf(sy_c_CombinableWands_Omultiply__sem__assertion__rel,type,
    multip8073008153270241703on_rel: produc6251377953716854561_state > produc6251377953716854561_state > $o ).

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_Fun_Oid_001t__PartialHeapSA__Ostate,type,
    id_state: state > state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    id_Pro1402184692240358763on_val: produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    id_Pro8715877757508063718_state: produc3142500478612311029_state > produc3142500478612311029_state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    id_Pro689249233197853983_state: produc3882902684156697774_state > produc3882902684156697774_state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_Mt__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_J,type,
    id_Pro2933125508726022968_state: produc7272280185086996969_state > produc7272280185086996969_state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__PartialHeapSA__Ostate_J,type,
    id_Pro5916511905602125660_state: produc4844402304972151915_state > produc4844402304972151915_state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__PosRat__Oprat_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    id_Pro8860473219607035410_state: produc6251377953716854561_state > produc6251377953716854561_state ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    id_Pro2088550933252328842on_val: produc1317171644805857433on_val > produc1317171644805857433on_val ).

thf(sy_c_Fun__Def_OTHE__default_001t__PartialHeapSA__Ostate,type,
    fun_TH8221542053827138557_state: state > ( state > $o ) > state ).

thf(sy_c_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    if_option_state: $o > option_state > option_state > option_state ).

thf(sy_c_If_001t__PartialHeapSA__Ostate,type,
    if_state: $o > state > state > state ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_M_Eo_J_J,type,
    sup_su4175861339346331443_val_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Option__Ooption_It__PartialHeapSA__Ostate_J_M_062_It__Option__Ooption_It__PartialHeapSA__Ostate_J_M_Eo_J_J,type,
    sup_su2989194110892726342tate_o: ( option_state > option_state > $o ) > ( option_state > option_state > $o ) > option_state > option_state > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__PartialHeapSA__Ostate_M_062_I_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_M_Eo_J_J,type,
    sup_su1108609720783396685tate_o: ( state > ( state > state ) > $o ) > ( state > ( state > state ) > $o ) > state > ( state > state ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__PartialHeapSA__Ostate_M_062_It__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_M_Eo_J_J,type,
    sup_su1162340662190093652tate_o: ( state > produc3882902684156697774_state > $o ) > ( state > produc3882902684156697774_state > $o ) > state > produc3882902684156697774_state > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
    sup_sup_state_o: ( state > $o ) > ( state > $o ) > state > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__PosRat__Oprat_M_062_It__PartialHeapSA__Ostate_M_Eo_J_J,type,
    sup_sup_prat_state_o: ( prat > state > $o ) > ( prat > state > $o ) > prat > state > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__PosRat__Oprat_M_062_It__Set__Oset_It__PartialHeapSA__Ostate_J_M_Eo_J_J,type,
    sup_su7614437069348798106tate_o: ( prat > set_state > $o ) > ( prat > set_state > $o ) > prat > set_state > $o ).

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thf(sy_c_PartialHeapSA_Ogreater,type,
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thf(sy_c_PartialHeapSA_Ogreater__set,type,
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thf(sy_c_PartialHeapSA_Ominus,type,
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thf(sy_c_PartialHeapSA_Oplus,type,
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thf(sy_c_PartialHeapSA_Ostable,type,
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thf(sy_c_PartialHeapSA_Ostate_OAbs__state,type,
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thf(sy_c_PartialHeapSA_Ostate_ORep__state,type,
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thf(sy_c_PartialHeapSA_Ounit,type,
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thf(sy_c_PartialHeapSA_Ovalid__state,type,
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thf(sy_c_Product__Type_OPair_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__PackageLogic__Oaassertion_It__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
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thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
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thf(sy_c_Product__Type_OPair_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
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    member2023868432420333456on_val: produc3517408837545735783on_val > set_Pr4590028993149660487on_val > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__PartialHeapSA__Ostate_J_Mt__Set__Oset_It__PartialHeapSA__Ostate_J_J,type,
    member1976142906092959210_state: produc597591266584407361_state > set_Pr5618112136499017889_state > $o ).

thf(sy_v_A,type,
    a: set_state ).

thf(sy_v_B,type,
    b: set_state ).

thf(sy_v__092_060alpha_062____,type,
    alpha: prat ).

thf(sy_v__092_060beta_062____,type,
    beta: prat ).

thf(sy_v_a____,type,
    a2: state ).

thf(sy_v_ba____,type,
    ba: state ).

thf(sy_v_p____,type,
    p: prat ).

thf(sy_v_w____,type,
    w: state ).

thf(sy_v_wa____,type,
    wa: state ).

thf(sy_v_wb____,type,
    wb: state ).

thf(sy_v_x____,type,
    x: state ).

thf(sy_v_xa____,type,
    xa: state ).

thf(sy_v_xb____,type,
    xb: state ).

% Relevant facts (1275)
thf(fact_0__092_060open_062_092_060not_062_Ascalable_Awb_Aa_092_060close_062,axiom,
    ~ ( scalable @ wb @ a2 ) ).

% \<open>\<not> scalable wb a\<close>
thf(fact_1_R__compatible__same,axiom,
    ! [A: state,W: state] :
      ( ( defined @ A @ W )
     => ( ( r @ A @ W )
        = W ) ) ).

% R_compatible_same
thf(fact_2__092_060open_062R_Aa_Awa_A_124_D_124_Aa_092_060close_062,axiom,
    defined @ ( r @ a2 @ wa ) @ a2 ).

% \<open>R a wa |#| a\<close>
thf(fact_3_calculation_I2_J,axiom,
    defined @ a2 @ ( multiply @ p @ w ) ).

% calculation(2)
thf(fact_4_r2,axiom,
    ( ( ( get_h @ ( r @ a2 @ wb ) )
      = ( get_h @ wb ) )
    & ( ( get_m @ ( r @ a2 @ wb ) )
      = ( comp_min_mask @ ( get_m @ a2 ) @ ( get_m @ wb ) ) ) ) ).

% r2
thf(fact_5__092_060open_062_092_060not_062_Ascalable_Aw_Aa_092_060close_062,axiom,
    ~ ( scalable @ w @ a2 ) ).

% \<open>\<not> scalable w a\<close>
thf(fact_6__092_060open_062_092_060not_062_Ascalable_Awa_Aa_092_060close_062,axiom,
    ~ ( scalable @ wa @ a2 ) ).

% \<open>\<not> scalable wa a\<close>
thf(fact_7_assms_I2_J,axiom,
    intuitionistic @ b ).

% assms(2)
thf(fact_8_assms_I1_J,axiom,
    combinable @ b ).

% assms(1)
thf(fact_9_scalable__def,axiom,
    ( scalable
    = ( ^ [W2: state,A2: state] :
        ! [X: state] :
          ( ( member_state @ X @ ( scaled @ W2 ) )
         => ~ ( defined @ A2 @ X ) ) ) ) ).

% scalable_def
thf(fact_10_calculation_I5_J,axiom,
    ( ( some_state @ ba )
    = ( plus @ ( r @ a2 @ wa ) @ a2 ) ) ).

% calculation(5)
thf(fact_11__092_060open_062wb_A_092_060in_062_Acwand_AA_AB_092_060close_062,axiom,
    member_state @ wb @ ( cwand @ a @ b ) ).

% \<open>wb \<in> cwand A B\<close>
thf(fact_12__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ba_O_ASome_Aba_A_061_AR_Aa_Awa_A_092_060oplus_062_Aa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Ba: state] :
        ( ( some_state @ Ba )
       != ( plus @ ( r @ a2 @ wa ) @ a2 ) ) ).

% \<open>\<And>thesis. (\<And>ba. Some ba = R a wa \<oplus> a \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_13_R__smaller,axiom,
    ! [W: state,A: state] : ( greater @ W @ ( r @ A @ W ) ) ).

% R_smaller
thf(fact_14__092_060open_062wa_A_092_060in_062_Acwand_AA_AB_092_060close_062,axiom,
    member_state @ wa @ ( cwand @ a @ b ) ).

% \<open>wa \<in> cwand A B\<close>
thf(fact_15_asm1,axiom,
    ( ( member_state @ a2 @ a )
    & ( ( some_state @ x )
      = ( plus @ ( r @ a2 @ w ) @ a2 ) ) ) ).

% asm1
thf(fact_16_r1,axiom,
    ( ( ( get_h @ ( r @ a2 @ wa ) )
      = ( get_h @ wa ) )
    & ( ( get_m @ ( r @ a2 @ wa ) )
      = ( comp_min_mask @ ( get_m @ a2 ) @ ( get_m @ wa ) ) ) ) ).

% r1
thf(fact_17_r3,axiom,
    ( ( ( get_h @ ( r @ a2 @ w ) )
      = ( get_h @ w ) )
    & ( ( get_m @ ( r @ a2 @ w ) )
      = ( comp_min_mask @ ( get_m @ a2 ) @ ( get_m @ w ) ) ) ) ).

% r3
thf(fact_18_r_I1_J,axiom,
    ! [A: state,X2: state] :
      ( ( ( member_state @ A @ a )
        & ( ( some_state @ X2 )
          = ( plus @ ( r @ A @ wa ) @ A ) ) )
     => ( member_state @ X2 @ b ) ) ).

% r(1)
thf(fact_19_r_I2_J,axiom,
    ! [A: state,X2: state] :
      ( ( ( member_state @ A @ a )
        & ( ( some_state @ X2 )
          = ( plus @ ( r @ A @ wb ) @ A ) ) )
     => ( member_state @ X2 @ b ) ) ).

% r(2)
thf(fact_20__092_060open_062_092_060And_062thesis_O_A_I_092_060lbrakk_062_092_060And_062a_Ax_O_Aa_A_092_060in_062_AA_A_092_060and_062_ASome_Ax_A_061_AR_Aa_Awa_A_092_060oplus_062_Aa_A_092_060Longrightarrow_062_Ax_A_092_060in_062_AB_059_A_092_060And_062a_Ax_O_Aa_A_092_060in_062_AA_A_092_060and_062_ASome_Ax_A_061_AR_Aa_Awb_A_092_060oplus_062_Aa_A_092_060Longrightarrow_062_Ax_A_092_060in_062_AB_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ( ! [A3: state,X3: state] :
          ( ( ( member_state @ A3 @ a )
            & ( ( some_state @ X3 )
              = ( plus @ ( r @ A3 @ wa ) @ A3 ) ) )
         => ( member_state @ X3 @ b ) )
     => ~ ! [A3: state,X3: state] :
            ( ( ( member_state @ A3 @ a )
              & ( ( some_state @ X3 )
                = ( plus @ ( r @ A3 @ wb ) @ A3 ) ) )
           => ( member_state @ X3 @ b ) ) ) ).

% \<open>\<And>thesis. (\<lbrakk>\<And>a x. a \<in> A \<and> Some x = R a wa \<oplus> a \<Longrightarrow> x \<in> B; \<And>a x. a \<in> A \<and> Some x = R a wb \<oplus> a \<Longrightarrow> x \<in> B\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_21__092_060open_062Some_Aw_A_061_Axa_A_092_060oplus_062_Axb_092_060close_062,axiom,
    ( ( some_state @ w )
    = ( plus @ xa @ xb ) ) ).

% \<open>Some w = xa \<oplus> xb\<close>
thf(fact_22_intuitionistic__def,axiom,
    ( intuitionistic
    = ( ^ [A4: set_state] :
        ! [A2: state,B: state] :
          ( ( ( greater @ A2 @ B )
            & ( member_state @ B @ A4 ) )
         => ( member_state @ A2 @ A4 ) ) ) ) ).

% intuitionistic_def
thf(fact_23_non__scalable__R__charact,axiom,
    ! [W: state,A: state] :
      ( ~ ( scalable @ W @ A )
     => ( ( ( get_m @ ( r @ A @ W ) )
          = ( comp_min_mask @ ( get_m @ A ) @ ( get_m @ W ) ) )
        & ( ( get_h @ ( r @ A @ W ) )
          = ( get_h @ W ) ) ) ) ).

% non_scalable_R_charact
thf(fact_24_in__cwand,axiom,
    ! [A5: set_state,W: state,B2: set_state] :
      ( ! [A6: state,X4: state] :
          ( ( ( member_state @ A6 @ A5 )
            & ( ( some_state @ X4 )
              = ( plus @ ( r @ A6 @ W ) @ A6 ) ) )
         => ( member_state @ X4 @ B2 ) )
     => ( member_state @ W @ ( cwand @ A5 @ B2 ) ) ) ).

% in_cwand
thf(fact_25_w__in__scaled,axiom,
    ! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).

% w_in_scaled
thf(fact_26__092_060open_062xb_A_092_060succeq_062_Amultiply_A_092_060beta_062_Awb_092_060close_062,axiom,
    greater @ xb @ ( multiply @ beta @ wb ) ).

% \<open>xb \<succeq> multiply \<beta> wb\<close>
thf(fact_27__092_060open_062xa_A_092_060succeq_062_Amultiply_A_092_060alpha_062_Awa_092_060close_062,axiom,
    greater @ xa @ ( multiply @ alpha @ wa ) ).

% \<open>xa \<succeq> multiply \<alpha> wa\<close>
thf(fact_28_PartialSA_Odefined__sum__move,axiom,
    ! [A: state,B3: state,X2: state,Y: state,A7: state] :
      ( ( defined @ A @ B3 )
     => ( ( ( some_state @ B3 )
          = ( plus @ X2 @ Y ) )
       => ( ( ( some_state @ A7 )
            = ( plus @ A @ X2 ) )
         => ( defined @ A7 @ Y ) ) ) ) ).

% PartialSA.defined_sum_move
thf(fact_29_PartialSA_Obigger__sum,axiom,
    ! [Phi: state,A: state,B3: state,Phi2: state] :
      ( ( ( some_state @ Phi )
        = ( plus @ A @ B3 ) )
     => ( ( greater @ Phi2 @ Phi )
       => ? [B4: state] :
            ( ( greater @ B4 @ B3 )
            & ( ( some_state @ Phi2 )
              = ( plus @ A @ B4 ) ) ) ) ) ).

% PartialSA.bigger_sum
thf(fact_30_PartialSA_Ogreater__def,axiom,
    ( greater
    = ( ^ [A2: state,B: state] :
        ? [C: state] :
          ( ( some_state @ A2 )
          = ( plus @ B @ C ) ) ) ) ).

% PartialSA.greater_def
thf(fact_31_PartialSA_Ogreater__equiv,axiom,
    ( greater
    = ( ^ [A2: state,B: state] :
        ? [C: state] :
          ( ( some_state @ A2 )
          = ( plus @ C @ B ) ) ) ) ).

% PartialSA.greater_equiv
thf(fact_32_PartialSA_Oaddition__bigger,axiom,
    ! [A7: state,A: state,X5: state,B3: state,X2: state] :
      ( ( greater @ A7 @ A )
     => ( ( ( some_state @ X5 )
          = ( plus @ A7 @ B3 ) )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B3 ) )
         => ( greater @ X5 @ X2 ) ) ) ) ).

% PartialSA.addition_bigger
thf(fact_33_PartialSA_Obigger__sum__smaller,axiom,
    ! [C2: state,A: state,B3: state,A7: state] :
      ( ( ( some_state @ C2 )
        = ( plus @ A @ B3 ) )
     => ( ( greater @ A @ A7 )
       => ? [B4: state] :
            ( ( greater @ B4 @ B3 )
            & ( ( some_state @ C2 )
              = ( plus @ A7 @ B4 ) ) ) ) ) ).

% PartialSA.bigger_sum_smaller
thf(fact_34_option_Oinject,axiom,
    ! [X22: state,Y2: state] :
      ( ( ( some_state @ X22 )
        = ( some_state @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_35_state__ext,axiom,
    ! [A: state,B3: state] :
      ( ( ( get_h @ A )
        = ( get_h @ B3 ) )
     => ( ( ( get_m @ A )
          = ( get_m @ B3 ) )
       => ( A = B3 ) ) ) ).

% state_ext
thf(fact_36_PartialSA_Osmaller__compatible,axiom,
    ! [A7: state,B3: state,A: state] :
      ( ( defined @ A7 @ B3 )
     => ( ( greater @ A7 @ A )
       => ( defined @ A @ B3 ) ) ) ).

% PartialSA.smaller_compatible
thf(fact_37_asso1,axiom,
    ! [A: state,B3: state,Ab: state,C2: state,Bc: state] :
      ( ( ( ( plus @ A @ B3 )
          = ( some_state @ Ab ) )
        & ( ( plus @ B3 @ C2 )
          = ( some_state @ Bc ) ) )
     => ( ( plus @ Ab @ C2 )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_38__092_060open_062xb_A_092_060in_062_Amultiply__sem__assertion_A_092_060beta_062_A_Icwand_AA_AB_J_092_060close_062,axiom,
    member_state @ xb @ ( multip8064567061438756306ertion @ beta @ ( cwand @ a @ b ) ) ).

% \<open>xb \<in> multiply_sem_assertion \<beta> (cwand A B)\<close>
thf(fact_39__092_060open_062xa_A_092_060in_062_Amultiply__sem__assertion_A_092_060alpha_062_A_Icwand_AA_AB_J_092_060close_062,axiom,
    member_state @ xa @ ( multip8064567061438756306ertion @ alpha @ ( cwand @ a @ b ) ) ).

% \<open>xa \<in> multiply_sem_assertion \<alpha> (cwand A B)\<close>
thf(fact_40__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062wa_Awb_O_A_092_060lbrakk_062wa_A_092_060in_062_Acwand_AA_AB_059_Awb_A_092_060in_062_Acwand_AA_AB_059_Axa_A_092_060succeq_062_Amultiply_A_092_060alpha_062_Awa_059_Axb_A_092_060succeq_062_Amultiply_A_092_060beta_062_Awb_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Wa: state] :
        ( ( member_state @ Wa @ ( cwand @ a @ b ) )
       => ! [Wb: state] :
            ( ( member_state @ Wb @ ( cwand @ a @ b ) )
           => ( ( greater @ xa @ ( multiply @ alpha @ Wa ) )
             => ~ ( greater @ xb @ ( multiply @ beta @ Wb ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>wa wb. \<lbrakk>wa \<in> cwand A B; wb \<in> cwand A B; xa \<succeq> multiply \<alpha> wa; xb \<succeq> multiply \<beta> wb\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_41_commutative,axiom,
    ( plus
    = ( ^ [A2: state,B: state] : ( plus @ B @ A2 ) ) ) ).

% commutative
thf(fact_42_PartialSA_Osucc__antisym,axiom,
    ! [A: state,B3: state] :
      ( ( greater @ A @ B3 )
     => ( ( greater @ B3 @ A )
       => ( A = B3 ) ) ) ).

% PartialSA.succ_antisym
thf(fact_43_PartialSA_Osucc__trans,axiom,
    ! [A: state,B3: state,C2: state] :
      ( ( greater @ A @ B3 )
     => ( ( greater @ B3 @ C2 )
       => ( greater @ A @ C2 ) ) ) ).

% PartialSA.succ_trans
thf(fact_44_PartialSA_Osucc__refl,axiom,
    ! [A: state] : ( greater @ A @ A ) ).

% PartialSA.succ_refl
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: produc3882902684156697774_state,P: produc3882902684156697774_state > $o] :
      ( ( member6715010751053357271_state @ A @ ( collec394671971739345689_state @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: produc7272280185086996969_state,P: produc7272280185086996969_state > $o] :
      ( ( member7461423308196221312_state @ A @ ( collec906845016635587262_state @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: state,P: state > $o] :
      ( ( member_state @ A @ ( collect_state @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ A @ ( collec7475294502966882545on_val @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A5: set_state] :
      ( ( collect_state
        @ ^ [X: state] : ( member_state @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_53_Collect__cong,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collec906845016635587262_state @ P )
        = ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_cong
thf(fact_54_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_55_Collect__cong,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X4: produc9003781588162623004on_val] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collec7475294502966882545on_val @ P )
        = ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_cong
thf(fact_56_positivity,axiom,
    ! [A: state,B3: state,C2: state] :
      ( ( ( plus @ A @ B3 )
        = ( some_state @ C2 ) )
     => ( ( ( some_state @ C2 )
          = ( plus @ C2 @ C2 ) )
       => ( ( some_state @ A )
          = ( plus @ A @ A ) ) ) ) ).

% positivity
thf(fact_57__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xa_Axb_O_A_092_060lbrakk_062Some_Aw_A_061_Axa_A_092_060oplus_062_Axb_059_Axa_A_092_060in_062_Amultiply__sem__assertion_A_092_060alpha_062_A_Icwand_AA_AB_J_059_Axb_A_092_060in_062_Amultiply__sem__assertion_A_092_060beta_062_A_Icwand_AA_AB_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Xa: state,Xb: state] :
        ( ( ( some_state @ w )
          = ( plus @ Xa @ Xb ) )
       => ( ( member_state @ Xa @ ( multip8064567061438756306ertion @ alpha @ ( cwand @ a @ b ) ) )
         => ~ ( member_state @ Xb @ ( multip8064567061438756306ertion @ beta @ ( cwand @ a @ b ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>xa xb. \<lbrakk>Some w = xa \<oplus> xb; xa \<in> multiply_sem_assertion \<alpha> (cwand A B); xb \<in> multiply_sem_assertion \<beta> (cwand A B)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_58_asm,axiom,
    member_state @ w @ ( add_set @ ( multip8064567061438756306ertion @ alpha @ ( cwand @ a @ b ) ) @ ( multip8064567061438756306ertion @ beta @ ( cwand @ a @ b ) ) ) ).

% asm
thf(fact_59_wandI,axiom,
    ! [A5: set_state,W: state,B2: set_state] :
      ( ! [A6: state,X4: state] :
          ( ( ( member_state @ A6 @ A5 )
            & ( ( some_state @ X4 )
              = ( plus @ A6 @ W ) ) )
         => ( member_state @ X4 @ B2 ) )
     => ( member_state @ W @ ( wand @ A5 @ B2 ) ) ) ).

% wandI
thf(fact_60_PartialSA_Osmaller__pure__sum__smaller,axiom,
    ! [Y: state,A: state,B3: state,X2: state] :
      ( ( greater @ Y @ A )
     => ( ( greater @ Y @ B3 )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B3 ) )
         => ( ( sep_pure_state @ plus @ B3 )
           => ( greater @ Y @ X2 ) ) ) ) ) ).

% PartialSA.smaller_pure_sum_smaller
thf(fact_61_PartialSA_Ointuitionistic__def,axiom,
    ! [A5: state > $o] :
      ( ( packag8361946002163212404_state @ plus @ A5 )
      = ( ! [Phi3: state,Phi4: state] :
            ( ( ( greater @ Phi3 @ Phi4 )
              & ( A5 @ Phi4 ) )
           => ( A5 @ Phi3 ) ) ) ) ).

% PartialSA.intuitionistic_def
thf(fact_62_PartialSA_Omono__prop__def,axiom,
    ! [P: state > $o] :
      ( ( sep_mono_prop_state @ plus @ P )
      = ( ! [X: state,Y3: state] :
            ( ( ( greater @ Y3 @ X )
              & ( P @ X ) )
           => ( P @ Y3 ) ) ) ) ).

% PartialSA.mono_prop_def
thf(fact_63_PartialSA_Omono__propI,axiom,
    ! [P: state > $o] :
      ( ! [X4: state,Y4: state] :
          ( ( ( greater @ Y4 @ X4 )
            & ( P @ X4 ) )
         => ( P @ Y4 ) )
     => ( sep_mono_prop_state @ plus @ P ) ) ).

% PartialSA.mono_propI
thf(fact_64_PartialSA_Oprove__last__completeness,axiom,
    ! [A7: state,A: state,Nf1: state,F2: state] :
      ( ( greater @ A7 @ A )
     => ( ( ( some_state @ A )
          = ( plus @ Nf1 @ F2 ) )
       => ( greater @ ( minus @ A7 @ Nf1 ) @ F2 ) ) ) ).

% PartialSA.prove_last_completeness
thf(fact_65_PartialSA_Ominus__and__plus,axiom,
    ! [Omega: state,Omega2: state,R: state,A: state] :
      ( ( ( some_state @ Omega )
        = ( plus @ Omega2 @ R ) )
     => ( ( greater @ Omega2 @ A )
       => ( ( some_state @ ( minus @ Omega @ A ) )
          = ( plus @ ( minus @ Omega2 @ A ) @ R ) ) ) ) ).

% PartialSA.minus_and_plus
thf(fact_66_PartialSA_Ominus__bigger,axiom,
    ! [X2: state,A: state,B3: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ A @ B3 ) )
     => ( greater @ ( minus @ X2 @ A ) @ B3 ) ) ).

% PartialSA.minus_bigger
thf(fact_67_PartialSA_Ominus__some,axiom,
    ! [A: state,B3: state] :
      ( ( greater @ A @ B3 )
     => ( ( some_state @ A )
        = ( plus @ B3 @ ( minus @ A @ B3 ) ) ) ) ).

% PartialSA.minus_some
thf(fact_68_PartialSA_Oadd__set__commm,axiom,
    ( add_set
    = ( ^ [A4: set_state,B5: set_state] : ( add_set @ B5 @ A4 ) ) ) ).

% PartialSA.add_set_commm
thf(fact_69_PartialSA_Oadd__set__asso,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( add_set @ ( add_set @ A5 @ B2 ) @ C3 )
      = ( add_set @ A5 @ ( add_set @ B2 @ C3 ) ) ) ).

% PartialSA.add_set_asso
thf(fact_70_PartialSA_Ominus__default,axiom,
    ! [B3: state,A: state] :
      ( ~ ( greater @ B3 @ A )
     => ( ( minus @ B3 @ A )
        = B3 ) ) ).

% PartialSA.minus_default
thf(fact_71_PartialSA_Ominus__smaller,axiom,
    ! [X2: state,A: state] :
      ( ( greater @ X2 @ A )
     => ( greater @ X2 @ ( minus @ X2 @ A ) ) ) ).

% PartialSA.minus_smaller
thf(fact_72_PartialSA_Ogreater__minus__trans,axiom,
    ! [Y: state,X2: state,A: state] :
      ( ( greater @ Y @ X2 )
     => ( ( greater @ X2 @ A )
       => ( greater @ ( minus @ Y @ A ) @ ( minus @ X2 @ A ) ) ) ) ).

% PartialSA.greater_minus_trans
thf(fact_73_in__multiply__refl,axiom,
    ! [X2: state,A5: set_state,P2: prat] :
      ( ( member_state @ X2 @ A5 )
     => ( member_state @ ( multiply @ P2 @ X2 ) @ ( multip8064567061438756306ertion @ P2 @ A5 ) ) ) ).

% in_multiply_refl
thf(fact_74_PartialSA_Opure__def,axiom,
    ! [A: state] :
      ( ( sep_pure_state @ plus @ A )
      = ( ( some_state @ A )
        = ( plus @ A @ A ) ) ) ).

% PartialSA.pure_def
thf(fact_75_PartialSA_Opure__stable,axiom,
    ! [A: state,B3: state,C2: state] :
      ( ( sep_pure_state @ plus @ A )
     => ( ( sep_pure_state @ plus @ B3 )
       => ( ( ( some_state @ C2 )
            = ( plus @ A @ B3 ) )
         => ( sep_pure_state @ plus @ C2 ) ) ) ) ).

% PartialSA.pure_stable
thf(fact_76_PartialSA_Opure__smaller,axiom,
    ! [A: state,B3: state] :
      ( ( sep_pure_state @ plus @ A )
     => ( ( greater @ A @ B3 )
       => ( sep_pure_state @ plus @ B3 ) ) ) ).

% PartialSA.pure_smaller
thf(fact_77_PartialSA_Oadd__set__elem,axiom,
    ! [Phi: state,A5: set_state,B2: set_state] :
      ( ( member_state @ Phi @ ( add_set @ A5 @ B2 ) )
      = ( ? [A2: state,B: state] :
            ( ( ( some_state @ Phi )
              = ( plus @ A2 @ B ) )
            & ( member_state @ A2 @ A5 )
            & ( member_state @ B @ B2 ) ) ) ) ).

% PartialSA.add_set_elem
thf(fact_78_PartialSA_Ox__elem__set__product,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ( member_state @ X2 @ ( add_set @ A5 @ B2 ) )
      = ( ? [A2: state,B: state] :
            ( ( member_state @ A2 @ A5 )
            & ( member_state @ B @ B2 )
            & ( ( some_state @ X2 )
              = ( plus @ A2 @ B ) ) ) ) ) ).

% PartialSA.x_elem_set_product
thf(fact_79_in__multiply__sem,axiom,
    ! [X2: state,P2: prat,A5: set_state] :
      ( ( member_state @ X2 @ ( multip8064567061438756306ertion @ P2 @ A5 ) )
     => ? [X4: state] :
          ( ( member_state @ X4 @ A5 )
          & ( greater @ X2 @ ( multiply @ P2 @ X4 ) ) ) ) ).

% in_multiply_sem
thf(fact_80_PartialSA_Ominus__sum,axiom,
    ! [A: state,B3: state,C2: state,X2: state] :
      ( ( ( some_state @ A )
        = ( plus @ B3 @ C2 ) )
     => ( ( greater @ X2 @ A )
       => ( ( minus @ X2 @ A )
          = ( minus @ ( minus @ X2 @ B3 ) @ C2 ) ) ) ) ).

% PartialSA.minus_sum
thf(fact_81_PartialSA_Omono__pruner__def,axiom,
    ! [P2: state > $o] :
      ( ( packag2456304381420842418_state @ plus @ P2 )
      = ( ! [Phi3: state,Phi4: state,R2: state] :
            ( ( ( sep_pure_state @ plus @ R2 )
              & ( P2 @ Phi4 )
              & ( ( some_state @ Phi3 )
                = ( plus @ Phi4 @ R2 ) ) )
           => ( P2 @ Phi3 ) ) ) ) ).

% PartialSA.mono_pruner_def
thf(fact_82_sep__algebra_Omono__prop_Ocong,axiom,
    sep_mono_prop_state = sep_mono_prop_state ).

% sep_algebra.mono_prop.cong
thf(fact_83_package__logic_Ointuitionistic_Ocong,axiom,
    packag8361946002163212404_state = packag8361946002163212404_state ).

% package_logic.intuitionistic.cong
thf(fact_84_PartialSA_OminusI,axiom,
    ! [B3: state,A: state,X2: state] :
      ( ( ( some_state @ B3 )
        = ( plus @ A @ X2 ) )
     => ( ( greater @ X2 @ ( core @ B3 ) )
       => ( X2
          = ( minus @ B3 @ A ) ) ) ) ).

% PartialSA.minusI
thf(fact_85_PartialSA_Ominus__equiv__def,axiom,
    ! [B3: state,A: state] :
      ( ( greater @ B3 @ A )
     => ( ( ( some_state @ B3 )
          = ( plus @ A @ ( minus @ B3 @ A ) ) )
        & ( greater @ ( minus @ B3 @ A ) @ ( core @ B3 ) ) ) ) ).

% PartialSA.minus_equiv_def
thf(fact_86_sep__algebra_Opure_Ocong,axiom,
    sep_pure_state = sep_pure_state ).

% sep_algebra.pure.cong
thf(fact_87_PartialSA_Ox__elem__set__product__splus,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ( member_state @ X2 @ ( add_set @ A5 @ B2 ) )
      = ( ? [A2: state,B: state] :
            ( ( member_state @ A2 @ A5 )
            & ( member_state @ B @ B2 )
            & ( ( some_state @ X2 )
              = ( sep_splus_state @ plus @ ( some_state @ A2 ) @ ( some_state @ B ) ) ) ) ) ) ).

% PartialSA.x_elem_set_product_splus
thf(fact_88_binary__same,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( binary @ A5 )
     => ( ( intuitionistic @ B2 )
       => ( ord_le2494988322063910608_state @ ( wand @ A5 @ B2 ) @ ( cwand @ A5 @ B2 ) ) ) ) ).

% binary_same
thf(fact_89_R__def,axiom,
    ( r
    = ( ^ [A2: state,W2: state] : ( if_state @ ( scalable @ W2 @ A2 ) @ W2 @ ( abs_state @ ( produc74620467203698198on_val @ ( comp_min_mask @ ( get_m @ A2 ) @ ( get_m @ W2 ) ) @ ( get_h @ W2 ) ) ) ) ) ) ).

% R_def
thf(fact_90_plus__charact_I1_J,axiom,
    ! [A: state,B3: state,X2: state] :
      ( ( ( plus @ A @ B3 )
        = ( some_state @ X2 ) )
     => ( ( get_m @ X2 )
        = ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B3 ) ) ) ) ).

% plus_charact(1)
thf(fact_91_PartialSA_Oequiv__stable__sum,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( sep_equiv_state @ plus @ A5 @ B2 )
     => ( sep_equiv_state @ plus @ ( add_set @ A5 @ C3 ) @ ( add_set @ B2 @ C3 ) ) ) ).

% PartialSA.equiv_stable_sum
thf(fact_92_package__logic_Omono__pruner_Ocong,axiom,
    packag2456304381420842418_state = packag2456304381420842418_state ).

% package_logic.mono_pruner.cong
thf(fact_93_sep__algebra_Osplus_Ocong,axiom,
    sep_splus_state = sep_splus_state ).

% sep_algebra.splus.cong
thf(fact_94_sep__algebra_Oequiv_Ocong,axiom,
    sep_equiv_state = sep_equiv_state ).

% sep_algebra.equiv.cong
thf(fact_95_valid__state_Ocases,axiom,
    ! [X2: produc9003781588162623004on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( X2
         != ( produc74620467203698198on_val @ Pi @ H ) ) ).

% valid_state.cases
thf(fact_96_PartialSA_Ocore__mono,axiom,
    ! [A: state,B3: state] :
      ( ( greater @ A @ B3 )
     => ( greater @ ( core @ A ) @ ( core @ B3 ) ) ) ).

% PartialSA.core_mono
thf(fact_97_core__defined_I1_J,axiom,
    ! [Phi: state] :
      ( ( get_h @ ( core @ Phi ) )
      = ( get_h @ Phi ) ) ).

% core_defined(1)
thf(fact_98_PartialSA_Ominus__core,axiom,
    ! [A: state,B3: state] :
      ( ( core @ ( minus @ A @ B3 ) )
      = ( core @ A ) ) ).

% PartialSA.minus_core
thf(fact_99_PartialSA_Ominus__core__weaker,axiom,
    ! [A: state,B3: state] :
      ( ( core @ ( minus @ A @ B3 ) )
      = ( minus @ ( core @ A ) @ ( core @ B3 ) ) ) ).

% PartialSA.minus_core_weaker
thf(fact_100_PartialSA_Osplus__comm,axiom,
    ! [A: option_state,B3: option_state] :
      ( ( sep_splus_state @ plus @ A @ B3 )
      = ( sep_splus_state @ plus @ B3 @ A ) ) ).

% PartialSA.splus_comm
thf(fact_101_PartialSA_Osplus__asso,axiom,
    ! [A: option_state,B3: option_state,C2: option_state] :
      ( ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ A @ B3 ) @ C2 )
      = ( sep_splus_state @ plus @ A @ ( sep_splus_state @ plus @ B3 @ C2 ) ) ) ).

% PartialSA.splus_asso
thf(fact_102_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X: state] : ( plus @ X @ ( core @ X ) ) ) ) ).

% core_is_smaller
thf(fact_103_core__is__pure,axiom,
    ! [X2: state] :
      ( ( some_state @ ( core @ X2 ) )
      = ( plus @ ( core @ X2 ) @ ( core @ X2 ) ) ) ).

% core_is_pure
thf(fact_104_cancellative,axiom,
    ! [A: state,B3: state,X2: state,Y: state] :
      ( ( ( some_state @ A )
        = ( plus @ B3 @ X2 ) )
     => ( ( ( some_state @ A )
          = ( plus @ B3 @ Y ) )
       => ( ( ( core @ X2 )
            = ( core @ Y ) )
         => ( X2 = Y ) ) ) ) ).

% cancellative
thf(fact_105_core__sum,axiom,
    ! [C2: state,A: state,B3: state] :
      ( ( ( some_state @ C2 )
        = ( plus @ A @ B3 ) )
     => ( ( some_state @ ( core @ C2 ) )
        = ( plus @ ( core @ A ) @ ( core @ B3 ) ) ) ) ).

% core_sum
thf(fact_106_core__max,axiom,
    ! [X2: state,C2: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ X2 @ C2 ) )
     => ? [R3: state] :
          ( ( some_state @ ( core @ X2 ) )
          = ( plus @ C2 @ R3 ) ) ) ).

% core_max
thf(fact_107_PartialSA_Osmaller__compatible__core,axiom,
    ! [Y: state,X2: state] :
      ( ( greater @ Y @ X2 )
     => ( defined @ X2 @ ( core @ Y ) ) ) ).

% PartialSA.smaller_compatible_core
thf(fact_108_PartialSA_Osplus__develop,axiom,
    ! [A: state,B3: state,C2: state,D: state] :
      ( ( ( some_state @ A )
        = ( plus @ B3 @ C2 ) )
     => ( ( plus @ A @ D )
        = ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ ( some_state @ B3 ) @ ( some_state @ C2 ) ) @ ( some_state @ D ) ) ) ) ).

% PartialSA.splus_develop
thf(fact_109_PartialSA_Osplus_Osimps_I3_J,axiom,
    ! [A: state,B3: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ A ) @ ( some_state @ B3 ) )
      = ( plus @ A @ B3 ) ) ).

% PartialSA.splus.simps(3)
thf(fact_110_PartialSA_Ogreater__than__sum__exists,axiom,
    ! [A: state,B3: state,B1: state,B22: state] :
      ( ( greater @ A @ B3 )
     => ( ( ( some_state @ B3 )
          = ( plus @ B1 @ B22 ) )
       => ? [R3: state] :
            ( ( ( some_state @ A )
              = ( plus @ R3 @ B22 ) )
            & ( greater @ ( core @ R3 ) @ ( core @ A ) )
            & ( greater @ R3 @ B1 ) ) ) ) ).

% PartialSA.greater_than_sum_exists
thf(fact_111_PartialSA_Obigger__core__sum__defined,axiom,
    ! [A: state,B3: state] :
      ( ( greater @ ( core @ A ) @ B3 )
     => ( ( some_state @ A )
        = ( plus @ A @ B3 ) ) ) ).

% PartialSA.bigger_core_sum_defined
thf(fact_112_PartialSA_Osmaller__than__core,axiom,
    ! [Y: state,X2: state,Z: state] :
      ( ( greater @ Y @ X2 )
     => ( ( ( some_state @ Z )
          = ( plus @ X2 @ ( core @ Y ) ) )
       => ( ( core @ Z )
          = ( core @ Y ) ) ) ) ).

% PartialSA.smaller_than_core
thf(fact_113_PartialSA_Ominus__unique,axiom,
    ! [B3: state,A: state,X2: state,Y: state] :
      ( ( ( ( some_state @ B3 )
          = ( plus @ A @ X2 ) )
        & ( greater @ X2 @ ( core @ B3 ) ) )
     => ( ( ( ( some_state @ B3 )
            = ( plus @ A @ Y ) )
          & ( greater @ Y @ ( core @ B3 ) ) )
       => ( X2 = Y ) ) ) ).

% PartialSA.minus_unique
thf(fact_114_PartialSA_Ominus__exists,axiom,
    ! [B3: state,A: state] :
      ( ( greater @ B3 @ A )
     => ? [X4: state] :
          ( ( ( some_state @ B3 )
            = ( plus @ A @ X4 ) )
          & ( greater @ X4 @ ( core @ B3 ) ) ) ) ).

% PartialSA.minus_exists
thf(fact_115_PartialSA_Oextract__core,axiom,
    ! [B3: state,A: state,X2: state] :
      ( ( ( ( some_state @ B3 )
          = ( plus @ A @ X2 ) )
        & ( greater @ X2 @ ( core @ B3 ) ) )
     => ( ( core @ X2 )
        = ( core @ B3 ) ) ) ).

% PartialSA.extract_core
thf(fact_116_cwand__stronger,axiom,
    ! [A5: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( cwand @ A5 @ B2 ) @ ( wand @ A5 @ B2 ) ) ).

% cwand_stronger
thf(fact_117_PartialSA_Ominus__equiv__def__any__elem,axiom,
    ! [X2: state,A: state,B3: state] :
      ( ( ( some_state @ X2 )
        = ( plus @ A @ B3 ) )
     => ( ( some_state @ ( minus @ X2 @ A ) )
        = ( plus @ B3 @ ( core @ X2 ) ) ) ) ).

% PartialSA.minus_equiv_def_any_elem
thf(fact_118_binary__def,axiom,
    ( binary
    = ( ^ [A4: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ( member_state @ ( abs_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ X ) ) @ ( get_h @ X ) ) ) @ A4 ) ) ) ) ).

% binary_def
thf(fact_119_double__inclusion,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ A5 )
       => ( A5 = B2 ) ) ) ).

% double_inclusion
thf(fact_120_subsetI,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X4 @ A5 )
         => ( member7461423308196221312_state @ X4 @ B2 ) )
     => ( ord_le4032880777312351359_state @ A5 @ B2 ) ) ).

% subsetI
thf(fact_121_subsetI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X4 @ A5 )
         => ( member2879030624002546611on_val @ X4 @ B2 ) )
     => ( ord_le6900622908513426738on_val @ A5 @ B2 ) ) ).

% subsetI
thf(fact_122_subsetI,axiom,
    ! [A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X4 @ A5 )
         => ( member6715010751053357271_state @ X4 @ B2 ) )
     => ( ord_le6483787752022044334_state @ A5 @ B2 ) ) ).

% subsetI
thf(fact_123_subsetI,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A5 )
         => ( member_state @ X4 @ B2 ) )
     => ( ord_le2494988322063910608_state @ A5 @ B2 ) ) ).

% subsetI
thf(fact_124_old_Oprod_Oinject,axiom,
    ! [A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,A7: produc9003781588162623004on_val,B6: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B3 )
        = ( produc4958730763592889809on_val @ A7 @ B6 ) )
      = ( ( A = A7 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_125_old_Oprod_Oinject,axiom,
    ! [A: option_state,B3: option_state,A7: option_state,B6: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B3 )
        = ( produc9160152616361873709_state @ A7 @ B6 ) )
      = ( ( A = A7 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_126_old_Oprod_Oinject,axiom,
    ! [A: prat,B3: state,A7: prat,B6: state] :
      ( ( ( produc6205719116230455971_state @ A @ B3 )
        = ( produc6205719116230455971_state @ A7 @ B6 ) )
      = ( ( A = A7 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_127_old_Oprod_Oinject,axiom,
    ! [A: prat,B3: set_state,A7: prat,B6: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B3 )
        = ( produc9080284787853255769_state @ A7 @ B6 ) )
      = ( ( A = A7 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_128_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,A7: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B3 )
        = ( produc74620467203698198on_val @ A7 @ B6 ) )
      = ( ( A = A7 )
        & ( B3 = B6 ) ) ) ).

% old.prod.inject
thf(fact_129_prod_Oinject,axiom,
    ! [X1: produc9003781588162623004on_val,X22: produc9003781588162623004on_val,Y1: produc9003781588162623004on_val,Y2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ X1 @ X22 )
        = ( produc4958730763592889809on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_130_prod_Oinject,axiom,
    ! [X1: option_state,X22: option_state,Y1: option_state,Y2: option_state] :
      ( ( ( produc9160152616361873709_state @ X1 @ X22 )
        = ( produc9160152616361873709_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_131_prod_Oinject,axiom,
    ! [X1: prat,X22: state,Y1: prat,Y2: state] :
      ( ( ( produc6205719116230455971_state @ X1 @ X22 )
        = ( produc6205719116230455971_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_132_prod_Oinject,axiom,
    ! [X1: prat,X22: set_state,Y1: prat,Y2: set_state] :
      ( ( ( produc9080284787853255769_state @ X1 @ X22 )
        = ( produc9080284787853255769_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_133_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X22 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_134_multiply_Osimps,axiom,
    ( multiply
    = ( ^ [P3: prat,Phi4: state] : ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P3 @ ( get_m @ Phi4 ) ) @ ( get_h @ Phi4 ) ) ) ) ) ).

% multiply.simps
thf(fact_135_multiply_Oelims,axiom,
    ! [X2: prat,Xa2: state,Y: state] :
      ( ( ( multiply @ X2 @ Xa2 )
        = Y )
     => ( Y
        = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X2 @ ( get_m @ Xa2 ) ) @ ( get_h @ Xa2 ) ) ) ) ) ).

% multiply.elims
thf(fact_136_dual__order_Orefl,axiom,
    ! [A: set_state] : ( ord_le2494988322063910608_state @ A @ A ) ).

% dual_order.refl
thf(fact_137_order__refl,axiom,
    ! [X2: set_state] : ( ord_le2494988322063910608_state @ X2 @ X2 ) ).

% order_refl
thf(fact_138_core__def,axiom,
    ( core
    = ( ^ [Phi4: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi4 ) ) ) ) ) ).

% core_def
thf(fact_139_ssubsetI,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ! [Pi: state,H: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H ) @ A5 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_140_ssubsetI,axiom,
    ! [A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ! [Pi: state,H: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H ) @ A5 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_141_ssubsetI,axiom,
    ! [A5: set_Pr4305388979797279737on_val,B2: set_Pr4305388979797279737on_val] :
      ( ! [Pi: produc9003781588162623004on_val,H: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ A5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ B2 ) )
     => ( ord_le7224449410216889241on_val @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_142_ssubsetI,axiom,
    ! [A5: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
      ( ! [Pi: option_state,H: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H ) @ A5 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H ) @ B2 ) )
     => ( ord_le6423325748750870005_state @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_143_ssubsetI,axiom,
    ! [A5: set_Pr6039890582625207499_state,B2: set_Pr6039890582625207499_state] :
      ( ! [Pi: prat,H: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H ) @ A5 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H ) @ B2 ) )
     => ( ord_le262197048489888875_state @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_144_ssubsetI,axiom,
    ! [A5: set_Pr6847263425858009217_state,B2: set_Pr6847263425858009217_state] :
      ( ! [Pi: prat,H: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H ) @ A5 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H ) @ B2 ) )
     => ( ord_le5761494056910127137_state @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_145_ssubsetI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ A5 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ A5 @ B2 ) ) ).

% ssubsetI
thf(fact_146_mult__distr__masks,axiom,
    ! [A: prat,B3: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B3 @ C2 ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B3 ) @ ( multiply_mask @ A @ C2 ) ) ) ).

% mult_distr_masks
thf(fact_147_minus__empty,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( Pi2
      = ( add_ma6586698345353345811t_char @ Pi2 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_148_add__states_Ocases,axiom,
    ! [X2: produc1317171644805857433on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi3: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
          ( X2
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi3 @ H2 ) ) ) ).

% add_states.cases
thf(fact_149_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
    = ( ^ [X: set_state,Y3: set_state] :
          ( ( ord_le2494988322063910608_state @ X @ Y3 )
          & ( ord_le2494988322063910608_state @ Y3 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_150_ord__eq__le__trans,axiom,
    ! [A: set_state,B3: set_state,C2: set_state] :
      ( ( A = B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ord_le2494988322063910608_state @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_151_ord__le__eq__trans,axiom,
    ! [A: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le2494988322063910608_state @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_152_order__antisym,axiom,
    ! [X2: set_state,Y: set_state] :
      ( ( ord_le2494988322063910608_state @ X2 @ Y )
     => ( ( ord_le2494988322063910608_state @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_153_order_Otrans,axiom,
    ! [A: set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ord_le2494988322063910608_state @ A @ C2 ) ) ) ).

% order.trans
thf(fact_154_order__trans,axiom,
    ! [X2: set_state,Y: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ X2 @ Y )
     => ( ( ord_le2494988322063910608_state @ Y @ Z )
       => ( ord_le2494988322063910608_state @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_155_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ B @ A2 )
          & ( ord_le2494988322063910608_state @ A2 @ B ) ) ) ) ).

% dual_order.eq_iff
thf(fact_156_dual__order_Oantisym,axiom,
    ! [B3: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A )
     => ( ( ord_le2494988322063910608_state @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_157_dual__order_Otrans,axiom,
    ! [B3: set_state,A: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A )
     => ( ( ord_le2494988322063910608_state @ C2 @ B3 )
       => ( ord_le2494988322063910608_state @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_158_antisym,axiom,
    ! [A: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( ord_le2494988322063910608_state @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_159_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
    = ( ^ [A2: set_state,B: set_state] :
          ( ( ord_le2494988322063910608_state @ A2 @ B )
          & ( ord_le2494988322063910608_state @ B @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_160_order__subst1,axiom,
    ! [A: set_state,F: set_state > set_state,B3: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ ( F @ B3 ) )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ! [X4: set_state,Y4: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y4 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_le2494988322063910608_state @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_161_order__subst2,axiom,
    ! [A: set_state,B3: set_state,F: set_state > set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( ord_le2494988322063910608_state @ ( F @ B3 ) @ C2 )
       => ( ! [X4: set_state,Y4: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y4 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_162_order__eq__refl,axiom,
    ! [X2: set_state,Y: set_state] :
      ( ( X2 = Y )
     => ( ord_le2494988322063910608_state @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_163_ord__eq__le__subst,axiom,
    ! [A: set_state,F: set_state > set_state,B3: set_state,C2: set_state] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_le2494988322063910608_state @ B3 @ C2 )
       => ( ! [X4: set_state,Y4: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y4 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_le2494988322063910608_state @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_164_ord__le__eq__subst,axiom,
    ! [A: set_state,B3: set_state,F: set_state > set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X4: set_state,Y4: set_state] :
              ( ( ord_le2494988322063910608_state @ X4 @ Y4 )
             => ( ord_le2494988322063910608_state @ ( F @ X4 ) @ ( F @ Y4 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_165_order__antisym__conv,axiom,
    ! [Y: set_state,X2: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X2 )
     => ( ( ord_le2494988322063910608_state @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_166_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A6: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A6 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_167_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3142500478612311029_state] :
      ~ ! [A6: option_state,B7: option_state] :
          ( Y
         != ( produc9160152616361873709_state @ A6 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_168_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4844402304972151915_state] :
      ~ ! [A6: prat,B7: state] :
          ( Y
         != ( produc6205719116230455971_state @ A6 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_169_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6251377953716854561_state] :
      ~ ! [A6: prat,B7: set_state] :
          ( Y
         != ( produc9080284787853255769_state @ A6 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_170_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ~ ! [A6: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc74620467203698198on_val @ A6 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_171_surj__pair,axiom,
    ! [P2: produc1317171644805857433on_val] :
    ? [X4: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
      ( P2
      = ( produc4958730763592889809on_val @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_172_surj__pair,axiom,
    ! [P2: produc3142500478612311029_state] :
    ? [X4: option_state,Y4: option_state] :
      ( P2
      = ( produc9160152616361873709_state @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_173_surj__pair,axiom,
    ! [P2: produc4844402304972151915_state] :
    ? [X4: prat,Y4: state] :
      ( P2
      = ( produc6205719116230455971_state @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_174_surj__pair,axiom,
    ! [P2: produc6251377953716854561_state] :
    ? [X4: prat,Y4: set_state] :
      ( P2
      = ( produc9080284787853255769_state @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_175_surj__pair,axiom,
    ! [P2: produc9003781588162623004on_val] :
    ? [X4: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
      ( P2
      = ( produc74620467203698198on_val @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_176_prod__cases,axiom,
    ! [P: produc1317171644805857433on_val > $o,P2: produc1317171644805857433on_val] :
      ( ! [A6: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] : ( P @ ( produc4958730763592889809on_val @ A6 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_177_prod__cases,axiom,
    ! [P: produc3142500478612311029_state > $o,P2: produc3142500478612311029_state] :
      ( ! [A6: option_state,B7: option_state] : ( P @ ( produc9160152616361873709_state @ A6 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_178_prod__cases,axiom,
    ! [P: produc4844402304972151915_state > $o,P2: produc4844402304972151915_state] :
      ( ! [A6: prat,B7: state] : ( P @ ( produc6205719116230455971_state @ A6 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_179_prod__cases,axiom,
    ! [P: produc6251377953716854561_state > $o,P2: produc6251377953716854561_state] :
      ( ! [A6: prat,B7: set_state] : ( P @ ( produc9080284787853255769_state @ A6 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_180_prod__cases,axiom,
    ! [P: produc9003781588162623004on_val > $o,P2: produc9003781588162623004on_val] :
      ( ! [A6: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] : ( P @ ( produc74620467203698198on_val @ A6 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_181_Pair__inject,axiom,
    ! [A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,A7: produc9003781588162623004on_val,B6: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B3 )
        = ( produc4958730763592889809on_val @ A7 @ B6 ) )
     => ~ ( ( A = A7 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_182_Pair__inject,axiom,
    ! [A: option_state,B3: option_state,A7: option_state,B6: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B3 )
        = ( produc9160152616361873709_state @ A7 @ B6 ) )
     => ~ ( ( A = A7 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_183_Pair__inject,axiom,
    ! [A: prat,B3: state,A7: prat,B6: state] :
      ( ( ( produc6205719116230455971_state @ A @ B3 )
        = ( produc6205719116230455971_state @ A7 @ B6 ) )
     => ~ ( ( A = A7 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_184_Pair__inject,axiom,
    ! [A: prat,B3: set_state,A7: prat,B6: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B3 )
        = ( produc9080284787853255769_state @ A7 @ B6 ) )
     => ~ ( ( A = A7 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_185_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,A7: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B3 )
        = ( produc74620467203698198on_val @ A7 @ B6 ) )
     => ~ ( ( A = A7 )
         => ( B3 != B6 ) ) ) ).

% Pair_inject
thf(fact_186_prod__cases3,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A6: produc9003781588162623004on_val,B7: produc1457211279475724562t_char > prat,C4: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A6 @ ( produc74620467203698198on_val @ B7 @ C4 ) ) ) ).

% prod_cases3
thf(fact_187_prod__induct3,axiom,
    ! [P: produc1317171644805857433on_val > $o,X2: produc1317171644805857433on_val] :
      ( ! [A6: produc9003781588162623004on_val,B7: produc1457211279475724562t_char > prat,C4: produc1457211279475724562t_char > option_val] : ( P @ ( produc4958730763592889809on_val @ A6 @ ( produc74620467203698198on_val @ B7 @ C4 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_188_in__mono,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,X2: produc7272280185086996969_state] :
      ( ( ord_le4032880777312351359_state @ A5 @ B2 )
     => ( ( member7461423308196221312_state @ X2 @ A5 )
       => ( member7461423308196221312_state @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_189_in__mono,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val,X2: produc9003781588162623004on_val] :
      ( ( ord_le6900622908513426738on_val @ A5 @ B2 )
     => ( ( member2879030624002546611on_val @ X2 @ A5 )
       => ( member2879030624002546611on_val @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_190_in__mono,axiom,
    ! [A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state,X2: produc3882902684156697774_state] :
      ( ( ord_le6483787752022044334_state @ A5 @ B2 )
     => ( ( member6715010751053357271_state @ X2 @ A5 )
       => ( member6715010751053357271_state @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_191_in__mono,axiom,
    ! [A5: set_state,B2: set_state,X2: state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ( member_state @ X2 @ A5 )
       => ( member_state @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_192_subsetD,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C2: produc7272280185086996969_state] :
      ( ( ord_le4032880777312351359_state @ A5 @ B2 )
     => ( ( member7461423308196221312_state @ C2 @ A5 )
       => ( member7461423308196221312_state @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_193_subsetD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val,C2: produc9003781588162623004on_val] :
      ( ( ord_le6900622908513426738on_val @ A5 @ B2 )
     => ( ( member2879030624002546611on_val @ C2 @ A5 )
       => ( member2879030624002546611on_val @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_194_subsetD,axiom,
    ! [A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state,C2: produc3882902684156697774_state] :
      ( ( ord_le6483787752022044334_state @ A5 @ B2 )
     => ( ( member6715010751053357271_state @ C2 @ A5 )
       => ( member6715010751053357271_state @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_195_subsetD,axiom,
    ! [A5: set_state,B2: set_state,C2: state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ( member_state @ C2 @ A5 )
       => ( member_state @ C2 @ B2 ) ) ) ).

% subsetD
thf(fact_196_equalityE,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( A5 = B2 )
     => ~ ( ( ord_le2494988322063910608_state @ A5 @ B2 )
         => ~ ( ord_le2494988322063910608_state @ B2 @ A5 ) ) ) ).

% equalityE
thf(fact_197_subset__eq,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
        ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A4 )
         => ( member7461423308196221312_state @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_198_subset__eq,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
        ! [X: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X @ A4 )
         => ( member2879030624002546611on_val @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_199_subset__eq,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A4: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
        ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A4 )
         => ( member6715010751053357271_state @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_200_subset__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A4: set_state,B5: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ( member_state @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_201_equalityD1,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( A5 = B2 )
     => ( ord_le2494988322063910608_state @ A5 @ B2 ) ) ).

% equalityD1
thf(fact_202_equalityD2,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( A5 = B2 )
     => ( ord_le2494988322063910608_state @ B2 @ A5 ) ) ).

% equalityD2
thf(fact_203_subset__iff,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
        ! [T: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ T @ A4 )
         => ( member7461423308196221312_state @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_204_subset__iff,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
        ! [T: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ T @ A4 )
         => ( member2879030624002546611on_val @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_205_subset__iff,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A4: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
        ! [T: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ T @ A4 )
         => ( member6715010751053357271_state @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_206_subset__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A4: set_state,B5: set_state] :
        ! [T: state] :
          ( ( member_state @ T @ A4 )
         => ( member_state @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_207_subset__refl,axiom,
    ! [A5: set_state] : ( ord_le2494988322063910608_state @ A5 @ A5 ) ).

% subset_refl
thf(fact_208_Collect__mono,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_mono
thf(fact_209_Collect__mono,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X4: produc9003781588162623004on_val] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_mono
thf(fact_210_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_211_subset__trans,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C3 )
       => ( ord_le2494988322063910608_state @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_212_set__eq__subset,axiom,
    ( ( ^ [Y5: set_state,Z2: set_state] : ( Y5 = Z2 ) )
    = ( ^ [A4: set_state,B5: set_state] :
          ( ( ord_le2494988322063910608_state @ A4 @ B5 )
          & ( ord_le2494988322063910608_state @ B5 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_213_Collect__mono__iff,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_214_Collect__mono__iff,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) )
      = ( ! [X: produc9003781588162623004on_val] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_215_Collect__mono__iff,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) )
      = ( ! [X: state] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_216_core__defined_I2_J,axiom,
    ! [Phi: state] :
      ( ( get_m @ ( core @ Phi ) )
      = empty_3446695950879338768t_char ) ).

% core_defined(2)
thf(fact_217_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B3 ) @ C2 )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B3 @ C2 ) ) ) ).

% add_masks_asso
thf(fact_218_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A2: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B @ A2 ) ) ) ).

% add_masks_comm
thf(fact_219_add__masks__cancellative,axiom,
    ! [B3: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B3 @ C2 )
        = ( add_ma6586698345353345811t_char @ B3 @ D ) )
     => ( C2 = D ) ) ).

% add_masks_cancellative
thf(fact_220_valid__bin,axiom,
    ! [A: state] : ( valid_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ A ) ) @ ( get_h @ A ) ) ) ).

% valid_bin
thf(fact_221_multiply_Opelims,axiom,
    ! [X2: prat,Xa2: state,Y: state] :
      ( ( ( multiply @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X2 @ Xa2 ) )
       => ~ ( ( Y
              = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X2 @ ( get_m @ Xa2 ) ) @ ( get_h @ Xa2 ) ) ) )
           => ~ ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X2 @ Xa2 ) ) ) ) ) ).

% multiply.pelims
thf(fact_222_PartialSA_Omono__prop__set__equiv,axiom,
    ! [P: state > $o,A5: set_state,B2: set_state] :
      ( ( sep_mono_prop_state @ plus @ P )
     => ( ( sep_equiv_state @ plus @ A5 @ B2 )
       => ( ( sep_setify_state @ P @ A5 )
          = ( sep_setify_state @ P @ B2 ) ) ) ) ).

% PartialSA.mono_prop_set_equiv
thf(fact_223_PartialSA_Omono__pure__condI,axiom,
    ! [B3: state > $o] :
      ( ! [Phi5: state] :
          ( ( B3 @ Phi5 )
          = ( B3 @ ( core @ Phi5 ) ) )
     => ( ! [Phi5: state,Phi6: state,R3: state] :
            ( ( ( sep_pure_state @ plus @ R3 )
              & ( ( some_state @ Phi6 )
                = ( plus @ Phi5 @ R3 ) )
              & ~ ( B3 @ Phi5 ) )
           => ~ ( B3 @ Phi6 ) )
       => ( packag6595153354952283300_state @ plus @ core @ B3 ) ) ) ).

% PartialSA.mono_pure_condI
thf(fact_224_PartialSA_Omono__pure__cond__def,axiom,
    ! [B3: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ B3 )
      = ( ! [Phi4: state] :
            ( ( B3 @ Phi4 )
            = ( B3 @ ( core @ Phi4 ) ) )
        & ! [Phi3: state,Phi4: state,R2: state] :
            ( ( ( sep_pure_state @ plus @ R2 )
              & ( ( some_state @ Phi3 )
                = ( plus @ Phi4 @ R2 ) )
              & ~ ( B3 @ Phi4 ) )
           => ~ ( B3 @ Phi3 ) ) ) ) ).

% PartialSA.mono_pure_cond_def
thf(fact_225_PartialSA_Oequiv__up__closed__subset,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( sep_up_closed_state @ plus @ A5 )
     => ( ( sep_equiv_state @ plus @ B2 @ C3 )
       => ( ( ord_le2494988322063910608_state @ B2 @ A5 )
          = ( ord_le2494988322063910608_state @ C3 @ A5 ) ) ) ) ).

% PartialSA.equiv_up_closed_subset
thf(fact_226_get__h__m,axiom,
    ( rep_state
    = ( ^ [X: state] : ( produc74620467203698198on_val @ ( get_m @ X ) @ ( get_h @ X ) ) ) ) ).

% get_h_m
thf(fact_227_multiply_Ocases,axiom,
    ! [X2: produc4844402304972151915_state] :
      ~ ! [P4: prat,Phi5: state] :
          ( X2
         != ( produc6205719116230455971_state @ P4 @ Phi5 ) ) ).

% multiply.cases
thf(fact_228_PartialSA_Osetify__def,axiom,
    ( sep_setify_state
    = ( ^ [P5: state > $o,A4: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ( P5 @ X ) ) ) ) ).

% PartialSA.setify_def
thf(fact_229_Rep__state__inject,axiom,
    ! [X2: state,Y: state] :
      ( ( ( rep_state @ X2 )
        = ( rep_state @ Y ) )
      = ( X2 = Y ) ) ).

% Rep_state_inject
thf(fact_230_sep__algebra_Oup__closed_Ocong,axiom,
    sep_up_closed_state = sep_up_closed_state ).

% sep_algebra.up_closed.cong
thf(fact_231_package__logic_Omono__pure__cond_Ocong,axiom,
    packag6595153354952283300_state = packag6595153354952283300_state ).

% package_logic.mono_pure_cond.cong
thf(fact_232_Rep__state__inverse,axiom,
    ! [X2: state] :
      ( ( abs_state @ ( rep_state @ X2 ) )
      = X2 ) ).

% Rep_state_inverse
thf(fact_233_PartialSA_Oup__closed__def,axiom,
    ! [A5: set_state] :
      ( ( sep_up_closed_state @ plus @ A5 )
      = ( ! [Phi3: state] :
            ( ? [X: state] :
                ( ( member_state @ X @ A5 )
                & ( greater @ Phi3 @ X ) )
           => ( member_state @ Phi3 @ A5 ) ) ) ) ).

% PartialSA.up_closed_def
thf(fact_234_PartialSA_Oup__closedI,axiom,
    ! [A5: set_state] :
      ( ! [Phi6: state,Phi5: state] :
          ( ( ( greater @ Phi6 @ Phi5 )
            & ( member_state @ Phi5 @ A5 ) )
         => ( member_state @ Phi6 @ A5 ) )
     => ( sep_up_closed_state @ plus @ A5 ) ) ).

% PartialSA.up_closedI
thf(fact_235_PartialSA_Oup__closed__sum,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( sep_up_closed_state @ plus @ A5 )
     => ( sep_up_closed_state @ plus @ ( add_set @ A5 @ B2 ) ) ) ).

% PartialSA.up_closed_sum
thf(fact_236_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B3 ) @ H3 ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H3 ) ) ) ).

% valid_state_decompose
thf(fact_237_PartialSA_Oup__close__equiv,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( sep_up_closed_state @ plus @ A5 )
     => ( ( sep_up_closed_state @ plus @ B2 )
       => ( ( sep_equiv_state @ plus @ A5 @ B2 )
          = ( A5 = B2 ) ) ) ) ).

% PartialSA.up_close_equiv
thf(fact_238_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_239_subrelI,axiom,
    ! [R: set_Pr4231240917283394591_state,S: set_Pr4231240917283394591_state] :
      ( ! [X4: state,Y4: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X4 @ Y4 ) @ R )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X4 @ Y4 ) @ S ) )
     => ( ord_le4032880777312351359_state @ R @ S ) ) ).

% subrelI
thf(fact_240_subrelI,axiom,
    ! [R: set_Pr2391701600042252046_state,S: set_Pr2391701600042252046_state] :
      ( ! [X4: state,Y4: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X4 @ Y4 ) @ R )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X4 @ Y4 ) @ S ) )
     => ( ord_le6483787752022044334_state @ R @ S ) ) ).

% subrelI
thf(fact_241_subrelI,axiom,
    ! [R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ! [X4: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X4 @ Y4 ) @ R )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X4 @ Y4 ) @ S ) )
     => ( ord_le7224449410216889241on_val @ R @ S ) ) ).

% subrelI
thf(fact_242_subrelI,axiom,
    ! [R: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
      ( ! [X4: option_state,Y4: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X4 @ Y4 ) @ R )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X4 @ Y4 ) @ S ) )
     => ( ord_le6423325748750870005_state @ R @ S ) ) ).

% subrelI
thf(fact_243_subrelI,axiom,
    ! [R: set_Pr6039890582625207499_state,S: set_Pr6039890582625207499_state] :
      ( ! [X4: prat,Y4: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X4 @ Y4 ) @ R )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X4 @ Y4 ) @ S ) )
     => ( ord_le262197048489888875_state @ R @ S ) ) ).

% subrelI
thf(fact_244_subrelI,axiom,
    ! [R: set_Pr6847263425858009217_state,S: set_Pr6847263425858009217_state] :
      ( ! [X4: prat,Y4: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X4 @ Y4 ) @ R )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X4 @ Y4 ) @ S ) )
     => ( ord_le5761494056910127137_state @ R @ S ) ) ).

% subrelI
thf(fact_245_subrelI,axiom,
    ! [R: set_Pr7649655576376376530on_val,S: set_Pr7649655576376376530on_val] :
      ( ! [X4: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y4 ) @ R )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y4 ) @ S ) )
     => ( ord_le6900622908513426738on_val @ R @ S ) ) ).

% subrelI
thf(fact_246_PartialSA_Omono__prop__set,axiom,
    ! [A5: set_state,B2: set_state,P: state > $o] :
      ( ( greater_set @ A5 @ B2 )
     => ( ( sep_setify_state @ P @ B2 )
       => ( ( sep_mono_prop_state @ plus @ P )
         => ( sep_setify_state @ P @ A5 ) ) ) ) ).

% PartialSA.mono_prop_set
thf(fact_247_PartialSA_Oup__closed__bigger__subset,axiom,
    ! [B2: set_state,A5: set_state] :
      ( ( sep_up_closed_state @ plus @ B2 )
     => ( ( greater_set @ A5 @ B2 )
       => ( ord_le2494988322063910608_state @ A5 @ B2 ) ) ) ).

% PartialSA.up_closed_bigger_subset
thf(fact_248_PartialSA_Omono__pure__cond__conj,axiom,
    ! [Pc: state > $o,B3: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ Pc )
     => ( ( packag6595153354952283300_state @ plus @ core @ B3 )
       => ( packag6595153354952283300_state @ plus @ core @ ( packag1330488657391168505_state @ Pc @ B3 ) ) ) ) ).

% PartialSA.mono_pure_cond_conj
thf(fact_249_PartialSA_Oupper__closure__up__closed,axiom,
    ! [A5: set_state] : ( sep_up_closed_state @ plus @ ( sep_up1246176804924251236_state @ plus @ A5 ) ) ).

% PartialSA.upper_closure_up_closed
thf(fact_250_PartialSA_Omax__projection__prop__pure__core,axiom,
    sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).

% PartialSA.max_projection_prop_pure_core
thf(fact_251_PartialSA_Obigger__the,axiom,
    ! [A: state,X5: state,Y: state,X2: state] :
      ( ( ( some_state @ A )
        = ( plus @ X5 @ Y ) )
     => ( ( greater @ X5 @ X2 )
       => ( greater @ ( the_state @ ( plus @ ( core @ A ) @ X5 ) ) @ ( the_state @ ( plus @ ( core @ A ) @ X2 ) ) ) ) ) ).

% PartialSA.bigger_the
thf(fact_252_sep__algebra_Omax__projection__prop_Ocong,axiom,
    sep_ma8214210560313151521_state = sep_ma8214210560313151521_state ).

% sep_algebra.max_projection_prop.cong
thf(fact_253_PartialSA_Obool__conj__def,axiom,
    ( packag1330488657391168505_state
    = ( ^ [A2: state > $o,B: state > $o,X: state] :
          ( ( A2 @ X )
          & ( B @ X ) ) ) ) ).

% PartialSA.bool_conj_def
thf(fact_254_PartialSA_Olarger__set__refl,axiom,
    ! [A5: set_state] : ( greater_set @ A5 @ A5 ) ).

% PartialSA.larger_set_refl
thf(fact_255_PartialSA_Osucc__set__trans,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( greater_set @ A5 @ B2 )
     => ( ( greater_set @ B2 @ C3 )
       => ( greater_set @ A5 @ C3 ) ) ) ).

% PartialSA.succ_set_trans
thf(fact_256_sep__algebra_Oupper__closure_Ocong,axiom,
    sep_up1246176804924251236_state = sep_up1246176804924251236_state ).

% sep_algebra.upper_closure.cong
thf(fact_257_option_Osel,axiom,
    ! [X22: state] :
      ( ( the_state @ ( some_state @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_258_PartialSA_Ompp__invo,axiom,
    ! [P: state > $o,F: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( F @ ( F @ X2 ) )
        = ( F @ X2 ) ) ) ).

% PartialSA.mpp_invo
thf(fact_259_PartialSA_Ompp__prop,axiom,
    ! [P: state > $o,F: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( P @ ( F @ X2 ) ) ) ).

% PartialSA.mpp_prop
thf(fact_260_PartialSA_Ogreater__setI,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ! [A6: state] :
          ( ( member_state @ A6 @ A5 )
         => ? [X3: state] :
              ( ( member_state @ X3 @ B2 )
              & ( greater @ A6 @ X3 ) ) )
     => ( greater_set @ A5 @ B2 ) ) ).

% PartialSA.greater_setI
thf(fact_261_PartialSA_Ogreater__set__def,axiom,
    ( greater_set
    = ( ^ [A4: set_state,B5: set_state] :
        ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ? [Y3: state] :
              ( ( member_state @ Y3 @ B5 )
              & ( greater @ X @ Y3 ) ) ) ) ) ).

% PartialSA.greater_set_def
thf(fact_262_PartialSA_Obigger__set,axiom,
    ! [A8: set_state,A5: set_state,B2: set_state] :
      ( ( greater_set @ A8 @ A5 )
     => ( greater_set @ ( add_set @ A8 @ B2 ) @ ( add_set @ A5 @ B2 ) ) ) ).

% PartialSA.bigger_set
thf(fact_263_PartialSA_Osub__bigger,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( greater_set @ A5 @ B2 ) ) ).

% PartialSA.sub_bigger
thf(fact_264_PartialSA_Omax__projection__propE_I3_J,axiom,
    ! [P: state > $o,F: state > state,P2: state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( ( P @ P2 )
          & ( greater @ X2 @ P2 ) )
       => ( greater @ ( F @ X2 ) @ P2 ) ) ) ).

% PartialSA.max_projection_propE(3)
thf(fact_265_PartialSA_OmppI,axiom,
    ! [P: state > $o,F: state > state,A: state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A @ X2 )
       => ( ( P @ X2 )
         => ( ( greater @ X2 @ ( F @ A ) )
           => ( X2
              = ( F @ A ) ) ) ) ) ) ).

% PartialSA.mppI
thf(fact_266_PartialSA_Ompp__mono,axiom,
    ! [P: state > $o,F: state > state,A: state,B3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A @ B3 )
       => ( greater @ ( F @ A ) @ ( F @ B3 ) ) ) ) ).

% PartialSA.mpp_mono
thf(fact_267_PartialSA_Ompp__smaller,axiom,
    ! [P: state > $o,F: state > state,X2: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( greater @ X2 @ ( F @ X2 ) ) ) ).

% PartialSA.mpp_smaller
thf(fact_268_PartialSA_Omax__projection__propI,axiom,
    ! [F: state > state,P: state > $o] :
      ( ! [X4: state] : ( greater @ X4 @ ( F @ X4 ) )
     => ( ! [X4: state] : ( P @ ( F @ X4 ) )
       => ( ! [X4: state,P4: state] :
              ( ( ( P @ P4 )
                & ( greater @ X4 @ P4 ) )
             => ( greater @ ( F @ X4 ) @ P4 ) )
         => ( sep_ma8214210560313151521_state @ plus @ P @ F ) ) ) ) ).

% PartialSA.max_projection_propI
thf(fact_269_PartialSA_Omax__projection__prop__def,axiom,
    ! [P: state > $o,F: state > state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
      = ( ! [X: state] :
            ( ( greater @ X @ ( F @ X ) )
            & ( P @ ( F @ X ) )
            & ! [P3: state] :
                ( ( ( P @ P3 )
                  & ( greater @ X @ P3 ) )
               => ( greater @ ( F @ X ) @ P3 ) ) ) ) ) ).

% PartialSA.max_projection_prop_def
thf(fact_270_PartialSA_Ompp__compatible,axiom,
    ! [P: state > $o,F: state > state,A: state,B3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( defined @ A @ B3 )
       => ( defined @ ( F @ A ) @ ( F @ B3 ) ) ) ) ).

% PartialSA.mpp_compatible
thf(fact_271_PartialSA_OequivI,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( greater_set @ A5 @ B2 )
     => ( ( greater_set @ B2 @ A5 )
       => ( sep_equiv_state @ plus @ A5 @ B2 ) ) ) ).

% PartialSA.equivI
thf(fact_272_PartialSA_Oequiv__def,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( sep_equiv_state @ plus @ A5 @ B2 )
      = ( ( greater_set @ A5 @ B2 )
        & ( greater_set @ B2 @ A5 ) ) ) ).

% PartialSA.equiv_def
thf(fact_273_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_274_multiply__sem__assertion_Oelims,axiom,
    ! [X2: prat,Xa2: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X2 @ Xa2 )
        = Y )
     => ( Y
        = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X2 ) @ Xa2 ) ) ) ) ).

% multiply_sem_assertion.elims
thf(fact_275_multiply__sem__assertion_Osimps,axiom,
    ( multip8064567061438756306ertion
    = ( ^ [P3: prat,P5: set_state] : ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ P3 ) @ P5 ) ) ) ) ).

% multiply_sem_assertion.simps
thf(fact_276_same__fstI,axiom,
    ! [P: prat > $o,X2: prat,Y6: state,Y: state,R4: prat > set_Pr1785066336555260981_state] :
      ( ( P @ X2 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ Y6 @ Y ) @ ( R4 @ X2 ) )
       => ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ X2 @ Y6 ) @ ( produc6205719116230455971_state @ X2 @ Y ) ) @ ( same_fst_prat_state @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_277_same__fstI,axiom,
    ! [P: prat > $o,X2: prat,Y6: set_state,Y: set_state,R4: prat > set_Pr5618112136499017889_state] :
      ( ( P @ X2 )
     => ( ( member1976142906092959210_state @ ( produc1834047914073684089_state @ Y6 @ Y ) @ ( R4 @ X2 ) )
       => ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ X2 @ Y6 ) @ ( produc9080284787853255769_state @ X2 @ Y ) ) @ ( same_f5333510722715586990_state @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_278_same__fstI,axiom,
    ! [P: ( produc1457211279475724562t_char > prat ) > $o,X2: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R4: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P @ X2 )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y6 @ Y ) @ ( R4 @ X2 ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X2 @ Y6 ) @ ( produc74620467203698198on_val @ X2 @ Y ) ) @ ( same_f8138205353876899841on_val @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_279_same__fstI,axiom,
    ! [P: produc9003781588162623004on_val > $o,X2: produc9003781588162623004on_val,Y6: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R4: produc9003781588162623004on_val > set_Pr4305388979797279737on_val] :
      ( ( P @ X2 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ Y ) @ ( R4 @ X2 ) )
       => ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ X2 @ Y6 ) @ ( produc4958730763592889809on_val @ X2 @ Y ) ) @ ( same_f7521981028318775590on_val @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_280_same__fstI,axiom,
    ! [P: option_state > $o,X2: option_state,Y6: option_state,Y: option_state,R4: option_state > set_Pr1688445902015331925_state] :
      ( ( P @ X2 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y6 @ Y ) @ ( R4 @ X2 ) )
       => ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ X2 @ Y6 ) @ ( produc9160152616361873709_state @ X2 @ Y ) ) @ ( same_f2704544105346032002_state @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_281_PartialSA_Oup__closed__plus__UNIV,axiom,
    ! [A5: set_state] : ( sep_up_closed_state @ plus @ ( add_set @ A5 @ top_top_set_state ) ) ).

% PartialSA.up_closed_plus_UNIV
thf(fact_282_relChain__def,axiom,
    ( bNF_Ca2927435689774128306_state
    = ( ^ [R2: set_Pr4305388979797279737on_val,As: produc9003781588162623004on_val > set_state] :
        ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R2 )
         => ( ord_le2494988322063910608_state @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_283_relChain__def,axiom,
    ( bNF_Ca2815325664443363616_state
    = ( ^ [R2: set_Pr1688445902015331925_state,As: option_state > set_state] :
        ! [I: option_state,J: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ I @ J ) @ R2 )
         => ( ord_le2494988322063910608_state @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_284_unit__charact_I1_J,axiom,
    ( ( get_h @ unit )
    = empty_7512026568161686585ar_val ) ).

% unit_charact(1)
thf(fact_285_add__masks__equiv__valid__null,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > prat] :
      ( ( valid_null @ ( add_ma6586698345353345811t_char @ A @ B3 ) )
      = ( ( valid_null @ A )
        & ( valid_null @ B3 ) ) ) ).

% add_masks_equiv_valid_null
thf(fact_286_image__eqI,axiom,
    ! [B3: state,F: state > state,X2: state,A5: set_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_state @ X2 @ A5 )
       => ( member_state @ B3 @ ( image_state_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_287_image__eqI,axiom,
    ! [B3: produc3882902684156697774_state,F: state > produc3882902684156697774_state,X2: state,A5: set_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_state @ X2 @ A5 )
       => ( member6715010751053357271_state @ B3 @ ( image_5311902671526399277_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_288_image__eqI,axiom,
    ! [B3: state,F: produc3882902684156697774_state > state,X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member6715010751053357271_state @ X2 @ A5 )
       => ( member_state @ B3 @ ( image_6759111139246981411_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_289_image__eqI,axiom,
    ! [B3: produc7272280185086996969_state,F: state > produc7272280185086996969_state,X2: state,A5: set_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_state @ X2 @ A5 )
       => ( member7461423308196221312_state @ B3 @ ( image_632992669292831658_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_290_image__eqI,axiom,
    ! [B3: state,F: produc7272280185086996969_state > state,X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member7461423308196221312_state @ X2 @ A5 )
       => ( member_state @ B3 @ ( image_2710133271811360426_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_291_image__eqI,axiom,
    ! [B3: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member6715010751053357271_state @ X2 @ A5 )
       => ( member6715010751053357271_state @ B3 @ ( image_508898238547643525_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_292_image__eqI,axiom,
    ! [B3: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member7461423308196221312_state @ X2 @ A5 )
       => ( member6715010751053357271_state @ B3 @ ( image_7466395622266389182_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_293_image__eqI,axiom,
    ! [B3: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member6715010751053357271_state @ X2 @ A5 )
       => ( member7461423308196221312_state @ B3 @ ( image_4364510454419742290_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_294_image__eqI,axiom,
    ! [B3: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member7461423308196221312_state @ X2 @ A5 )
       => ( member7461423308196221312_state @ B3 @ ( image_3064086290059306969_state @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_295_image__eqI,axiom,
    ! [B3: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val,X2: state,A5: set_state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( ( member_state @ X2 @ A5 )
       => ( member2879030624002546611on_val @ B3 @ ( image_4772210114854293981on_val @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_296_UNIV__I,axiom,
    ! [X2: produc7272280185086996969_state] : ( member7461423308196221312_state @ X2 @ top_to174962084833059535_state ) ).

% UNIV_I
thf(fact_297_UNIV__I,axiom,
    ! [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ top_to1561438921946393474on_val ) ).

% UNIV_I
thf(fact_298_UNIV__I,axiom,
    ! [X2: produc3882902684156697774_state] : ( member6715010751053357271_state @ X2 @ top_to1744471583079920734_state ) ).

% UNIV_I
thf(fact_299_UNIV__I,axiom,
    ! [X2: state] : ( member_state @ X2 @ top_top_set_state ) ).

% UNIV_I
thf(fact_300_imageI,axiom,
    ! [X2: state,A5: set_state,F: state > state] :
      ( ( member_state @ X2 @ A5 )
     => ( member_state @ ( F @ X2 ) @ ( image_state_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_301_imageI,axiom,
    ! [X2: state,A5: set_state,F: state > produc3882902684156697774_state] :
      ( ( member_state @ X2 @ A5 )
     => ( member6715010751053357271_state @ ( F @ X2 ) @ ( image_5311902671526399277_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_302_imageI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( member_state @ ( F @ X2 ) @ ( image_6759111139246981411_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_303_imageI,axiom,
    ! [X2: state,A5: set_state,F: state > produc7272280185086996969_state] :
      ( ( member_state @ X2 @ A5 )
     => ( member7461423308196221312_state @ ( F @ X2 ) @ ( image_632992669292831658_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_304_imageI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( member_state @ ( F @ X2 ) @ ( image_2710133271811360426_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_305_imageI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( member6715010751053357271_state @ ( F @ X2 ) @ ( image_508898238547643525_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_306_imageI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( member6715010751053357271_state @ ( F @ X2 ) @ ( image_7466395622266389182_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_307_imageI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( member7461423308196221312_state @ ( F @ X2 ) @ ( image_4364510454419742290_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_308_imageI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( member7461423308196221312_state @ ( F @ X2 ) @ ( image_3064086290059306969_state @ F @ A5 ) ) ) ).

% imageI
thf(fact_309_imageI,axiom,
    ! [X2: state,A5: set_state,F: state > produc9003781588162623004on_val] :
      ( ( member_state @ X2 @ A5 )
     => ( member2879030624002546611on_val @ ( F @ X2 ) @ ( image_4772210114854293981on_val @ F @ A5 ) ) ) ).

% imageI
thf(fact_310_rangeI,axiom,
    ! [F: state > state,X2: state] : ( member_state @ ( F @ X2 ) @ ( image_state_state @ F @ top_top_set_state ) ) ).

% rangeI
thf(fact_311_rangeI,axiom,
    ! [F: state > produc7272280185086996969_state,X2: state] : ( member7461423308196221312_state @ ( F @ X2 ) @ ( image_632992669292831658_state @ F @ top_top_set_state ) ) ).

% rangeI
thf(fact_312_rangeI,axiom,
    ! [F: state > produc9003781588162623004on_val,X2: state] : ( member2879030624002546611on_val @ ( F @ X2 ) @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) ) ).

% rangeI
thf(fact_313_rangeI,axiom,
    ! [F: state > produc3882902684156697774_state,X2: state] : ( member6715010751053357271_state @ ( F @ X2 ) @ ( image_5311902671526399277_state @ F @ top_top_set_state ) ) ).

% rangeI
thf(fact_314_UNIV__eq__I,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ! [X4: produc7272280185086996969_state] : ( member7461423308196221312_state @ X4 @ A5 )
     => ( top_to174962084833059535_state = A5 ) ) ).

% UNIV_eq_I
thf(fact_315_UNIV__eq__I,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ! [X4: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X4 @ A5 )
     => ( top_to1561438921946393474on_val = A5 ) ) ).

% UNIV_eq_I
thf(fact_316_UNIV__eq__I,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ! [X4: produc3882902684156697774_state] : ( member6715010751053357271_state @ X4 @ A5 )
     => ( top_to1744471583079920734_state = A5 ) ) ).

% UNIV_eq_I
thf(fact_317_UNIV__eq__I,axiom,
    ! [A5: set_state] :
      ( ! [X4: state] : ( member_state @ X4 @ A5 )
     => ( top_top_set_state = A5 ) ) ).

% UNIV_eq_I
thf(fact_318_image__iff,axiom,
    ! [Z: state,F: state > state,A5: set_state] :
      ( ( member_state @ Z @ ( image_state_state @ F @ A5 ) )
      = ( ? [X: state] :
            ( ( member_state @ X @ A5 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_319_range__eqI,axiom,
    ! [B3: state,F: state > state,X2: state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( member_state @ B3 @ ( image_state_state @ F @ top_top_set_state ) ) ) ).

% range_eqI
thf(fact_320_range__eqI,axiom,
    ! [B3: produc7272280185086996969_state,F: state > produc7272280185086996969_state,X2: state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( member7461423308196221312_state @ B3 @ ( image_632992669292831658_state @ F @ top_top_set_state ) ) ) ).

% range_eqI
thf(fact_321_range__eqI,axiom,
    ! [B3: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val,X2: state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( member2879030624002546611on_val @ B3 @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) ) ) ).

% range_eqI
thf(fact_322_range__eqI,axiom,
    ! [B3: produc3882902684156697774_state,F: state > produc3882902684156697774_state,X2: state] :
      ( ( B3
        = ( F @ X2 ) )
     => ( member6715010751053357271_state @ B3 @ ( image_5311902671526399277_state @ F @ top_top_set_state ) ) ) ).

% range_eqI
thf(fact_323_bex__imageD,axiom,
    ! [F: state > state,A5: set_state,P: state > $o] :
      ( ? [X3: state] :
          ( ( member_state @ X3 @ ( image_state_state @ F @ A5 ) )
          & ( P @ X3 ) )
     => ? [X4: state] :
          ( ( member_state @ X4 @ A5 )
          & ( P @ ( F @ X4 ) ) ) ) ).

% bex_imageD
thf(fact_324_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_325_ball__imageD,axiom,
    ! [F: state > state,A5: set_state,P: state > $o] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ ( image_state_state @ F @ A5 ) )
         => ( P @ X4 ) )
     => ! [X3: state] :
          ( ( member_state @ X3 @ A5 )
         => ( P @ ( F @ X3 ) ) ) ) ).

% ball_imageD
thf(fact_326_UNIV__witness,axiom,
    ? [X4: produc7272280185086996969_state] : ( member7461423308196221312_state @ X4 @ top_to174962084833059535_state ) ).

% UNIV_witness
thf(fact_327_UNIV__witness,axiom,
    ? [X4: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X4 @ top_to1561438921946393474on_val ) ).

% UNIV_witness
thf(fact_328_UNIV__witness,axiom,
    ? [X4: produc3882902684156697774_state] : ( member6715010751053357271_state @ X4 @ top_to1744471583079920734_state ) ).

% UNIV_witness
thf(fact_329_UNIV__witness,axiom,
    ? [X4: state] : ( member_state @ X4 @ top_top_set_state ) ).

% UNIV_witness
thf(fact_330_rev__image__eqI,axiom,
    ! [X2: state,A5: set_state,B3: state,F: state > state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_state @ B3 @ ( image_state_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_331_rev__image__eqI,axiom,
    ! [X2: state,A5: set_state,B3: produc3882902684156697774_state,F: state > produc3882902684156697774_state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member6715010751053357271_state @ B3 @ ( image_5311902671526399277_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_332_rev__image__eqI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B3: state,F: produc3882902684156697774_state > state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_state @ B3 @ ( image_6759111139246981411_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_333_rev__image__eqI,axiom,
    ! [X2: state,A5: set_state,B3: produc7272280185086996969_state,F: state > produc7272280185086996969_state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member7461423308196221312_state @ B3 @ ( image_632992669292831658_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_334_rev__image__eqI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B3: state,F: produc7272280185086996969_state > state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member_state @ B3 @ ( image_2710133271811360426_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_335_rev__image__eqI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member6715010751053357271_state @ B3 @ ( image_508898238547643525_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_336_rev__image__eqI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B3: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member6715010751053357271_state @ B3 @ ( image_7466395622266389182_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_337_rev__image__eqI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B3: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member7461423308196221312_state @ B3 @ ( image_4364510454419742290_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member7461423308196221312_state @ B3 @ ( image_3064086290059306969_state @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_339_rev__image__eqI,axiom,
    ! [X2: state,A5: set_state,B3: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val] :
      ( ( member_state @ X2 @ A5 )
     => ( ( B3
          = ( F @ X2 ) )
       => ( member2879030624002546611on_val @ B3 @ ( image_4772210114854293981on_val @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_340_range__subsetD,axiom,
    ! [F: state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,I2: state] :
      ( ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ top_top_set_state ) @ B2 )
     => ( member7461423308196221312_state @ ( F @ I2 ) @ B2 ) ) ).

% range_subsetD
thf(fact_341_range__subsetD,axiom,
    ! [F: state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,I2: state] :
      ( ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) @ B2 )
     => ( member2879030624002546611on_val @ ( F @ I2 ) @ B2 ) ) ).

% range_subsetD
thf(fact_342_range__subsetD,axiom,
    ! [F: state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,I2: state] :
      ( ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ top_top_set_state ) @ B2 )
     => ( member6715010751053357271_state @ ( F @ I2 ) @ B2 ) ) ).

% range_subsetD
thf(fact_343_range__subsetD,axiom,
    ! [F: state > state,B2: set_state,I2: state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ top_top_set_state ) @ B2 )
     => ( member_state @ ( F @ I2 ) @ B2 ) ) ).

% range_subsetD
thf(fact_344_top_Oextremum__uniqueI,axiom,
    ! [A: set_state] :
      ( ( ord_le2494988322063910608_state @ top_top_set_state @ A )
     => ( A = top_top_set_state ) ) ).

% top.extremum_uniqueI
thf(fact_345_top_Oextremum__unique,axiom,
    ! [A: set_state] :
      ( ( ord_le2494988322063910608_state @ top_top_set_state @ A )
      = ( A = top_top_set_state ) ) ).

% top.extremum_unique
thf(fact_346_top__greatest,axiom,
    ! [A: set_state] : ( ord_le2494988322063910608_state @ A @ top_top_set_state ) ).

% top_greatest
thf(fact_347_subset__image__iff,axiom,
    ! [B2: set_state,F: state > state,A5: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A5 ) )
      = ( ? [AA: set_state] :
            ( ( ord_le2494988322063910608_state @ AA @ A5 )
            & ( B2
              = ( image_state_state @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_348_image__subset__iff,axiom,
    ! [F: state > state,A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A5 ) @ B2 )
      = ( ! [X: state] :
            ( ( member_state @ X @ A5 )
           => ( member_state @ ( F @ X ) @ B2 ) ) ) ) ).

% image_subset_iff
thf(fact_349_subset__imageE,axiom,
    ! [B2: set_state,F: state > state,A5: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A5 ) )
     => ~ ! [C5: set_state] :
            ( ( ord_le2494988322063910608_state @ C5 @ A5 )
           => ( B2
             != ( image_state_state @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_350_image__subsetI,axiom,
    ! [A5: set_state,F: state > state,B2: set_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A5 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_351_image__subsetI,axiom,
    ! [A5: set_state,F: state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A5 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_352_image__subsetI,axiom,
    ! [A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state,B2: set_state] :
      ( ! [X4: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X4 @ A5 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_6759111139246981411_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_353_image__subsetI,axiom,
    ! [A5: set_state,F: state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A5 )
         => ( member7461423308196221312_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_354_image__subsetI,axiom,
    ! [A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state,B2: set_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X4 @ A5 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_355_image__subsetI,axiom,
    ! [A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X4 @ A5 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_508898238547643525_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_356_image__subsetI,axiom,
    ! [A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X4 @ A5 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_357_image__subsetI,axiom,
    ! [A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X4 @ A5 )
         => ( member7461423308196221312_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_4364510454419742290_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_358_image__subsetI,axiom,
    ! [A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X4 @ A5 )
         => ( member7461423308196221312_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_359_image__subsetI,axiom,
    ! [A5: set_state,F: state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X4: state] :
          ( ( member_state @ X4 @ A5 )
         => ( member2879030624002546611on_val @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ A5 ) @ B2 ) ) ).

% image_subsetI
thf(fact_360_image__mono,axiom,
    ! [A5: set_state,B2: set_state,F: state > state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A5 ) @ ( image_state_state @ F @ B2 ) ) ) ).

% image_mono
thf(fact_361_subset__UNIV,axiom,
    ! [A5: set_state] : ( ord_le2494988322063910608_state @ A5 @ top_top_set_state ) ).

% subset_UNIV
thf(fact_362_PartialSA_Ounit__core,axiom,
    ( ( core @ unit )
    = unit ) ).

% PartialSA.unit_core
thf(fact_363_PartialSA_Ounit__smaller,axiom,
    ! [Phi: state] : ( greater @ Phi @ unit ) ).

% PartialSA.unit_smaller
thf(fact_364_unit__neutral,axiom,
    ( some_state
    = ( ^ [A2: state] : ( plus @ A2 @ unit ) ) ) ).

% unit_neutral
thf(fact_365_unit__charact_I2_J,axiom,
    ( ( get_m @ unit )
    = empty_3446695950879338768t_char ) ).

% unit_charact(2)
thf(fact_366_all__subset__image,axiom,
    ! [F: state > state,A5: set_state,P: set_state > $o] :
      ( ( ! [B5: set_state] :
            ( ( ord_le2494988322063910608_state @ B5 @ ( image_state_state @ F @ A5 ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_state] :
            ( ( ord_le2494988322063910608_state @ B5 @ A5 )
           => ( P @ ( image_state_state @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_367_multiply__sem__assertion_Opelims,axiom,
    ! [X2: prat,Xa2: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X2 @ Xa2 ) )
       => ~ ( ( Y
              = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X2 ) @ Xa2 ) ) )
           => ~ ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X2 @ Xa2 ) ) ) ) ) ).

% multiply_sem_assertion.pelims
thf(fact_368_PartialSA_Omono__transformer__def,axiom,
    ! [T2: state > state] :
      ( ( packag4817583915586052229_state @ plus @ unit @ T2 )
      = ( ! [Phi4: state,Phi3: state] :
            ( ( greater @ Phi3 @ Phi4 )
           => ( greater @ ( T2 @ Phi3 ) @ ( T2 @ Phi4 ) ) )
        & ( ( T2 @ unit )
          = unit ) ) ) ).

% PartialSA.mono_transformer_def
thf(fact_369_PartialSA_Ovalid__package__setI,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state] :
      ( ! [A6: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( defined @ A6 @ U )
            & ( greater @ ( core @ A6 ) @ ( core @ U ) )
            & ( packag4817583915586052229_state @ plus @ unit @ T3 )
            & ( greater @ A6 @ ( core @ ( T3 @ F ) ) ) ) )
     => ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F ) ) ).

% PartialSA.valid_package_setI
thf(fact_370_minus__def,axiom,
    ( minus
    = ( sep_minus_state @ plus @ core ) ) ).

% minus_def
thf(fact_371_PartialSA_Osplus_Oelims,axiom,
    ! [X2: option_state,Xa2: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X2 @ Xa2 )
        = Y )
     => ( ( ( X2 = none_state )
         => ( Y != none_state ) )
       => ( ( ? [V: state] :
                ( X2
                = ( some_state @ V ) )
           => ( ( Xa2 = none_state )
             => ( Y != none_state ) ) )
         => ~ ! [A6: state] :
                ( ( X2
                  = ( some_state @ A6 ) )
               => ! [B7: state] :
                    ( ( Xa2
                      = ( some_state @ B7 ) )
                   => ( Y
                     != ( plus @ A6 @ B7 ) ) ) ) ) ) ) ).

% PartialSA.splus.elims
thf(fact_372_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_373_not__None__eq,axiom,
    ! [X2: option_state] :
      ( ( X2 != none_state )
      = ( ? [Y3: state] :
            ( X2
            = ( some_state @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_374_not__Some__eq,axiom,
    ! [X2: option_state] :
      ( ( ! [Y3: state] :
            ( X2
           != ( some_state @ Y3 ) ) )
      = ( X2 = none_state ) ) ).

% not_Some_eq
thf(fact_375_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_376_None__notin__image__Some,axiom,
    ! [A5: set_state] :
      ~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A5 ) ) ).

% None_notin_image_Some
thf(fact_377_multiply__sem__assertion_Ocases,axiom,
    ! [X2: produc6251377953716854561_state] :
      ~ ! [P4: prat,P6: set_state] :
          ( X2
         != ( produc9080284787853255769_state @ P4 @ P6 ) ) ).

% multiply_sem_assertion.cases
thf(fact_378_package__logic_Ovalid__package__set_Ocong,axiom,
    packag3877571396724326311_state = packag3877571396724326311_state ).

% package_logic.valid_package_set.cong
thf(fact_379_package__logic_Omono__transformer_Ocong,axiom,
    packag4817583915586052229_state = packag4817583915586052229_state ).

% package_logic.mono_transformer.cong
thf(fact_380_sep__algebra_Ominus_Ocong,axiom,
    sep_minus_state = sep_minus_state ).

% sep_algebra.minus.cong
thf(fact_381_notin__range__Some,axiom,
    ! [X2: option_state] :
      ( ( ~ ( member_option_state @ X2 @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) )
      = ( X2 = none_state ) ) ).

% notin_range_Some
thf(fact_382_top__set__def,axiom,
    ( top_to174962084833059535_state
    = ( collec906845016635587262_state @ top_to3660658859742706806tate_o ) ) ).

% top_set_def
thf(fact_383_top__set__def,axiom,
    ( top_to1561438921946393474on_val
    = ( collec7475294502966882545on_val @ top_to9203468014822947715_val_o ) ) ).

% top_set_def
thf(fact_384_top__set__def,axiom,
    ( top_top_set_state
    = ( collect_state @ top_top_state_o ) ) ).

% top_set_def
thf(fact_385_compatible__options_Ocases,axiom,
    ! [X2: produc3142500478612311029_state] :
      ( ! [A6: state,B7: state] :
          ( X2
         != ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) )
     => ( ! [Uv: option_state] :
            ( X2
           != ( produc9160152616361873709_state @ none_state @ Uv ) )
       => ~ ! [Uu: option_state] :
              ( X2
             != ( produc9160152616361873709_state @ Uu @ none_state ) ) ) ) ).

% compatible_options.cases
thf(fact_386_option_Odistinct_I1_J,axiom,
    ! [X22: state] :
      ( none_state
     != ( some_state @ X22 ) ) ).

% option.distinct(1)
thf(fact_387_option_OdiscI,axiom,
    ! [Option: option_state,X22: state] :
      ( ( Option
        = ( some_state @ X22 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_388_option_Oexhaust,axiom,
    ! [Y: option_state] :
      ( ( Y != none_state )
     => ~ ! [X23: state] :
            ( Y
           != ( some_state @ X23 ) ) ) ).

% option.exhaust
thf(fact_389_split__option__ex,axiom,
    ( ( ^ [P7: option_state > $o] :
        ? [X6: option_state] : ( P7 @ X6 ) )
    = ( ^ [P5: option_state > $o] :
          ( ( P5 @ none_state )
          | ? [X: state] : ( P5 @ ( some_state @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_390_split__option__all,axiom,
    ( ( ^ [P7: option_state > $o] :
        ! [X6: option_state] : ( P7 @ X6 ) )
    = ( ^ [P5: option_state > $o] :
          ( ( P5 @ none_state )
          & ! [X: state] : ( P5 @ ( some_state @ X ) ) ) ) ) ).

% split_option_all
thf(fact_391_combine__options__cases,axiom,
    ! [X2: option_state,P: option_state > option_state > $o,Y: option_state] :
      ( ( ( X2 = none_state )
       => ( P @ X2 @ Y ) )
     => ( ( ( Y = none_state )
         => ( P @ X2 @ Y ) )
       => ( ! [A6: state,B7: state] :
              ( ( X2
                = ( some_state @ A6 ) )
             => ( ( Y
                  = ( some_state @ B7 ) )
               => ( P @ X2 @ Y ) ) )
         => ( P @ X2 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_392_PartialSA_Osplus_Ocases,axiom,
    ! [X2: produc3142500478612311029_state] :
      ( ! [Uu: option_state] :
          ( X2
         != ( produc9160152616361873709_state @ none_state @ Uu ) )
     => ( ! [V: state] :
            ( X2
           != ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) )
       => ~ ! [A6: state,B7: state] :
              ( X2
             != ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) ) ) ) ).

% PartialSA.splus.cases
thf(fact_393_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_394_empty__heap__def,axiom,
    ( empty_7512026568161686585ar_val
    = ( ^ [Hl: produc1457211279475724562t_char] : none_val ) ) ).

% empty_heap_def
thf(fact_395_PartialSA_Oasso3,axiom,
    ! [A: state,B3: state,C2: state,Bc: state] :
      ( ( ( plus @ A @ B3 )
        = none_state )
     => ( ( ( plus @ B3 @ C2 )
          = ( some_state @ Bc ) )
       => ( ( plus @ A @ Bc )
          = none_state ) ) ) ).

% PartialSA.asso3
thf(fact_396_asso2,axiom,
    ! [A: state,B3: state,Ab: state,C2: state] :
      ( ( ( ( plus @ A @ B3 )
          = ( some_state @ Ab ) )
        & ( ( plus @ B3 @ C2 )
          = none_state ) )
     => ( ( plus @ Ab @ C2 )
        = none_state ) ) ).

% asso2
thf(fact_397_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_398_PartialSA_Odefined__def,axiom,
    ( defined
    = ( ^ [A2: state,B: state] :
          ( ( plus @ A2 @ B )
         != none_state ) ) ) ).

% PartialSA.defined_def
thf(fact_399_PartialSA_Osplus_Osimps_I1_J,axiom,
    ! [Uu2: option_state] :
      ( ( sep_splus_state @ plus @ none_state @ Uu2 )
      = none_state ) ).

% PartialSA.splus.simps(1)
thf(fact_400_plus__def,axiom,
    ( plus
    = ( ^ [A2: state,B: state] : ( if_option_state @ ( compatible @ ( rep_state @ A2 ) @ ( rep_state @ B ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A2 ) @ ( rep_state @ B ) ) ) ) @ none_state ) ) ) ).

% plus_def
thf(fact_401_compatible__options_Oelims_I2_J,axiom,
    ! [X2: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X2 @ Xa2 )
     => ( ! [A6: state] :
            ( ( X2
              = ( some_state @ A6 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( A6 != B7 ) ) )
       => ( ( X2 != none_state )
         => ( Xa2 = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_402_compatible__options_Oelims_I1_J,axiom,
    ! [X2: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X2 @ Xa2 )
        = Y )
     => ( ! [A6: state] :
            ( ( X2
              = ( some_state @ A6 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( Y
                  = ( A6 != B7 ) ) ) )
       => ( ( ( X2 = none_state )
           => ~ Y )
         => ~ ( ( Xa2 = none_state )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_403_PartialSA_Opackage__rhs__connectionI,axiom,
    ! [Phi: state,F: state,Phi2: state,F3: state,S2: set_Pr4231240917283394591_state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state] :
      ( ( ( plus @ Phi @ F )
        = ( plus @ Phi2 @ F3 ) )
     => ( ( stable @ F3 )
       => ( ( defined @ Phi @ F )
         => ( ( greater @ F3 @ F )
           => ( ! [A6: state,U: state,T3: state > state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
                 => ? [Au: state] :
                      ( ( ( some_state @ Au )
                        = ( plus @ A6 @ U ) )
                      & ( ( defined @ Au @ ( minus @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                       => ? [A9: state,U2: state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A9 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                            & ( greater @ ( core @ A9 ) @ ( core @ A6 ) )
                            & ( ( plus @ Au @ ( minus @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                              = ( plus @ A9 @ U2 ) )
                            & ( greater @ U2 @ U )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A5 @ A9 @ U2 @ U ) ) ) ) )
             => ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 ) ) ) ) ) ) ).

% PartialSA.package_rhs_connectionI
thf(fact_404_PartialSA_Opackage__rhs__connection__instantiate,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,A: state,U3: state,T2: state > state] :
      ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
     => ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ ( produc8632743443909064678_state @ U3 @ T2 ) ) @ S2 )
       => ~ ! [Au2: state] :
              ( ( ( some_state @ Au2 )
                = ( plus @ A @ U3 ) )
             => ~ ( ( defined @ Au2 @ ( minus @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                 => ? [A10: state,U4: state] :
                      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A10 @ ( produc8632743443909064678_state @ U4 @ T2 ) ) @ S3 )
                      & ( greater @ ( core @ A10 ) @ ( core @ A ) )
                      & ( ( plus @ Au2 @ ( minus @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                        = ( plus @ A10 @ U4 ) )
                      & ( greater @ U4 @ U3 )
                      & ( packag8070727096299822599_state @ plus @ core @ Pc @ A5 @ A10 @ U4 @ U3 ) ) ) ) ) ) ).

% PartialSA.package_rhs_connection_instantiate
thf(fact_405_package__logic_Opackage__sat_Ocong,axiom,
    packag8070727096299822599_state = packag8070727096299822599_state ).

% package_logic.package_sat.cong
thf(fact_406_package__logic_Opackage__rhs__connection_Ocong,axiom,
    packag2054414653763826108_state = packag2054414653763826108_state ).

% package_logic.package_rhs_connection.cong
thf(fact_407_stable__unit,axiom,
    stable @ unit ).

% stable_unit
thf(fact_408_compatible__options_Osimps_I1_J,axiom,
    ! [A: state,B3: state] :
      ( ( compat2278460363914054422_state @ ( some_state @ A ) @ ( some_state @ B3 ) )
      = ( A = B3 ) ) ).

% compatible_options.simps(1)
thf(fact_409_compatible__options_Oelims_I3_J,axiom,
    ! [X2: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X2 @ Xa2 )
     => ~ ! [A6: state] :
            ( ( X2
              = ( some_state @ A6 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( A6 = B7 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_410_stable__sum,axiom,
    ! [A: state,B3: state,X2: state] :
      ( ( stable @ A )
     => ( ( stable @ B3 )
       => ( ( ( some_state @ X2 )
            = ( plus @ A @ B3 ) )
         => ( stable @ X2 ) ) ) ) ).

% stable_sum
thf(fact_411_compatible__options_Osimps_I3_J,axiom,
    ! [Uu2: option_state] : ( compat2278460363914054422_state @ Uu2 @ none_state ) ).

% compatible_options.simps(3)
thf(fact_412_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_state] : ( compat2278460363914054422_state @ none_state @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_413_these__image__Some__eq,axiom,
    ! [A5: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A5 ) )
      = A5 ) ).

% these_image_Some_eq
thf(fact_414_PartialSA_Opackage__satI,axiom,
    ! [Pc: state > $o,A7: state,U5: state,U3: state,A5: aassertion_state] :
      ( ( ( Pc @ ( core @ A7 ) )
       => ? [X3: state] :
            ( ( ( some_state @ X3 )
              = ( plus @ ( core @ A7 ) @ ( minus @ U5 @ U3 ) ) )
            & ( package_sat_state @ plus @ A5 @ X3 ) ) )
     => ( packag8070727096299822599_state @ plus @ core @ Pc @ A5 @ A7 @ U5 @ U3 ) ) ).

% PartialSA.package_satI
thf(fact_415_PartialSA_Opackage__sat__def,axiom,
    ! [Pc: state > $o,A5: aassertion_state,A7: state,U5: state,U3: state] :
      ( ( packag8070727096299822599_state @ plus @ core @ Pc @ A5 @ A7 @ U5 @ U3 )
      = ( ( Pc @ ( core @ A7 ) )
       => ? [X: state] :
            ( ( ( some_state @ X )
              = ( plus @ ( core @ A7 ) @ ( minus @ U5 @ U3 ) ) )
            & ( package_sat_state @ plus @ A5 @ X ) ) ) ) ).

% PartialSA.package_sat_def
thf(fact_416_PartialSA_Opackage__rhs__proof,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
       => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( stable @ F )
             => ( ( defined @ Phi @ F )
               => ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
                  & ( packag3877571396724326311_state @ plus @ core @ unit @ S3 @ F3 ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs_proof
thf(fact_417_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_418_add__states_Oelims,axiom,
    ! [X2: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X2 @ Xa2 )
        = Y )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X2
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ! [Pi3: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
                ( ( Xa2
                  = ( produc74620467203698198on_val @ Pi3 @ H2 ) )
               => ( Y
                 != ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi3 ) @ ( map_ad2908576761495157028ar_val @ H @ H2 ) ) ) ) ) ) ).

% add_states.elims
thf(fact_419_insert__absorb2,axiom,
    ! [X2: state,A5: set_state] :
      ( ( insert_state @ X2 @ ( insert_state @ X2 @ A5 ) )
      = ( insert_state @ X2 @ A5 ) ) ).

% insert_absorb2
thf(fact_420_insert__iff,axiom,
    ! [A: state,B3: state,A5: set_state] :
      ( ( member_state @ A @ ( insert_state @ B3 @ A5 ) )
      = ( ( A = B3 )
        | ( member_state @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_421_insert__iff,axiom,
    ! [A: produc7272280185086996969_state,B3: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ B3 @ A5 ) )
      = ( ( A = B3 )
        | ( member7461423308196221312_state @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_422_insert__iff,axiom,
    ! [A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ B3 @ A5 ) )
      = ( ( A = B3 )
        | ( member2879030624002546611on_val @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_423_insert__iff,axiom,
    ! [A: produc3882902684156697774_state,B3: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ B3 @ A5 ) )
      = ( ( A = B3 )
        | ( member6715010751053357271_state @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_424_insertCI,axiom,
    ! [A: state,B2: set_state,B3: state] :
      ( ( ~ ( member_state @ A @ B2 )
       => ( A = B3 ) )
     => ( member_state @ A @ ( insert_state @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_425_insertCI,axiom,
    ! [A: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state] :
      ( ( ~ ( member7461423308196221312_state @ A @ B2 )
       => ( A = B3 ) )
     => ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_426_insertCI,axiom,
    ! [A: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,B3: produc9003781588162623004on_val] :
      ( ( ~ ( member2879030624002546611on_val @ A @ B2 )
       => ( A = B3 ) )
     => ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_427_insertCI,axiom,
    ! [A: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state] :
      ( ( ~ ( member6715010751053357271_state @ A @ B2 )
       => ( A = B3 ) )
     => ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_428_insert__image,axiom,
    ! [X2: state,A5: set_state,F: state > state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( insert_state @ ( F @ X2 ) @ ( image_state_state @ F @ A5 ) )
        = ( image_state_state @ F @ A5 ) ) ) ).

% insert_image
thf(fact_429_insert__image,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( insert_state @ ( F @ X2 ) @ ( image_2710133271811360426_state @ F @ A5 ) )
        = ( image_2710133271811360426_state @ F @ A5 ) ) ) ).

% insert_image
thf(fact_430_insert__image,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > state] :
      ( ( member2879030624002546611on_val @ X2 @ A5 )
     => ( ( insert_state @ ( F @ X2 ) @ ( image_1205529093420515805_state @ F @ A5 ) )
        = ( image_1205529093420515805_state @ F @ A5 ) ) ) ).

% insert_image
thf(fact_431_insert__image,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( insert_state @ ( F @ X2 ) @ ( image_6759111139246981411_state @ F @ A5 ) )
        = ( image_6759111139246981411_state @ F @ A5 ) ) ) ).

% insert_image
thf(fact_432_image__insert,axiom,
    ! [F: state > state,A: state,B2: set_state] :
      ( ( image_state_state @ F @ ( insert_state @ A @ B2 ) )
      = ( insert_state @ ( F @ A ) @ ( image_state_state @ F @ B2 ) ) ) ).

% image_insert
thf(fact_433_insert__subset,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( insert175804457004744729_state @ X2 @ A5 ) @ B2 )
      = ( ( member7461423308196221312_state @ X2 @ B2 )
        & ( ord_le4032880777312351359_state @ A5 @ B2 ) ) ) ).

% insert_subset
thf(fact_434_insert__subset,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X2 @ A5 ) @ B2 )
      = ( ( member2879030624002546611on_val @ X2 @ B2 )
        & ( ord_le6900622908513426738on_val @ A5 @ B2 ) ) ) ).

% insert_subset
thf(fact_435_insert__subset,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( ord_le6483787752022044334_state @ ( insert94804854012721790_state @ X2 @ A5 ) @ B2 )
      = ( ( member6715010751053357271_state @ X2 @ B2 )
        & ( ord_le6483787752022044334_state @ A5 @ B2 ) ) ) ).

% insert_subset
thf(fact_436_insert__subset,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( insert_state @ X2 @ A5 ) @ B2 )
      = ( ( member_state @ X2 @ B2 )
        & ( ord_le2494988322063910608_state @ A5 @ B2 ) ) ) ).

% insert_subset
thf(fact_437_these__insert__None,axiom,
    ! [A5: set_option_state] :
      ( ( these_state @ ( insert_option_state @ none_state @ A5 ) )
      = ( these_state @ A5 ) ) ).

% these_insert_None
thf(fact_438_these__insert__Some,axiom,
    ! [X2: state,A5: set_option_state] :
      ( ( these_state @ ( insert_option_state @ ( some_state @ X2 ) @ A5 ) )
      = ( insert_state @ X2 @ ( these_state @ A5 ) ) ) ).

% these_insert_Some
thf(fact_439_package__logic_Owf__assertion_Ocong,axiom,
    packag7298478212905792770_state = packag7298478212905792770_state ).

% package_logic.wf_assertion.cong
thf(fact_440_package__logic_Opackage__rhs_Ocong,axiom,
    packag3171892606229212990_state = packag3171892606229212990_state ).

% package_logic.package_rhs.cong
thf(fact_441_package__logic_Osat_Ocong,axiom,
    package_sat_state = package_sat_state ).

% package_logic.sat.cong
thf(fact_442_mk__disjoint__insert,axiom,
    ! [A: state,A5: set_state] :
      ( ( member_state @ A @ A5 )
     => ? [B8: set_state] :
          ( ( A5
            = ( insert_state @ A @ B8 ) )
          & ~ ( member_state @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_443_mk__disjoint__insert,axiom,
    ! [A: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ A @ A5 )
     => ? [B8: set_Pr4231240917283394591_state] :
          ( ( A5
            = ( insert175804457004744729_state @ A @ B8 ) )
          & ~ ( member7461423308196221312_state @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_444_mk__disjoint__insert,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ A5 )
     => ? [B8: set_Pr7649655576376376530on_val] :
          ( ( A5
            = ( insert3030518898131756108on_val @ A @ B8 ) )
          & ~ ( member2879030624002546611on_val @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_445_mk__disjoint__insert,axiom,
    ! [A: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ A @ A5 )
     => ? [B8: set_Pr2391701600042252046_state] :
          ( ( A5
            = ( insert94804854012721790_state @ A @ B8 ) )
          & ~ ( member6715010751053357271_state @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_446_insert__commute,axiom,
    ! [X2: state,Y: state,A5: set_state] :
      ( ( insert_state @ X2 @ ( insert_state @ Y @ A5 ) )
      = ( insert_state @ Y @ ( insert_state @ X2 @ A5 ) ) ) ).

% insert_commute
thf(fact_447_insert__eq__iff,axiom,
    ! [A: state,A5: set_state,B3: state,B2: set_state] :
      ( ~ ( member_state @ A @ A5 )
     => ( ~ ( member_state @ B3 @ B2 )
       => ( ( ( insert_state @ A @ A5 )
            = ( insert_state @ B3 @ B2 ) )
          = ( ( ( A = B3 )
             => ( A5 = B2 ) )
            & ( ( A != B3 )
             => ? [C6: set_state] :
                  ( ( A5
                    = ( insert_state @ B3 @ C6 ) )
                  & ~ ( member_state @ B3 @ C6 )
                  & ( B2
                    = ( insert_state @ A @ C6 ) )
                  & ~ ( member_state @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_448_insert__eq__iff,axiom,
    ! [A: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ~ ( member7461423308196221312_state @ A @ A5 )
     => ( ~ ( member7461423308196221312_state @ B3 @ B2 )
       => ( ( ( insert175804457004744729_state @ A @ A5 )
            = ( insert175804457004744729_state @ B3 @ B2 ) )
          = ( ( ( A = B3 )
             => ( A5 = B2 ) )
            & ( ( A != B3 )
             => ? [C6: set_Pr4231240917283394591_state] :
                  ( ( A5
                    = ( insert175804457004744729_state @ B3 @ C6 ) )
                  & ~ ( member7461423308196221312_state @ B3 @ C6 )
                  & ( B2
                    = ( insert175804457004744729_state @ A @ C6 ) )
                  & ~ ( member7461423308196221312_state @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_449_insert__eq__iff,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B3: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ~ ( member2879030624002546611on_val @ A @ A5 )
     => ( ~ ( member2879030624002546611on_val @ B3 @ B2 )
       => ( ( ( insert3030518898131756108on_val @ A @ A5 )
            = ( insert3030518898131756108on_val @ B3 @ B2 ) )
          = ( ( ( A = B3 )
             => ( A5 = B2 ) )
            & ( ( A != B3 )
             => ? [C6: set_Pr7649655576376376530on_val] :
                  ( ( A5
                    = ( insert3030518898131756108on_val @ B3 @ C6 ) )
                  & ~ ( member2879030624002546611on_val @ B3 @ C6 )
                  & ( B2
                    = ( insert3030518898131756108on_val @ A @ C6 ) )
                  & ~ ( member2879030624002546611on_val @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_450_insert__eq__iff,axiom,
    ! [A: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ~ ( member6715010751053357271_state @ A @ A5 )
     => ( ~ ( member6715010751053357271_state @ B3 @ B2 )
       => ( ( ( insert94804854012721790_state @ A @ A5 )
            = ( insert94804854012721790_state @ B3 @ B2 ) )
          = ( ( ( A = B3 )
             => ( A5 = B2 ) )
            & ( ( A != B3 )
             => ? [C6: set_Pr2391701600042252046_state] :
                  ( ( A5
                    = ( insert94804854012721790_state @ B3 @ C6 ) )
                  & ~ ( member6715010751053357271_state @ B3 @ C6 )
                  & ( B2
                    = ( insert94804854012721790_state @ A @ C6 ) )
                  & ~ ( member6715010751053357271_state @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_451_insert__absorb,axiom,
    ! [A: state,A5: set_state] :
      ( ( member_state @ A @ A5 )
     => ( ( insert_state @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_452_insert__absorb,axiom,
    ! [A: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ A @ A5 )
     => ( ( insert175804457004744729_state @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_453_insert__absorb,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ A5 )
     => ( ( insert3030518898131756108on_val @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_454_insert__absorb,axiom,
    ! [A: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ A @ A5 )
     => ( ( insert94804854012721790_state @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_455_insert__ident,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ~ ( member_state @ X2 @ A5 )
     => ( ~ ( member_state @ X2 @ B2 )
       => ( ( ( insert_state @ X2 @ A5 )
            = ( insert_state @ X2 @ B2 ) )
          = ( A5 = B2 ) ) ) ) ).

% insert_ident
thf(fact_456_insert__ident,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ~ ( member7461423308196221312_state @ X2 @ A5 )
     => ( ~ ( member7461423308196221312_state @ X2 @ B2 )
       => ( ( ( insert175804457004744729_state @ X2 @ A5 )
            = ( insert175804457004744729_state @ X2 @ B2 ) )
          = ( A5 = B2 ) ) ) ) ).

% insert_ident
thf(fact_457_insert__ident,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ~ ( member2879030624002546611on_val @ X2 @ A5 )
     => ( ~ ( member2879030624002546611on_val @ X2 @ B2 )
       => ( ( ( insert3030518898131756108on_val @ X2 @ A5 )
            = ( insert3030518898131756108on_val @ X2 @ B2 ) )
          = ( A5 = B2 ) ) ) ) ).

% insert_ident
thf(fact_458_insert__ident,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ~ ( member6715010751053357271_state @ X2 @ A5 )
     => ( ~ ( member6715010751053357271_state @ X2 @ B2 )
       => ( ( ( insert94804854012721790_state @ X2 @ A5 )
            = ( insert94804854012721790_state @ X2 @ B2 ) )
          = ( A5 = B2 ) ) ) ) ).

% insert_ident
thf(fact_459_Set_Oset__insert,axiom,
    ! [X2: state,A5: set_state] :
      ( ( member_state @ X2 @ A5 )
     => ~ ! [B8: set_state] :
            ( ( A5
              = ( insert_state @ X2 @ B8 ) )
           => ( member_state @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_460_Set_Oset__insert,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ~ ! [B8: set_Pr4231240917283394591_state] :
            ( ( A5
              = ( insert175804457004744729_state @ X2 @ B8 ) )
           => ( member7461423308196221312_state @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_461_Set_Oset__insert,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ X2 @ A5 )
     => ~ ! [B8: set_Pr7649655576376376530on_val] :
            ( ( A5
              = ( insert3030518898131756108on_val @ X2 @ B8 ) )
           => ( member2879030624002546611on_val @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_462_Set_Oset__insert,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ~ ! [B8: set_Pr2391701600042252046_state] :
            ( ( A5
              = ( insert94804854012721790_state @ X2 @ B8 ) )
           => ( member6715010751053357271_state @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_463_insertI2,axiom,
    ! [A: state,B2: set_state,B3: state] :
      ( ( member_state @ A @ B2 )
     => ( member_state @ A @ ( insert_state @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_464_insertI2,axiom,
    ! [A: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,B3: produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ A @ B2 )
     => ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_465_insertI2,axiom,
    ! [A: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,B3: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ A @ B2 )
     => ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_466_insertI2,axiom,
    ! [A: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,B3: produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ A @ B2 )
     => ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_467_insertI1,axiom,
    ! [A: state,B2: set_state] : ( member_state @ A @ ( insert_state @ A @ B2 ) ) ).

% insertI1
thf(fact_468_insertI1,axiom,
    ! [A: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] : ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ A @ B2 ) ) ).

% insertI1
thf(fact_469_insertI1,axiom,
    ! [A: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] : ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ A @ B2 ) ) ).

% insertI1
thf(fact_470_insertI1,axiom,
    ! [A: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] : ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ A @ B2 ) ) ).

% insertI1
thf(fact_471_insertE,axiom,
    ! [A: state,B3: state,A5: set_state] :
      ( ( member_state @ A @ ( insert_state @ B3 @ A5 ) )
     => ( ( A != B3 )
       => ( member_state @ A @ A5 ) ) ) ).

% insertE
thf(fact_472_insertE,axiom,
    ! [A: produc7272280185086996969_state,B3: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ B3 @ A5 ) )
     => ( ( A != B3 )
       => ( member7461423308196221312_state @ A @ A5 ) ) ) ).

% insertE
thf(fact_473_insertE,axiom,
    ! [A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ B3 @ A5 ) )
     => ( ( A != B3 )
       => ( member2879030624002546611on_val @ A @ A5 ) ) ) ).

% insertE
thf(fact_474_insertE,axiom,
    ! [A: produc3882902684156697774_state,B3: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ B3 @ A5 ) )
     => ( ( A != B3 )
       => ( member6715010751053357271_state @ A @ A5 ) ) ) ).

% insertE
thf(fact_475_insert__UNIV,axiom,
    ! [X2: state] :
      ( ( insert_state @ X2 @ top_top_set_state )
      = top_top_set_state ) ).

% insert_UNIV
thf(fact_476_insert__mono,axiom,
    ! [C3: set_state,D2: set_state,A: state] :
      ( ( ord_le2494988322063910608_state @ C3 @ D2 )
     => ( ord_le2494988322063910608_state @ ( insert_state @ A @ C3 ) @ ( insert_state @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_477_subset__insert,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ~ ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( ord_le4032880777312351359_state @ A5 @ ( insert175804457004744729_state @ X2 @ B2 ) )
        = ( ord_le4032880777312351359_state @ A5 @ B2 ) ) ) ).

% subset_insert
thf(fact_478_subset__insert,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ~ ( member2879030624002546611on_val @ X2 @ A5 )
     => ( ( ord_le6900622908513426738on_val @ A5 @ ( insert3030518898131756108on_val @ X2 @ B2 ) )
        = ( ord_le6900622908513426738on_val @ A5 @ B2 ) ) ) ).

% subset_insert
thf(fact_479_subset__insert,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ~ ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( ord_le6483787752022044334_state @ A5 @ ( insert94804854012721790_state @ X2 @ B2 ) )
        = ( ord_le6483787752022044334_state @ A5 @ B2 ) ) ) ).

% subset_insert
thf(fact_480_subset__insert,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ~ ( member_state @ X2 @ A5 )
     => ( ( ord_le2494988322063910608_state @ A5 @ ( insert_state @ X2 @ B2 ) )
        = ( ord_le2494988322063910608_state @ A5 @ B2 ) ) ) ).

% subset_insert
thf(fact_481_subset__insertI,axiom,
    ! [B2: set_state,A: state] : ( ord_le2494988322063910608_state @ B2 @ ( insert_state @ A @ B2 ) ) ).

% subset_insertI
thf(fact_482_subset__insertI2,axiom,
    ! [A5: set_state,B2: set_state,B3: state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ord_le2494988322063910608_state @ A5 @ ( insert_state @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_483_empty__heap__neutral,axiom,
    ! [A: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ A @ empty_7512026568161686585ar_val )
      = A ) ).

% empty_heap_neutral
thf(fact_484_PartialSA_Owf__assertion__sat__larger__pure,axiom,
    ! [A5: aassertion_state,Phi: state,Phi2: state,R: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ A5 )
     => ( ( package_sat_state @ plus @ A5 @ Phi )
       => ( ( ( some_state @ Phi2 )
            = ( plus @ Phi @ R ) )
         => ( ( sep_pure_state @ plus @ R )
           => ( package_sat_state @ plus @ A5 @ Phi2 ) ) ) ) ) ).

% PartialSA.wf_assertion_sat_larger_pure
thf(fact_485_PartialSA_Owf__assertion__and__the,axiom,
    ! [A: state,B3: state,A5: aassertion_state] :
      ( ( defined @ ( core @ A ) @ B3 )
     => ( ( package_sat_state @ plus @ A5 @ B3 )
       => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
         => ( package_sat_state @ plus @ A5 @ ( the_state @ ( plus @ ( core @ A ) @ B3 ) ) ) ) ) ) ).

% PartialSA.wf_assertion_and_the
thf(fact_486_in__these__eq,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_op8023886964038656367_state] :
      ( ( member7461423308196221312_state @ X2 @ ( these_4347185934205712492_state @ A5 ) )
      = ( member3894984953261475024_state @ ( some_P6297086604321900740_state @ X2 ) @ A5 ) ) ).

% in_these_eq
thf(fact_487_in__these__eq,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_op6240181410663953058on_val] :
      ( ( member2879030624002546611on_val @ X2 @ ( these_3919539662624406175on_val @ A5 ) )
      = ( member4602090469187516291on_val @ ( some_P2581650809551911671on_val @ X2 ) @ A5 ) ) ).

% in_these_eq
thf(fact_488_in__these__eq,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_op3387789762728285652_state] :
      ( ( member6715010751053357271_state @ X2 @ ( these_7452845017025203435_state @ A5 ) )
      = ( member5919342556599141533_state @ ( some_P5383513176685068947_state @ X2 ) @ A5 ) ) ).

% in_these_eq
thf(fact_489_in__these__eq,axiom,
    ! [X2: state,A5: set_option_state] :
      ( ( member_state @ X2 @ ( these_state @ A5 ) )
      = ( member_option_state @ ( some_state @ X2 ) @ A5 ) ) ).

% in_these_eq
thf(fact_490_plus__charact_I2_J,axiom,
    ! [A: state,B3: state,X2: state] :
      ( ( ( plus @ A @ B3 )
        = ( some_state @ X2 ) )
     => ( ( get_h @ X2 )
        = ( map_ad2908576761495157028ar_val @ ( get_h @ A ) @ ( get_h @ B3 ) ) ) ) ).

% plus_charact(2)
thf(fact_491_add__states_Osimps,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val,Pi4: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
      ( ( add_states @ ( produc74620467203698198on_val @ Pi2 @ H3 ) @ ( produc74620467203698198on_val @ Pi4 @ H4 ) )
      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi2 @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H3 @ H4 ) ) ) ).

% add_states.simps
thf(fact_492_map__add__find__right,axiom,
    ! [N2: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,Xx: val,M2: produc1457211279475724562t_char > option_val] :
      ( ( ( N2 @ K )
        = ( some_val @ Xx ) )
     => ( ( map_ad2908576761495157028ar_val @ M2 @ N2 @ K )
        = ( some_val @ Xx ) ) ) ).

% map_add_find_right
thf(fact_493_map__add__SomeD,axiom,
    ! [M2: produc1457211279475724562t_char > option_val,N2: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X2: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M2 @ N2 @ K )
        = ( some_val @ X2 ) )
     => ( ( ( N2 @ K )
          = ( some_val @ X2 ) )
        | ( ( ( N2 @ K )
            = none_val )
          & ( ( M2 @ K )
            = ( some_val @ X2 ) ) ) ) ) ).

% map_add_SomeD
thf(fact_494_map__add__Some__iff,axiom,
    ! [M2: produc1457211279475724562t_char > option_val,N2: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X2: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M2 @ N2 @ K )
        = ( some_val @ X2 ) )
      = ( ( ( N2 @ K )
          = ( some_val @ X2 ) )
        | ( ( ( N2 @ K )
            = none_val )
          & ( ( M2 @ K )
            = ( some_val @ X2 ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_495_insert__subsetI,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,X7: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( ord_le4032880777312351359_state @ X7 @ A5 )
       => ( ord_le4032880777312351359_state @ ( insert175804457004744729_state @ X2 @ X7 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_496_insert__subsetI,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,X7: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ X2 @ A5 )
     => ( ( ord_le6900622908513426738on_val @ X7 @ A5 )
       => ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X2 @ X7 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_497_insert__subsetI,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,X7: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( ord_le6483787752022044334_state @ X7 @ A5 )
       => ( ord_le6483787752022044334_state @ ( insert94804854012721790_state @ X2 @ X7 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_498_insert__subsetI,axiom,
    ! [X2: state,A5: set_state,X7: set_state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( ord_le2494988322063910608_state @ X7 @ A5 )
       => ( ord_le2494988322063910608_state @ ( insert_state @ X2 @ X7 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_499_PartialSA_Osat__star__exists__bigger,axiom,
    ! [A5: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ A5 @ B2 ) @ Phi )
     => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
       => ( ( packag7298478212905792770_state @ plus @ core @ B2 )
         => ? [A6: state,B7: state] :
              ( ( greater @ ( core @ A6 ) @ ( core @ Phi ) )
              & ( greater @ ( core @ B7 ) @ ( core @ Phi ) )
              & ( ( some_state @ Phi )
                = ( plus @ A6 @ B7 ) )
              & ( package_sat_state @ plus @ A5 @ A6 )
              & ( package_sat_state @ plus @ B2 @ B7 ) ) ) ) ) ).

% PartialSA.sat_star_exists_bigger
thf(fact_500_add__states_Opelims,axiom,
    ! [X2: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X2 @ Xa2 ) )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X2
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ! [Pi3: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
                  ( ( Xa2
                    = ( produc74620467203698198on_val @ Pi3 @ H2 ) )
                 => ( ( Y
                      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi3 ) @ ( map_ad2908576761495157028ar_val @ H @ H2 ) ) )
                   => ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi3 @ H2 ) ) ) ) ) ) ) ) ).

% add_states.pelims
thf(fact_501_aassertion_Oinject_I1_J,axiom,
    ! [X11: aassertion_state,X12: aassertion_state,Y11: aassertion_state,Y12: aassertion_state] :
      ( ( ( aStar_state @ X11 @ X12 )
        = ( aStar_state @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% aassertion.inject(1)
thf(fact_502_ssubst__Pair__rhs,axiom,
    ! [R: state,S: produc3882902684156697774_state,R4: set_Pr4231240917283394591_state,S4: produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_503_ssubst__Pair__rhs,axiom,
    ! [R: state,S: state > state,R4: set_Pr2391701600042252046_state,S4: state > state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_504_ssubst__Pair__rhs,axiom,
    ! [R: produc9003781588162623004on_val,S: produc9003781588162623004on_val,R4: set_Pr4305388979797279737on_val,S4: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_505_ssubst__Pair__rhs,axiom,
    ! [R: option_state,S: option_state,R4: set_Pr1688445902015331925_state,S4: option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_506_ssubst__Pair__rhs,axiom,
    ! [R: prat,S: state,R4: set_Pr6039890582625207499_state,S4: state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_507_ssubst__Pair__rhs,axiom,
    ! [R: prat,S: set_state,R4: set_Pr6847263425858009217_state,S4: set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_508_ssubst__Pair__rhs,axiom,
    ! [R: produc1457211279475724562t_char > prat,S: produc1457211279475724562t_char > option_val,R4: set_Pr7649655576376376530on_val,S4: produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S ) @ R4 )
     => ( ( S4 = S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S4 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_509_PartialSA_Osat_Osimps_I1_J,axiom,
    ! [Aa: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ Aa @ B2 ) @ Phi )
      = ( ? [A2: state,B: state] :
            ( ( ( some_state @ Phi )
              = ( plus @ A2 @ B ) )
            & ( package_sat_state @ plus @ Aa @ A2 )
            & ( package_sat_state @ plus @ B2 @ B ) ) ) ) ).

% PartialSA.sat.simps(1)
thf(fact_510_PartialSA_Owf__assertion_Osimps_I1_J,axiom,
    ! [Aa: aassertion_state,B2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aStar_state @ Aa @ B2 ) )
      = ( ( packag7298478212905792770_state @ plus @ core @ Aa )
        & ( packag7298478212905792770_state @ plus @ core @ B2 ) ) ) ).

% PartialSA.wf_assertion.simps(1)
thf(fact_511_PartialSA_Opackage__rhs_Ointros_I1_J,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,Ba2: aassertion_state,Phi7: state,F4: state,S5: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ Ba2 @ Phi7 @ F4 @ S5 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aStar_state @ A5 @ Ba2 ) @ Phi7 @ F4 @ S5 ) ) ) ).

% PartialSA.package_rhs.intros(1)
thf(fact_512_PartialSA_OAStar,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,B2: aassertion_state,Phi7: state,F4: state,S5: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ B2 @ Phi7 @ F4 @ S5 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aStar_state @ A5 @ B2 ) @ Phi7 @ F4 @ S5 ) ) ) ).

% PartialSA.AStar
thf(fact_513_PartialSA_OAImp,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,B3: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ ( packag1330488657391168505_state @ Pc @ B3 ) @ A5 @ Phi2 @ F3 @ S3 )
     => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aImp_state @ B3 @ A5 ) @ Phi2 @ F3 @ S3 ) ) ).

% PartialSA.AImp
thf(fact_514_PartialSA_Ogeneral__completeness,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state,A5: aassertion_state,Phi: state,Phi2: state] :
      ( ! [A6: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ! [X4: state] :
              ( ( ( some_state @ X4 )
                = ( plus @ A6 @ ( T3 @ F ) ) )
             => ( package_sat_state @ plus @ A5 @ X4 ) ) )
     => ( ( ( some_state @ Phi )
          = ( plus @ Phi2 @ F ) )
       => ( ( stable @ F )
         => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ unit )
           => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ unit @ S2
                  @ ^ [Uu3: state] : $true
                  @ A5
                  @ Phi2
                  @ F
                  @ X_1 ) ) ) ) ) ) ).

% PartialSA.general_completeness
thf(fact_515_PartialSA_Ois__footprint__standard__def,axiom,
    ! [W: state,A5: aassertion_state,B2: aassertion_state] :
      ( ( packag7096211993527611489_state @ plus @ W @ A5 @ B2 )
      = ( ! [A2: state,B: state] :
            ( ( ( package_sat_state @ plus @ A5 @ A2 )
              & ( ( some_state @ B )
                = ( plus @ A2 @ W ) ) )
           => ( package_sat_state @ plus @ B2 @ B ) ) ) ) ).

% PartialSA.is_footprint_standard_def
thf(fact_516_PartialSA_Ois__footprint__general__def,axiom,
    ! [W: state,R4: state > state > state,A5: aassertion_state,B2: aassertion_state] :
      ( ( packag6192661653625646554_state @ plus @ W @ R4 @ A5 @ B2 )
      = ( ! [A2: state,B: state] :
            ( ( ( package_sat_state @ plus @ A5 @ A2 )
              & ( ( some_state @ B )
                = ( plus @ A2 @ ( R4 @ A2 @ W ) ) ) )
           => ( package_sat_state @ plus @ B2 @ B ) ) ) ) ).

% PartialSA.is_footprint_general_def
thf(fact_517_PartialSA_Ois__footprint__standardI,axiom,
    ! [A5: aassertion_state,W: state,B2: aassertion_state] :
      ( ! [A6: state,B7: state] :
          ( ( ( package_sat_state @ plus @ A5 @ A6 )
            & ( ( some_state @ B7 )
              = ( plus @ A6 @ W ) ) )
         => ( package_sat_state @ plus @ B2 @ B7 ) )
     => ( packag7096211993527611489_state @ plus @ W @ A5 @ B2 ) ) ).

% PartialSA.is_footprint_standardI
thf(fact_518_aassertion_Oinject_I2_J,axiom,
    ! [X21: state > $o,X222: aassertion_state,Y21: state > $o,Y22: aassertion_state] :
      ( ( ( aImp_state @ X21 @ X222 )
        = ( aImp_state @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% aassertion.inject(2)
thf(fact_519_image__ident,axiom,
    ! [Y7: set_state] :
      ( ( image_state_state
        @ ^ [X: state] : X
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_520_map__add__def,axiom,
    ( map_ad2908576761495157028ar_val
    = ( ^ [M1: produc1457211279475724562t_char > option_val,M22: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char] : ( case_o7794706883426356101al_val @ ( M1 @ X ) @ some_val @ ( M22 @ X ) ) ) ) ).

% map_add_def
thf(fact_521_insert__Collect,axiom,
    ! [A: produc7272280185086996969_state,P: produc7272280185086996969_state > $o] :
      ( ( insert175804457004744729_state @ A @ ( collec906845016635587262_state @ P ) )
      = ( collec906845016635587262_state
        @ ^ [U6: produc7272280185086996969_state] :
            ( ( U6 != A )
           => ( P @ U6 ) ) ) ) ).

% insert_Collect
thf(fact_522_insert__Collect,axiom,
    ! [A: state,P: state > $o] :
      ( ( insert_state @ A @ ( collect_state @ P ) )
      = ( collect_state
        @ ^ [U6: state] :
            ( ( U6 != A )
           => ( P @ U6 ) ) ) ) ).

% insert_Collect
thf(fact_523_insert__Collect,axiom,
    ! [A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( insert3030518898131756108on_val @ A @ ( collec7475294502966882545on_val @ P ) )
      = ( collec7475294502966882545on_val
        @ ^ [U6: produc9003781588162623004on_val] :
            ( ( U6 != A )
           => ( P @ U6 ) ) ) ) ).

% insert_Collect
thf(fact_524_insert__compr,axiom,
    ( insert94804854012721790_state
    = ( ^ [A2: produc3882902684156697774_state,B5: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [X: produc3882902684156697774_state] :
              ( ( X = A2 )
              | ( member6715010751053357271_state @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_525_insert__compr,axiom,
    ( insert175804457004744729_state
    = ( ^ [A2: produc7272280185086996969_state,B5: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [X: produc7272280185086996969_state] :
              ( ( X = A2 )
              | ( member7461423308196221312_state @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_526_insert__compr,axiom,
    ( insert_state
    = ( ^ [A2: state,B5: set_state] :
          ( collect_state
          @ ^ [X: state] :
              ( ( X = A2 )
              | ( member_state @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_527_insert__compr,axiom,
    ( insert3030518898131756108on_val
    = ( ^ [A2: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [X: produc9003781588162623004on_val] :
              ( ( X = A2 )
              | ( member2879030624002546611on_val @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_528_prop__restrict,axiom,
    ! [X2: produc3882902684156697774_state,Z3: set_Pr2391701600042252046_state,X7: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ( member6715010751053357271_state @ X2 @ Z3 )
     => ( ( ord_le6483787752022044334_state @ Z3
          @ ( collec394671971739345689_state
            @ ^ [X: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_529_prop__restrict,axiom,
    ! [X2: produc7272280185086996969_state,Z3: set_Pr4231240917283394591_state,X7: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( member7461423308196221312_state @ X2 @ Z3 )
     => ( ( ord_le4032880777312351359_state @ Z3
          @ ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_530_prop__restrict,axiom,
    ! [X2: produc9003781588162623004on_val,Z3: set_Pr7649655576376376530on_val,X7: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ X2 @ Z3 )
     => ( ( ord_le6900622908513426738on_val @ Z3
          @ ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_531_prop__restrict,axiom,
    ! [X2: state,Z3: set_state,X7: set_state,P: state > $o] :
      ( ( member_state @ X2 @ Z3 )
     => ( ( ord_le2494988322063910608_state @ Z3
          @ ( collect_state
            @ ^ [X: state] :
                ( ( member_state @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_532_Collect__restrict,axiom,
    ! [X7: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ord_le6483787752022044334_state
      @ ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_533_Collect__restrict,axiom,
    ! [X7: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ord_le4032880777312351359_state
      @ ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_534_Collect__restrict,axiom,
    ! [X7: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_535_Collect__restrict,axiom,
    ! [X7: set_state,P: state > $o] :
      ( ord_le2494988322063910608_state
      @ ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_536_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > state,B2: set_state] :
      ( ! [X4: state] :
          ( ( P @ X4 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_537_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: state] :
          ( ( P @ X4 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_538_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: state] :
          ( ( P @ X4 )
         => ( member7461423308196221312_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_539_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > state,B2: set_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_540_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_541_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
         => ( member7461423308196221312_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_542_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X4: state] :
          ( ( P @ X4 )
         => ( member2879030624002546611on_val @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_543_image__Collect__subsetI,axiom,
    ! [P: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > state,B2: set_state] :
      ( ! [X4: produc9003781588162623004on_val] :
          ( ( P @ X4 )
         => ( member_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ ( collec7475294502966882545on_val @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_544_image__Collect__subsetI,axiom,
    ! [P: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X4: produc9003781588162623004on_val] :
          ( ( P @ X4 )
         => ( member6715010751053357271_state @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ ( collec7475294502966882545on_val @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_545_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X4: produc7272280185086996969_state] :
          ( ( P @ X4 )
         => ( member2879030624002546611on_val @ ( F @ X4 ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_3393013468800240652on_val @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_546_pred__equals__eq2,axiom,
    ! [R4: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_547_pred__equals__eq2,axiom,
    ! [R4: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_548_pred__equals__eq2,axiom,
    ! [R4: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( ( ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_549_pred__equals__eq2,axiom,
    ! [R4: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_550_pred__equals__eq2,axiom,
    ! [R4: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_551_pred__equals__eq2,axiom,
    ! [R4: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_552_pred__equals__eq2,axiom,
    ! [R4: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ R4 ) )
        = ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_553_imageE,axiom,
    ! [B3: state,F: state > state,A5: set_state] :
      ( ( member_state @ B3 @ ( image_state_state @ F @ A5 ) )
     => ~ ! [X4: state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_554_imageE,axiom,
    ! [B3: state,F: produc3882902684156697774_state > state,A5: set_Pr2391701600042252046_state] :
      ( ( member_state @ B3 @ ( image_6759111139246981411_state @ F @ A5 ) )
     => ~ ! [X4: produc3882902684156697774_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member6715010751053357271_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_555_imageE,axiom,
    ! [B3: produc3882902684156697774_state,F: state > produc3882902684156697774_state,A5: set_state] :
      ( ( member6715010751053357271_state @ B3 @ ( image_5311902671526399277_state @ F @ A5 ) )
     => ~ ! [X4: state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_556_imageE,axiom,
    ! [B3: state,F: produc7272280185086996969_state > state,A5: set_Pr4231240917283394591_state] :
      ( ( member_state @ B3 @ ( image_2710133271811360426_state @ F @ A5 ) )
     => ~ ! [X4: produc7272280185086996969_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member7461423308196221312_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_557_imageE,axiom,
    ! [B3: produc7272280185086996969_state,F: state > produc7272280185086996969_state,A5: set_state] :
      ( ( member7461423308196221312_state @ B3 @ ( image_632992669292831658_state @ F @ A5 ) )
     => ~ ! [X4: state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_558_imageE,axiom,
    ! [B3: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ B3 @ ( image_508898238547643525_state @ F @ A5 ) )
     => ~ ! [X4: produc3882902684156697774_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member6715010751053357271_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_559_imageE,axiom,
    ! [B3: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,A5: set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ B3 @ ( image_4364510454419742290_state @ F @ A5 ) )
     => ~ ! [X4: produc3882902684156697774_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member6715010751053357271_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_560_imageE,axiom,
    ! [B3: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,A5: set_Pr4231240917283394591_state] :
      ( ( member6715010751053357271_state @ B3 @ ( image_7466395622266389182_state @ F @ A5 ) )
     => ~ ! [X4: produc7272280185086996969_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member7461423308196221312_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_561_imageE,axiom,
    ! [B3: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ B3 @ ( image_3064086290059306969_state @ F @ A5 ) )
     => ~ ! [X4: produc7272280185086996969_state] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member7461423308196221312_state @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_562_imageE,axiom,
    ! [B3: state,F: produc9003781588162623004on_val > state,A5: set_Pr7649655576376376530on_val] :
      ( ( member_state @ B3 @ ( image_1205529093420515805_state @ F @ A5 ) )
     => ~ ! [X4: produc9003781588162623004on_val] :
            ( ( B3
              = ( F @ X4 ) )
           => ~ ( member2879030624002546611on_val @ X4 @ A5 ) ) ) ).

% imageE
thf(fact_563_image__image,axiom,
    ! [F: state > state,G: state > state,A5: set_state] :
      ( ( image_state_state @ F @ ( image_state_state @ G @ A5 ) )
      = ( image_state_state
        @ ^ [X: state] : ( F @ ( G @ X ) )
        @ A5 ) ) ).

% image_image
thf(fact_564_Compr__image__eq,axiom,
    ! [F: state > state,A5: set_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ ( image_state_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_state_state @ F
        @ ( collect_state
          @ ^ [X: state] :
              ( ( member_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_565_Compr__image__eq,axiom,
    ! [F: state > produc3882902684156697774_state,A5: set_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ ( image_5311902671526399277_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_5311902671526399277_state @ F
        @ ( collect_state
          @ ^ [X: state] :
              ( ( member_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_566_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > state,A5: set_Pr2391701600042252046_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ ( image_6759111139246981411_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_6759111139246981411_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_567_Compr__image__eq,axiom,
    ! [F: state > produc7272280185086996969_state,A5: set_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ ( image_632992669292831658_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_632992669292831658_state @ F
        @ ( collect_state
          @ ^ [X: state] :
              ( ( member_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_568_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > state,A5: set_Pr4231240917283394591_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ ( image_2710133271811360426_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_2710133271811360426_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_569_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ ( image_508898238547643525_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_508898238547643525_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_570_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > produc3882902684156697774_state,A5: set_Pr4231240917283394591_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ ( image_7466395622266389182_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_7466395622266389182_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_571_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > produc7272280185086996969_state,A5: set_Pr2391701600042252046_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ ( image_4364510454419742290_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_4364510454419742290_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_572_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ ( image_3064086290059306969_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_3064086290059306969_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_573_Compr__image__eq,axiom,
    ! [F: produc9003781588162623004on_val > state,A5: set_Pr7649655576376376530on_val,P: state > $o] :
      ( ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ ( image_1205529093420515805_state @ F @ A5 ) )
            & ( P @ X ) ) )
      = ( image_1205529093420515805_state @ F
        @ ( collec7475294502966882545on_val
          @ ^ [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ A5 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_574_case__optionE,axiom,
    ! [P: $o,Q: state > $o,X2: option_state] :
      ( ( case_option_o_state @ P @ Q @ X2 )
     => ( ( ( X2 = none_state )
         => ~ P )
       => ~ ! [Y4: state] :
              ( ( X2
                = ( some_state @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_575_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_state] :
      ( ( Option = none_state )
      = ( case_option_o_state @ $true
        @ ^ [Uu3: state] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_576_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
      = ( case_option_o_state @ $false
        @ ^ [Uu3: state] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_577_pred__subset__eq2,axiom,
    ! [R4: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ord_le6704609105599929480tate_o
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ R4 )
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ S2 ) )
      = ( ord_le4032880777312351359_state @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_578_pred__subset__eq2,axiom,
    ! [R4: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ord_le1106678366802930969tate_o
        @ ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ R4 )
        @ ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ S2 ) )
      = ( ord_le6483787752022044334_state @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_579_pred__subset__eq2,axiom,
    ! [R4: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( ord_le616650860478780142_val_o
        @ ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ R4 )
        @ ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ S2 ) )
      = ( ord_le7224449410216889241on_val @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_580_pred__subset__eq2,axiom,
    ! [R4: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( ord_le2052508915978315794tate_o
        @ ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 )
        @ ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ S2 ) )
      = ( ord_le6423325748750870005_state @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_581_pred__subset__eq2,axiom,
    ! [R4: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( ord_le9040380983013460636tate_o
        @ ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ R4 )
        @ ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ S2 ) )
      = ( ord_le262197048489888875_state @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_582_pred__subset__eq2,axiom,
    ! [R4: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( ord_le8830457724327557734tate_o
        @ ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ R4 )
        @ ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ S2 ) )
      = ( ord_le5761494056910127137_state @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_583_pred__subset__eq2,axiom,
    ! [R4: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ord_le3852884270828607079_val_o
        @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ R4 )
        @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ S2 ) )
      = ( ord_le6900622908513426738on_val @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_584_less__eq__set__def,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
          ( ord_le7281427340513060550tate_o
          @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ A4 )
          @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_585_less__eq__set__def,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
          ( ord_le6140313152057869779_val_o
          @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ A4 )
          @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_586_less__eq__set__def,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A4: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
          ( ord_le8077393285996101039tate_o
          @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ A4 )
          @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_587_less__eq__set__def,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A4: set_state,B5: set_state] :
          ( ord_less_eq_state_o
          @ ^ [X: state] : ( member_state @ X @ A4 )
          @ ^ [X: state] : ( member_state @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_588_Collect__subset,axiom,
    ! [A5: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ord_le6483787752022044334_state
      @ ( collec394671971739345689_state
        @ ^ [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ A5 )
            & ( P @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_589_Collect__subset,axiom,
    ! [A5: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ord_le4032880777312351359_state
      @ ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ A5 )
            & ( P @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_590_Collect__subset,axiom,
    ! [A5: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ A5 )
            & ( P @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_591_Collect__subset,axiom,
    ! [A5: set_state,P: state > $o] :
      ( ord_le2494988322063910608_state
      @ ( collect_state
        @ ^ [X: state] :
            ( ( member_state @ X @ A5 )
            & ( P @ X ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_592_package__logic_Ois__footprint__standard_Ocong,axiom,
    packag7096211993527611489_state = packag7096211993527611489_state ).

% package_logic.is_footprint_standard.cong
thf(fact_593_package__logic_Ois__footprint__general_Ocong,axiom,
    packag6192661653625646554_state = packag6192661653625646554_state ).

% package_logic.is_footprint_general.cong
thf(fact_594_pred__subset__eq,axiom,
    ! [R4: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ord_le7281427340513060550tate_o
        @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ R4 )
        @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ S2 ) )
      = ( ord_le4032880777312351359_state @ R4 @ S2 ) ) ).

% pred_subset_eq
thf(fact_595_pred__subset__eq,axiom,
    ! [R4: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ord_le6140313152057869779_val_o
        @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ R4 )
        @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ S2 ) )
      = ( ord_le6900622908513426738on_val @ R4 @ S2 ) ) ).

% pred_subset_eq
thf(fact_596_pred__subset__eq,axiom,
    ! [R4: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ord_le8077393285996101039tate_o
        @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ R4 )
        @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ S2 ) )
      = ( ord_le6483787752022044334_state @ R4 @ S2 ) ) ).

% pred_subset_eq
thf(fact_597_pred__subset__eq,axiom,
    ! [R4: set_state,S2: set_state] :
      ( ( ord_less_eq_state_o
        @ ^ [X: state] : ( member_state @ X @ R4 )
        @ ^ [X: state] : ( member_state @ X @ S2 ) )
      = ( ord_le2494988322063910608_state @ R4 @ S2 ) ) ).

% pred_subset_eq
thf(fact_598_rangeE,axiom,
    ! [B3: state,F: state > state] :
      ( ( member_state @ B3 @ ( image_state_state @ F @ top_top_set_state ) )
     => ~ ! [X4: state] :
            ( B3
           != ( F @ X4 ) ) ) ).

% rangeE
thf(fact_599_rangeE,axiom,
    ! [B3: produc7272280185086996969_state,F: state > produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ B3 @ ( image_632992669292831658_state @ F @ top_top_set_state ) )
     => ~ ! [X4: state] :
            ( B3
           != ( F @ X4 ) ) ) ).

% rangeE
thf(fact_600_rangeE,axiom,
    ! [B3: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ B3 @ ( image_4772210114854293981on_val @ F @ top_top_set_state ) )
     => ~ ! [X4: state] :
            ( B3
           != ( F @ X4 ) ) ) ).

% rangeE
thf(fact_601_rangeE,axiom,
    ! [B3: produc3882902684156697774_state,F: state > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ B3 @ ( image_5311902671526399277_state @ F @ top_top_set_state ) )
     => ~ ! [X4: state] :
            ( B3
           != ( F @ X4 ) ) ) ).

% rangeE
thf(fact_602_range__composition,axiom,
    ! [F: state > state,G: state > state] :
      ( ( image_state_state
        @ ^ [X: state] : ( F @ ( G @ X ) )
        @ top_top_set_state )
      = ( image_state_state @ F @ ( image_state_state @ G @ top_top_set_state ) ) ) ).

% range_composition
thf(fact_603_UNIV__def,axiom,
    ( top_to174962084833059535_state
    = ( collec906845016635587262_state
      @ ^ [X: produc7272280185086996969_state] : $true ) ) ).

% UNIV_def
thf(fact_604_UNIV__def,axiom,
    ( top_to1561438921946393474on_val
    = ( collec7475294502966882545on_val
      @ ^ [X: produc9003781588162623004on_val] : $true ) ) ).

% UNIV_def
thf(fact_605_UNIV__def,axiom,
    ( top_top_set_state
    = ( collect_state
      @ ^ [X: state] : $true ) ) ).

% UNIV_def
thf(fact_606_top__empty__eq2,axiom,
    ( top_to8386588197433551576tate_o
    = ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ top_to174962084833059535_state ) ) ) ).

% top_empty_eq2
thf(fact_607_top__empty__eq2,axiom,
    ( top_to53189150443868873tate_o
    = ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ top_to1744471583079920734_state ) ) ) ).

% top_empty_eq2
thf(fact_608_top__empty__eq2,axiom,
    ( top_to8462242132197052574_val_o
    = ( ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ top_to5796317595507894601on_val ) ) ) ).

% top_empty_eq2
thf(fact_609_top__empty__eq2,axiom,
    ( top_to2009358410527348162tate_o
    = ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ top_to5248747511432852389_state ) ) ) ).

% top_empty_eq2
thf(fact_610_top__empty__eq2,axiom,
    ( top_top_prat_state_o
    = ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ top_to5813313822981523995_state ) ) ) ).

% top_empty_eq2
thf(fact_611_top__empty__eq2,axiom,
    ( top_to8783976926004287510tate_o
    = ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ top_to4399397253935103441_state ) ) ) ).

% top_empty_eq2
thf(fact_612_top__empty__eq2,axiom,
    ( top_to1211379171580150967_val_o
    = ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ top_to1561438921946393474on_val ) ) ) ).

% top_empty_eq2
thf(fact_613_aassertion_Odistinct_I1_J,axiom,
    ! [X11: aassertion_state,X12: aassertion_state,X21: state > $o,X222: aassertion_state] :
      ( ( aStar_state @ X11 @ X12 )
     != ( aImp_state @ X21 @ X222 ) ) ).

% aassertion.distinct(1)
thf(fact_614_Option_Othese__def,axiom,
    ( these_state
    = ( ^ [A4: set_option_state] :
          ( image_3532137647693456075_state @ the_state
          @ ( collect_option_state
            @ ^ [X: option_state] :
                ( ( member_option_state @ X @ A4 )
                & ( X != none_state ) ) ) ) ) ) ).

% Option.these_def
thf(fact_615_Some__image__these__eq,axiom,
    ! [A5: set_option_state] :
      ( ( image_6076465424260689483_state @ some_state @ ( these_state @ A5 ) )
      = ( collect_option_state
        @ ^ [X: option_state] :
            ( ( member_option_state @ X @ A5 )
            & ( X != none_state ) ) ) ) ).

% Some_image_these_eq
thf(fact_616_PartialSA_Osat_Osimps_I2_J,axiom,
    ! [B3: state > $o,Aa: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aImp_state @ B3 @ Aa ) @ Phi )
      = ( ( B3 @ Phi )
       => ( package_sat_state @ plus @ Aa @ Phi ) ) ) ).

% PartialSA.sat.simps(2)
thf(fact_617_PartialSA_Owf__assertion_Osimps_I2_J,axiom,
    ! [B3: state > $o,Aa: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aImp_state @ B3 @ Aa ) )
      = ( ( packag6595153354952283300_state @ plus @ core @ B3 )
        & ( packag7298478212905792770_state @ plus @ core @ Aa ) ) ) ).

% PartialSA.wf_assertion.simps(2)
thf(fact_618_PartialSA_Ois__footprint__generalI,axiom,
    ! [A5: aassertion_state,R4: state > state > state,W: state,B2: aassertion_state] :
      ( ! [A6: state,B7: state] :
          ( ( ( package_sat_state @ plus @ A5 @ A6 )
            & ( ( some_state @ B7 )
              = ( plus @ A6 @ ( R4 @ A6 @ W ) ) ) )
         => ( package_sat_state @ plus @ B2 @ B7 ) )
     => ( packag6192661653625646554_state @ plus @ W @ R4 @ A5 @ B2 ) ) ).

% PartialSA.is_footprint_generalI
thf(fact_619_PartialSA_Ocompleteness,axiom,
    ! [B2: aassertion_state,W: state,R4: state > state > state,A5: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ( ( stable @ W )
          & ( packag6192661653625646554_state @ plus @ W @ R4 @ A5 @ B2 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ( ! [A6: state] :
                ( ( package_sat_state @ plus @ A5 @ A6 )
               => ( packag4817583915586052229_state @ plus @ unit @ ( R4 @ A6 ) ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ ( R4 @ A2 ) ) ) )
                      & ( package_sat_state @ plus @ A5 @ A2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B2
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% PartialSA.completeness
thf(fact_620_PartialSA_Osoundness,axiom,
    ! [B2: aassertion_state,A5: aassertion_state,S2: set_state,R4: state > state > state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ! [A6: state] :
            ( ( package_sat_state @ plus @ A5 @ A6 )
           => ( member_state @ A6 @ S2 ) )
       => ( ! [A6: state] :
              ( ( member_state @ A6 @ S2 )
             => ( packag4817583915586052229_state @ plus @ unit @ ( R4 @ A6 ) ) )
         => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ ( R4 @ A2 ) ) ) )
                    & ( member_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B2 ) )
                | ( packag4152378406607043437_state @ plus @ S3 ) )
             => ( ( stable @ W )
                & ( ( some_state @ Sigma )
                  = ( plus @ Sigma2 @ W ) )
                & ( packag6192661653625646554_state @ plus @ W @ R4 @ A5 @ B2 ) ) ) ) ) ) ) ).

% PartialSA.soundness
thf(fact_621_PartialSA_Ominus__def,axiom,
    ( minus
    = ( ^ [B: state,A2: state] :
          ( fun_TH8221542053827138557_state @ B
          @ ^ [X: state] :
              ( ( ( some_state @ B )
                = ( plus @ A2 @ X ) )
              & ( greater @ X @ ( core @ B ) ) ) ) ) ) ).

% PartialSA.minus_def
thf(fact_622_PartialSA_Owf__assertion_Oelims_I1_J,axiom,
    ! [X2: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X2 )
        = Y )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ( Y
              = ( ~ ( ( packag7298478212905792770_state @ plus @ core @ A11 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ( Y
                = ( ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ( Y
                  = ( ~ ( packag2456304381420842418_state @ plus @ A11 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(1)
thf(fact_623_PartialSA_Owf__assertion_Oelims_I2_J,axiom,
    ! [X2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ~ ( ( packag7298478212905792770_state @ plus @ core @ A11 )
                & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                  & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ~ ( packag2456304381420842418_state @ plus @ A11 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(2)
thf(fact_624_aassertion_Oinject_I3_J,axiom,
    ! [X32: state > $o,Y32: state > $o] :
      ( ( ( aSem_state @ X32 )
        = ( aSem_state @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% aassertion.inject(3)
thf(fact_625_package__logic_Opure__remains_Ocong,axiom,
    packag4152378406607043437_state = packag4152378406607043437_state ).

% package_logic.pure_remains.cong
thf(fact_626_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_3064086290059306969_state @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_627_setcompr__eq__image,axiom,
    ! [F: state > produc7272280185086996969_state,P: state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_632992669292831658_state @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_628_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc7272280185086996969_state,P: produc9003781588162623004on_val > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_3204238854701872908_state @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_629_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > state,P: produc7272280185086996969_state > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_2710133271811360426_state @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_630_setcompr__eq__image,axiom,
    ! [F: state > state,P: state > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_state_state @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_631_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > state,P: produc9003781588162623004on_val > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_1205529093420515805_state @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_632_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc9003781588162623004on_val,P: produc7272280185086996969_state > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_3393013468800240652on_val @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_633_setcompr__eq__image,axiom,
    ! [F: state > produc9003781588162623004on_val,P: state > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_4772210114854293981on_val @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_634_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image_4170889944715819327on_val @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_635_Setcompr__eq__image,axiom,
    ! [F: state > state,A5: set_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member_state @ X @ A5 ) ) )
      = ( image_state_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_636_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > state,A5: set_Pr2391701600042252046_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member6715010751053357271_state @ X @ A5 ) ) )
      = ( image_6759111139246981411_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_637_Setcompr__eq__image,axiom,
    ! [F: state > produc7272280185086996969_state,A5: set_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member_state @ X @ A5 ) ) )
      = ( image_632992669292831658_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_638_Setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > state,A5: set_Pr4231240917283394591_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member7461423308196221312_state @ X @ A5 ) ) )
      = ( image_2710133271811360426_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_639_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > produc7272280185086996969_state,A5: set_Pr2391701600042252046_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member6715010751053357271_state @ X @ A5 ) ) )
      = ( image_4364510454419742290_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_640_Setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member7461423308196221312_state @ X @ A5 ) ) )
      = ( image_3064086290059306969_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_641_Setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > state,A5: set_Pr7649655576376376530on_val] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member2879030624002546611on_val @ X @ A5 ) ) )
      = ( image_1205529093420515805_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_642_Setcompr__eq__image,axiom,
    ! [F: state > produc9003781588162623004on_val,A5: set_state] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X: state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member_state @ X @ A5 ) ) )
      = ( image_4772210114854293981on_val @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_643_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A5: set_Pr2391701600042252046_state] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member6715010751053357271_state @ X @ A5 ) ) )
      = ( image_4690842878458256517on_val @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_644_Setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A5: set_Pr7649655576376376530on_val] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member2879030624002546611on_val @ X @ A5 ) ) )
      = ( image_3204238854701872908_state @ F @ A5 ) ) ).

% Setcompr_eq_image
thf(fact_645_aassertion_Odistinct_I3_J,axiom,
    ! [X11: aassertion_state,X12: aassertion_state,X32: state > $o] :
      ( ( aStar_state @ X11 @ X12 )
     != ( aSem_state @ X32 ) ) ).

% aassertion.distinct(3)
thf(fact_646_aassertion_Odistinct_I5_J,axiom,
    ! [X21: state > $o,X222: aassertion_state,X32: state > $o] :
      ( ( aImp_state @ X21 @ X222 )
     != ( aSem_state @ X32 ) ) ).

% aassertion.distinct(5)
thf(fact_647_PartialSA_Osat_Osimps_I3_J,axiom,
    ! [A5: state > $o,Phi: state] :
      ( ( package_sat_state @ plus @ ( aSem_state @ A5 ) @ Phi )
      = ( A5 @ Phi ) ) ).

% PartialSA.sat.simps(3)
thf(fact_648_full__SetCompr__eq,axiom,
    ! [F: state > produc7272280185086996969_state] :
      ( ( collec906845016635587262_state
        @ ^ [U6: produc7272280185086996969_state] :
          ? [X: state] :
            ( U6
            = ( F @ X ) ) )
      = ( image_632992669292831658_state @ F @ top_top_set_state ) ) ).

% full_SetCompr_eq
thf(fact_649_full__SetCompr__eq,axiom,
    ! [F: state > state] :
      ( ( collect_state
        @ ^ [U6: state] :
          ? [X: state] :
            ( U6
            = ( F @ X ) ) )
      = ( image_state_state @ F @ top_top_set_state ) ) ).

% full_SetCompr_eq
thf(fact_650_full__SetCompr__eq,axiom,
    ! [F: state > produc9003781588162623004on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [U6: produc9003781588162623004on_val] :
          ? [X: state] :
            ( U6
            = ( F @ X ) ) )
      = ( image_4772210114854293981on_val @ F @ top_top_set_state ) ) ).

% full_SetCompr_eq
thf(fact_651_PartialSA_Osat_Ocases,axiom,
    ! [X2: produc2756261550280135502_state] :
      ( ! [A11: aassertion_state,B8: aassertion_state,Phi5: state] :
          ( X2
         != ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Phi5 ) )
     => ( ! [B7: state > $o,A11: aassertion_state,Phi5: state] :
            ( X2
           != ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Phi5 ) )
       => ~ ! [A11: state > $o,Phi5: state] :
              ( X2
             != ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Phi5 ) ) ) ) ).

% PartialSA.sat.cases
thf(fact_652_PartialSA_Owf__assertion_Ocases,axiom,
    ! [X2: aassertion_state] :
      ( ! [A11: aassertion_state,B8: aassertion_state] :
          ( X2
         != ( aStar_state @ A11 @ B8 ) )
     => ( ! [B7: state > $o,A11: aassertion_state] :
            ( X2
           != ( aImp_state @ B7 @ A11 ) )
       => ~ ! [A11: state > $o] :
              ( X2
             != ( aSem_state @ A11 ) ) ) ) ).

% PartialSA.wf_assertion.cases
thf(fact_653_aassertion_Oexhaust,axiom,
    ! [Y: aassertion_state] :
      ( ! [X112: aassertion_state,X122: aassertion_state] :
          ( Y
         != ( aStar_state @ X112 @ X122 ) )
     => ( ! [X212: state > $o,X223: aassertion_state] :
            ( Y
           != ( aImp_state @ X212 @ X223 ) )
       => ~ ! [X33: state > $o] :
              ( Y
             != ( aSem_state @ X33 ) ) ) ) ).

% aassertion.exhaust
thf(fact_654_PartialSA_Oadd__set__def,axiom,
    ( add_set
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi4: state,A2: state,B: state] :
              ( ( Uu3 = Phi4 )
              & ( member_state @ A2 @ A4 )
              & ( member_state @ B @ B5 )
              & ( ( some_state @ Phi4 )
                = ( plus @ A2 @ B ) ) ) ) ) ) ).

% PartialSA.add_set_def
thf(fact_655_PartialSA_Oupper__closure__def,axiom,
    ! [A5: set_state] :
      ( ( sep_up1246176804924251236_state @ plus @ A5 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Phi3: state,Phi4: state] :
            ( ( Uu3 = Phi3 )
            & ( greater @ Phi3 @ Phi4 )
            & ( member_state @ Phi4 @ A5 ) ) ) ) ).

% PartialSA.upper_closure_def
thf(fact_656_PartialSA_Owf__assertion_Osimps_I3_J,axiom,
    ! [A5: state > $o] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aSem_state @ A5 ) )
      = ( packag2456304381420842418_state @ plus @ A5 ) ) ).

% PartialSA.wf_assertion.simps(3)
thf(fact_657_PartialSA_Osat_Oelims_I3_J,axiom,
    ! [X2: aassertion_state,Xa2: state] :
      ( ~ ( package_sat_state @ plus @ X2 @ Xa2 )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ? [A3: state,B9: state] :
                ( ( ( some_state @ Xa2 )
                  = ( plus @ A3 @ B9 ) )
                & ( package_sat_state @ plus @ A11 @ A3 )
                & ( package_sat_state @ plus @ B8 @ B9 ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ( ( B7 @ Xa2 )
               => ( package_sat_state @ plus @ A11 @ Xa2 ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ( A11 @ Xa2 ) ) ) ) ) ).

% PartialSA.sat.elims(3)
thf(fact_658_PartialSA_Osat_Oelims_I2_J,axiom,
    ! [X2: aassertion_state,Xa2: state] :
      ( ( package_sat_state @ plus @ X2 @ Xa2 )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ~ ? [A6: state,B7: state] :
                  ( ( ( some_state @ Xa2 )
                    = ( plus @ A6 @ B7 ) )
                  & ( package_sat_state @ plus @ A11 @ A6 )
                  & ( package_sat_state @ plus @ B8 @ B7 ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ~ ( ( B7 @ Xa2 )
                 => ( package_sat_state @ plus @ A11 @ Xa2 ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ~ ( A11 @ Xa2 ) ) ) ) ) ).

% PartialSA.sat.elims(2)
thf(fact_659_PartialSA_Osat_Oelims_I1_J,axiom,
    ! [X2: aassertion_state,Xa2: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X2 @ Xa2 )
        = Y )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ( Y
              = ( ~ ? [A2: state,B: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( plus @ A2 @ B ) )
                      & ( package_sat_state @ plus @ A11 @ A2 )
                      & ( package_sat_state @ plus @ B8 @ B ) ) ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ( Y
                = ( ~ ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A11 @ Xa2 ) ) ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ( Y
                  = ( ~ ( A11 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.elims(1)
thf(fact_660_PartialSA_Owf__assertion_Oelims_I3_J,axiom,
    ! [X2: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( ( X2
              = ( aStar_state @ A11 @ B8 ) )
           => ( ( packag7298478212905792770_state @ plus @ core @ A11 )
              & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( X2
                = ( aImp_state @ B7 @ A11 ) )
             => ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) )
         => ~ ! [A11: state > $o] :
                ( ( X2
                  = ( aSem_state @ A11 ) )
               => ( packag2456304381420842418_state @ plus @ A11 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(3)
thf(fact_661_PartialSA_Osoundness__paper,axiom,
    ! [B2: aassertion_state,A5: aassertion_state,S2: set_state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ! [A6: state] :
            ( ( package_sat_state @ plus @ A5 @ A6 )
           => ( member_state @ A6 @ S2 ) )
       => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                  & ( member_state @ A2 @ S2 ) ) )
            @ ^ [Uu3: state] : $true
            @ B2
            @ Sigma2
            @ W
            @ S3 )
         => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ B2 ) )
              | ( packag4152378406607043437_state @ plus @ S3 ) )
           => ( ( stable @ W )
              & ( ( some_state @ Sigma )
                = ( plus @ Sigma2 @ W ) )
              & ( packag7096211993527611489_state @ plus @ W @ A5 @ B2 ) ) ) ) ) ) ).

% PartialSA.soundness_paper
thf(fact_662_PartialSA_Ocompleteness__paper,axiom,
    ! [B2: aassertion_state,W: state,A5: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ( ( stable @ W )
          & ( packag7096211993527611489_state @ plus @ W @ A5 @ B2 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ? [X_1: set_Pr4231240917283394591_state] :
              ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                    & ( package_sat_state @ plus @ A5 @ A2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B2
              @ Sigma2
              @ W
              @ X_1 ) ) ) ) ).

% PartialSA.completeness_paper
thf(fact_663_package__logic_Osoundness,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Unit: produc7272280185086996969_state,Stable: produc7272280185086996969_state > $o,B2: aasser7019605857572776418_state,A5: aasser7019605857572776418_state,S2: set_Pr4231240917283394591_state,R4: produc7272280185086996969_state > produc7272280185086996969_state > produc7272280185086996969_state,Sigma: produc7272280185086996969_state,Sigma2: produc7272280185086996969_state,W: produc7272280185086996969_state,S3: set_Pr3289333603297600445_state] :
      ( ( packag2204431621656667814_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6254535061403973169_state @ Plus @ Core @ B2 )
       => ( ! [A6: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A5 @ A6 )
             => ( member7461423308196221312_state @ A6 @ S2 ) )
         => ( ! [A6: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ A6 @ S2 )
               => ( packag3884151059541177780_state @ Plus @ Unit @ ( R4 @ A6 ) ) )
           => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec7677102015826254172_state
                  @ ^ [Uu3: produc3057782296003698823_state] :
                    ? [A2: produc7272280185086996969_state] :
                      ( ( Uu3
                        = ( produc1904032489304825401_state @ A2 @ ( produc7648782313880683123_state @ Unit @ ( R4 @ A2 ) ) ) )
                      & ( member7461423308196221312_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc7272280185086996969_state] : $true
                @ B2
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1160585701684337059_state @ Plus @ ( packag4865960244449834244_state @ Plus @ B2 ) )
                  | ( packag3875795242785447337_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P6297086604321900740_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag7261830501500456713_state @ Plus @ W @ R4 @ A5 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_664_package__logic_Osoundness,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Unit: produc9003781588162623004on_val,Stable: produc9003781588162623004on_val > $o,B2: aasser1163550960519286933on_val,A5: aasser1163550960519286933on_val,S2: set_Pr7649655576376376530on_val,R4: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val,Sigma: produc9003781588162623004on_val,Sigma2: produc9003781588162623004on_val,W: produc9003781588162623004on_val,S3: set_Pr1565700067222966883on_val] :
      ( ( packag4549880205214723289on_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5541956234368469604on_val @ Plus @ Core @ B2 )
       => ( ! [A6: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A5 @ A6 )
             => ( member2879030624002546611on_val @ A6 @ S2 ) )
         => ( ! [A6: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ A6 @ S2 )
               => ( packag5955012701910386663on_val @ Plus @ Unit @ ( R4 @ A6 ) ) )
           => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec3787697688312515330on_val
                  @ ^ [Uu3: produc217507841372899885on_val] :
                    ? [A2: produc9003781588162623004on_val] :
                      ( ( Uu3
                        = ( produc6985948203474072671on_val @ A2 @ ( produc6949276712527400972on_val @ Unit @ ( R4 @ A2 ) ) ) )
                      & ( member2879030624002546611on_val @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc9003781588162623004on_val] : $true
                @ B2
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1465513459401564630on_val @ Plus @ ( packag3677137384183703351on_val @ Plus @ B2 ) )
                  | ( packag161099568566171125on_val @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P2581650809551911671on_val @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag8228529196479615804on_val @ Plus @ W @ R4 @ A5 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_665_package__logic_Osoundness,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Unit: produc3882902684156697774_state,Stable: produc3882902684156697774_state > $o,B2: aasser3179708585289803787_state,A5: aasser3179708585289803787_state,S2: set_Pr2391701600042252046_state,R4: produc3882902684156697774_state > produc3882902684156697774_state > produc3882902684156697774_state,Sigma: produc3882902684156697774_state,Sigma2: produc3882902684156697774_state,W: produc3882902684156697774_state,S3: set_Pr5308428015838290133_state] :
      ( ( packag9124413901382861105_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8517968027831007846_state @ Plus @ Core @ B2 )
       => ( ! [A6: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A5 @ A6 )
             => ( member6715010751053357271_state @ A6 @ S2 ) )
         => ( ! [A6: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ A6 @ S2 )
               => ( packag5827594511758291363_state @ Plus @ Unit @ ( R4 @ A6 ) ) )
           => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec6806307004801468532_state
                  @ ^ [Uu3: produc6455357576889117855_state] :
                    ? [A2: produc3882902684156697774_state] :
                      ( ( Uu3
                        = ( produc5427679348311974809_state @ A2 @ ( produc3987245858496418492_state @ Unit @ ( R4 @ A2 ) ) ) )
                      & ( member6715010751053357271_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: produc3882902684156697774_state] : $true
                @ B2
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag1930333982449825652_state @ Plus @ ( packag2518615827549365843_state @ Plus @ B2 ) )
                  | ( packag1386629450763126729_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_P5383513176685068947_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag4438072966337169294_state @ Plus @ W @ R4 @ A5 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_666_package__logic_Osoundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,A5: aassertion_state,S2: set_state,R4: state > state > state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ! [A6: state] :
              ( ( package_sat_state @ Plus @ A5 @ A6 )
             => ( member_state @ A6 @ S2 ) )
         => ( ! [A6: state] :
                ( ( member_state @ A6 @ S2 )
               => ( packag4817583915586052229_state @ Plus @ Unit @ ( R4 @ A6 ) ) )
           => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ ( R4 @ A2 ) ) ) )
                      & ( member_state @ A2 @ S2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B2
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ B2 ) )
                  | ( packag4152378406607043437_state @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_state @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag6192661653625646554_state @ Plus @ W @ R4 @ A5 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_667_accp__subset,axiom,
    ! [R1: produc2756261550280135502_state > produc2756261550280135502_state > $o,R22: produc2756261550280135502_state > produc2756261550280135502_state > $o] :
      ( ( ord_le5370872128109154830tate_o @ R1 @ R22 )
     => ( ord_le5566168861750573711tate_o @ ( accp_P1644577663358802135_state @ R22 ) @ ( accp_P1644577663358802135_state @ R1 ) ) ) ).

% accp_subset
thf(fact_668_accp__subset,axiom,
    ! [R1: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,R22: produc1317171644805857433on_val > produc1317171644805857433on_val > $o] :
      ( ( ord_le2319172193063290574_val_o @ R1 @ R22 )
     => ( ord_le8528861841627966596_val_o @ ( accp_P6797916352740768290on_val @ R22 ) @ ( accp_P6797916352740768290on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_669_accp__subset,axiom,
    ! [R1: produc3142500478612311029_state > produc3142500478612311029_state > $o,R22: produc3142500478612311029_state > produc3142500478612311029_state > $o] :
      ( ( ord_le3734019104314464462tate_o @ R1 @ R22 )
     => ( ord_le5366583112655343912tate_o @ ( accp_P7183347322223220350_state @ R22 ) @ ( accp_P7183347322223220350_state @ R1 ) ) ) ).

% accp_subset
thf(fact_670_accp__subset,axiom,
    ! [R1: produc4844402304972151915_state > produc4844402304972151915_state > $o,R22: produc4844402304972151915_state > produc4844402304972151915_state > $o] :
      ( ( ord_le2599682012455951694tate_o @ R1 @ R22 )
     => ( ord_le5681876185599785522tate_o @ ( accp_P6233459008282737652_state @ R22 ) @ ( accp_P6233459008282737652_state @ R1 ) ) ) ).

% accp_subset
thf(fact_671_accp__subset,axiom,
    ! [R1: produc6251377953716854561_state > produc6251377953716854561_state > $o,R22: produc6251377953716854561_state > produc6251377953716854561_state > $o] :
      ( ( ord_le249087669307848398tate_o @ R1 @ R22 )
     => ( ord_le8699832056682887676tate_o @ ( accp_P6554740477754186410_state @ R22 ) @ ( accp_P6554740477754186410_state @ R1 ) ) ) ).

% accp_subset
thf(fact_672_PartialSA_OAddFromOutside,axiom,
    ! [Phi: state,Phi2: state,M2: state,F3: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi7: state,F4: state,S5: set_Pr4231240917283394591_state,F: state,S2: set_Pr4231240917283394591_state] :
      ( ( ( some_state @ Phi )
        = ( plus @ Phi2 @ M2 ) )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ A5 @ Phi7 @ F4 @ S5 )
       => ( ( stable @ M2 )
         => ( ( ( some_state @ F3 )
              = ( plus @ F @ M2 ) )
           => ( ( S3
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T4: state > state,R2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                      & ( ( some_state @ R2 )
                        = ( plus @ A2 @ ( minus @ ( T4 @ F3 ) @ ( T4 @ F ) ) ) )
                      & ( defined @ R2 @ U6 ) ) ) )
             => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi7 @ F4 @ S5 ) ) ) ) ) ) ).

% PartialSA.AddFromOutside
thf(fact_673_package__logic_Ounit__neutral,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( some_state @ A )
        = ( Plus @ A @ Unit ) ) ) ).

% package_logic.unit_neutral
thf(fact_674_package__logic_Ostable__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state,X2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A )
       => ( ( Stable @ B3 )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B3 ) )
           => ( Stable @ X2 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_675_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_676_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_677_package__logic_Obool__conj__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state > $o,B3: state > $o,X2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1330488657391168505_state @ A @ B3 @ X2 )
        = ( ( A @ X2 )
          & ( B3 @ X2 ) ) ) ) ).

% package_logic.bool_conj_def
thf(fact_678_package__logic_Osat_Osimps_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: state > $o,A5: aassertion_state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aImp_state @ B3 @ A5 ) @ Phi )
        = ( ( B3 @ Phi )
         => ( package_sat_state @ Plus @ A5 @ Phi ) ) ) ) ).

% package_logic.sat.simps(2)
thf(fact_679_package__logic_Osat_Osimps_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: state > $o,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aSem_state @ A5 ) @ Phi )
        = ( A5 @ Phi ) ) ) ).

% package_logic.sat.simps(3)
thf(fact_680_package__logic_Owf__assertion_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ ( aStar_state @ A5 @ B2 ) )
        = ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
          & ( packag7298478212905792770_state @ Plus @ Core @ B2 ) ) ) ) ).

% package_logic.wf_assertion.simps(1)
thf(fact_681_package__logic_OAStar,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,B2: aassertion_state,Phi7: state,F4: state,S5: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
       => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F3 @ S3 @ Pc @ B2 @ Phi7 @ F4 @ S5 )
         => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ ( aStar_state @ A5 @ B2 ) @ Phi7 @ F4 @ S5 ) ) ) ) ).

% package_logic.AStar
thf(fact_682_package__logic_Omono__pure__cond__conj,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,B3: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
       => ( ( packag6595153354952283300_state @ Plus @ Core @ B3 )
         => ( packag6595153354952283300_state @ Plus @ Core @ ( packag1330488657391168505_state @ Pc @ B3 ) ) ) ) ) ).

% package_logic.mono_pure_cond_conj
thf(fact_683_Rep__state,axiom,
    ! [X2: state] :
      ( member2879030624002546611on_val @ ( rep_state @ X2 )
      @ ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [Phi4: produc9003781588162623004on_val] :
            ( ( Uu3 = Phi4 )
            & ( valid_state @ Phi4 ) ) ) ) ).

% Rep_state
thf(fact_684_Rep__state__cases,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y @ ( collec7475294502966882545on_val @ valid_state ) )
     => ~ ! [X4: state] :
            ( Y
           != ( rep_state @ X4 ) ) ) ).

% Rep_state_cases
thf(fact_685_Rep__state__induct,axiom,
    ! [Y: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ! [X4: state] : ( P @ ( rep_state @ X4 ) )
       => ( P @ Y ) ) ) ).

% Rep_state_induct
thf(fact_686_Abs__state__inject,axiom,
    ! [X2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ X2
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ( member2879030624002546611on_val @ Y
          @ ( collec7475294502966882545on_val
            @ ^ [Uu3: produc9003781588162623004on_val] :
              ? [Phi4: produc9003781588162623004on_val] :
                ( ( Uu3 = Phi4 )
                & ( valid_state @ Phi4 ) ) ) )
       => ( ( ( abs_state @ X2 )
            = ( abs_state @ Y ) )
          = ( X2 = Y ) ) ) ) ).

% Abs_state_inject
thf(fact_687_Abs__state__induct,axiom,
    ! [P: state > $o,X2: state] :
      ( ! [Y4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ Y4
            @ ( collec7475294502966882545on_val
              @ ^ [Uu3: produc9003781588162623004on_val] :
                ? [Phi4: produc9003781588162623004on_val] :
                  ( ( Uu3 = Phi4 )
                  & ( valid_state @ Phi4 ) ) ) )
         => ( P @ ( abs_state @ Y4 ) ) )
     => ( P @ X2 ) ) ).

% Abs_state_induct
thf(fact_688_Abs__state__cases,axiom,
    ! [X2: state] :
      ~ ! [Y4: produc9003781588162623004on_val] :
          ( ( X2
            = ( abs_state @ Y4 ) )
         => ~ ( member2879030624002546611on_val @ Y4 @ ( collec7475294502966882545on_val @ valid_state ) ) ) ).

% Abs_state_cases
thf(fact_689_package__logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A5 @ B2 ) @ Phi )
        = ( ? [A2: state,B: state] :
              ( ( ( some_state @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( package_sat_state @ Plus @ A5 @ A2 )
              & ( package_sat_state @ Plus @ B2 @ B ) ) ) ) ) ).

% package_logic.sat.simps(1)
thf(fact_690_package__logic_Osat_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: produc2756261550280135502_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A11: aassertion_state,B8: aassertion_state,Phi5: state] :
            ( X2
           != ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Phi5 ) )
       => ( ! [B7: state > $o,A11: aassertion_state,Phi5: state] :
              ( X2
             != ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Phi5 ) )
         => ~ ! [A11: state > $o,Phi5: state] :
                ( X2
               != ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Phi5 ) ) ) ) ) ).

% package_logic.sat.cases
thf(fact_691_package__logic_Owf__assertion_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A11: aassertion_state,B8: aassertion_state] :
            ( X2
           != ( aStar_state @ A11 @ B8 ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( X2
             != ( aImp_state @ B7 @ A11 ) )
         => ~ ! [A11: state > $o] :
                ( X2
               != ( aSem_state @ A11 ) ) ) ) ) ).

% package_logic.wf_assertion.cases
thf(fact_692_package__logic_Omono__pure__condI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi5: state] :
            ( ( B3 @ Phi5 )
            = ( B3 @ ( Core @ Phi5 ) ) )
       => ( ! [Phi5: state,Phi6: state,R3: state] :
              ( ( ( sep_pure_state @ Plus @ R3 )
                & ( ( some_state @ Phi6 )
                  = ( Plus @ Phi5 @ R3 ) )
                & ~ ( B3 @ Phi5 ) )
             => ~ ( B3 @ Phi6 ) )
         => ( packag6595153354952283300_state @ Plus @ Core @ B3 ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_693_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ B3 )
        = ( ! [Phi4: state] :
              ( ( B3 @ Phi4 )
              = ( B3 @ ( Core @ Phi4 ) ) )
          & ! [Phi3: state,Phi4: state,R2: state] :
              ( ( ( sep_pure_state @ Plus @ R2 )
                & ( ( some_state @ Phi3 )
                  = ( Plus @ Phi4 @ R2 ) )
                & ~ ( B3 @ Phi4 ) )
             => ~ ( B3 @ Phi3 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_694_package__logic_Owf__assertion_Osimps_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B3: state > $o,A5: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ ( aImp_state @ B3 @ A5 ) )
        = ( ( packag6595153354952283300_state @ Plus @ Core @ B3 )
          & ( packag7298478212905792770_state @ Plus @ Core @ A5 ) ) ) ) ).

% package_logic.wf_assertion.simps(2)
thf(fact_695_package__logic_Ois__footprint__standard__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,A5: aassertion_state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7096211993527611489_state @ Plus @ W @ A5 @ B2 )
        = ( ! [A2: state,B: state] :
              ( ( ( package_sat_state @ Plus @ A5 @ A2 )
                & ( ( some_state @ B )
                  = ( Plus @ A2 @ W ) ) )
             => ( package_sat_state @ Plus @ B2 @ B ) ) ) ) ) ).

% package_logic.is_footprint_standard_def
thf(fact_696_package__logic_Ois__footprint__general__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,R4: state > state > state,A5: aassertion_state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6192661653625646554_state @ Plus @ W @ R4 @ A5 @ B2 )
        = ( ! [A2: state,B: state] :
              ( ( ( package_sat_state @ Plus @ A5 @ A2 )
                & ( ( some_state @ B )
                  = ( Plus @ A2 @ ( R4 @ A2 @ W ) ) ) )
             => ( package_sat_state @ Plus @ B2 @ B ) ) ) ) ) ).

% package_logic.is_footprint_general_def
thf(fact_697_package__logic_Ois__footprint__standardI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,W: state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,B7: state] :
            ( ( ( package_sat_state @ Plus @ A5 @ A6 )
              & ( ( some_state @ B7 )
                = ( Plus @ A6 @ W ) ) )
           => ( package_sat_state @ Plus @ B2 @ B7 ) )
       => ( packag7096211993527611489_state @ Plus @ W @ A5 @ B2 ) ) ) ).

% package_logic.is_footprint_standardI
thf(fact_698_package__logic_Ois__footprint__generalI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,R4: state > state > state,W: state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,B7: state] :
            ( ( ( package_sat_state @ Plus @ A5 @ A6 )
              & ( ( some_state @ B7 )
                = ( Plus @ A6 @ ( R4 @ A6 @ W ) ) ) )
           => ( package_sat_state @ Plus @ B2 @ B7 ) )
       => ( packag6192661653625646554_state @ Plus @ W @ R4 @ A5 @ B2 ) ) ) ).

% package_logic.is_footprint_generalI
thf(fact_699_package__logic_Omono__pruner__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,P2: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2456304381420842418_state @ Plus @ P2 )
        = ( ! [Phi3: state,Phi4: state,R2: state] :
              ( ( ( sep_pure_state @ Plus @ R2 )
                & ( P2 @ Phi4 )
                & ( ( some_state @ Phi3 )
                  = ( Plus @ Phi4 @ R2 ) ) )
             => ( P2 @ Phi3 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_700_package__logic_Owf__assertion_Osimps_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ ( aSem_state @ A5 ) )
        = ( packag2456304381420842418_state @ Plus @ A5 ) ) ) ).

% package_logic.wf_assertion.simps(3)
thf(fact_701_package__logic_OAImp,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,B3: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ ( packag1330488657391168505_state @ Pc @ B3 ) @ A5 @ Phi2 @ F3 @ S3 )
       => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ ( aImp_state @ B3 @ A5 ) @ Phi2 @ F3 @ S3 ) ) ) ).

% package_logic.AImp
thf(fact_702_Abs__state__inverse,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi4 )
              & ( valid_state @ Phi4 ) ) ) )
     => ( ( rep_state @ ( abs_state @ Y ) )
        = Y ) ) ).

% Abs_state_inverse
thf(fact_703_PartialSA_Opackage__logic__axioms,axiom,
    package_logic_state @ plus @ core @ unit @ stable ).

% PartialSA.package_logic_axioms
thf(fact_704_package__logic_Owf__assertion__sat__larger__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,Phi: state,Phi2: state,R: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
       => ( ( package_sat_state @ Plus @ A5 @ Phi )
         => ( ( ( some_state @ Phi2 )
              = ( Plus @ Phi @ R ) )
           => ( ( sep_pure_state @ Plus @ R )
             => ( package_sat_state @ Plus @ A5 @ Phi2 ) ) ) ) ) ) ).

% package_logic.wf_assertion_sat_larger_pure
thf(fact_705_package__logic_Opackage__satI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A7: state,U5: state,U3: state,A5: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Pc @ ( Core @ A7 ) )
         => ? [X3: state] :
              ( ( ( some_state @ X3 )
                = ( Plus @ ( Core @ A7 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_state @ Plus @ A5 @ X3 ) ) )
       => ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A5 @ A7 @ U5 @ U3 ) ) ) ).

% package_logic.package_satI
thf(fact_706_package__logic_Opackage__sat__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A5: aassertion_state,A7: state,U5: state,U3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A5 @ A7 @ U5 @ U3 )
        = ( ( Pc @ ( Core @ A7 ) )
         => ? [X: state] :
              ( ( ( some_state @ X )
                = ( Plus @ ( Core @ A7 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_state @ Plus @ A5 @ X ) ) ) ) ) ).

% package_logic.package_sat_def
thf(fact_707_package__logic_Ocompleteness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,W: state,A5: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag7096211993527611489_state @ Plus @ W @ A5 @ B2 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                      & ( package_sat_state @ Plus @ A5 @ A2 ) ) )
                @ ^ [Uu3: state] : $true
                @ B2
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% package_logic.completeness_paper
thf(fact_708_package__logic_Osat_Oelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X2 @ Xa2 )
          = Y )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( Y
                = ( ~ ? [A2: state,B: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( Plus @ A2 @ B ) )
                        & ( package_sat_state @ Plus @ A11 @ A2 )
                        & ( package_sat_state @ Plus @ B8 @ B ) ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( Y
                  = ( ~ ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( Y
                    = ( ~ ( A11 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.elims(1)
thf(fact_709_package__logic_Osat_Oelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X2 @ Xa2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ~ ? [A6: state,B7: state] :
                    ( ( ( some_state @ Xa2 )
                      = ( Plus @ A6 @ B7 ) )
                    & ( package_sat_state @ Plus @ A11 @ A6 )
                    & ( package_sat_state @ Plus @ B8 @ B7 ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ~ ( ( B7 @ Xa2 )
                   => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ~ ( A11 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(2)
thf(fact_710_package__logic_Osat_Oelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X2 @ Xa2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ? [A3: state,B9: state] :
                  ( ( ( some_state @ Xa2 )
                    = ( Plus @ A3 @ B9 ) )
                  & ( package_sat_state @ Plus @ A11 @ A3 )
                  & ( package_sat_state @ Plus @ B8 @ B9 ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( B7 @ Xa2 )
                 => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( A11 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(3)
thf(fact_711_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Unit: produc7272280185086996969_state,Stable: produc7272280185086996969_state > $o,B2: aasser7019605857572776418_state,A5: aasser7019605857572776418_state,S2: set_Pr4231240917283394591_state,Sigma: produc7272280185086996969_state,Sigma2: produc7272280185086996969_state,W: produc7272280185086996969_state,S3: set_Pr3289333603297600445_state] :
      ( ( packag2204431621656667814_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6254535061403973169_state @ Plus @ Core @ B2 )
       => ( ! [A6: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A5 @ A6 )
             => ( member7461423308196221312_state @ A6 @ S2 ) )
         => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec7677102015826254172_state
                @ ^ [Uu3: produc3057782296003698823_state] :
                  ? [A2: produc7272280185086996969_state] :
                    ( ( Uu3
                      = ( produc1904032489304825401_state @ A2 @ ( produc7648782313880683123_state @ Unit @ id_Pro2933125508726022968_state ) ) )
                    & ( member7461423308196221312_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc7272280185086996969_state] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1160585701684337059_state @ Plus @ ( packag4865960244449834244_state @ Plus @ B2 ) )
                | ( packag3875795242785447337_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P6297086604321900740_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag8077625386405451152_state @ Plus @ W @ A5 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_712_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Unit: produc9003781588162623004on_val,Stable: produc9003781588162623004on_val > $o,B2: aasser1163550960519286933on_val,A5: aasser1163550960519286933on_val,S2: set_Pr7649655576376376530on_val,Sigma: produc9003781588162623004on_val,Sigma2: produc9003781588162623004on_val,W: produc9003781588162623004on_val,S3: set_Pr1565700067222966883on_val] :
      ( ( packag4549880205214723289on_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5541956234368469604on_val @ Plus @ Core @ B2 )
       => ( ! [A6: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A5 @ A6 )
             => ( member2879030624002546611on_val @ A6 @ S2 ) )
         => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec3787697688312515330on_val
                @ ^ [Uu3: produc217507841372899885on_val] :
                  ? [A2: produc9003781588162623004on_val] :
                    ( ( Uu3
                      = ( produc6985948203474072671on_val @ A2 @ ( produc6949276712527400972on_val @ Unit @ id_Pro1402184692240358763on_val ) ) )
                    & ( member2879030624002546611on_val @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc9003781588162623004on_val] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1465513459401564630on_val @ Plus @ ( packag3677137384183703351on_val @ Plus @ B2 ) )
                | ( packag161099568566171125on_val @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P2581650809551911671on_val @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag6258220590826782659on_val @ Plus @ W @ A5 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_713_package__logic_Osoundness__paper,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Unit: produc3882902684156697774_state,Stable: produc3882902684156697774_state > $o,B2: aasser3179708585289803787_state,A5: aasser3179708585289803787_state,S2: set_Pr2391701600042252046_state,Sigma: produc3882902684156697774_state,Sigma2: produc3882902684156697774_state,W: produc3882902684156697774_state,S3: set_Pr5308428015838290133_state] :
      ( ( packag9124413901382861105_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8517968027831007846_state @ Plus @ Core @ B2 )
       => ( ! [A6: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A5 @ A6 )
             => ( member6715010751053357271_state @ A6 @ S2 ) )
         => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec6806307004801468532_state
                @ ^ [Uu3: produc6455357576889117855_state] :
                  ? [A2: produc3882902684156697774_state] :
                    ( ( Uu3
                      = ( produc5427679348311974809_state @ A2 @ ( produc3987245858496418492_state @ Unit @ id_Pro689249233197853983_state ) ) )
                    & ( member6715010751053357271_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: produc3882902684156697774_state] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag1930333982449825652_state @ Plus @ ( packag2518615827549365843_state @ Plus @ B2 ) )
                | ( packag1386629450763126729_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_P5383513176685068947_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag3130788337556013255_state @ Plus @ W @ A5 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_714_package__logic_Osoundness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,A5: aassertion_state,S2: set_state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ! [A6: state] :
              ( ( package_sat_state @ Plus @ A5 @ A6 )
             => ( member_state @ A6 @ S2 ) )
         => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                    & ( member_state @ A2 @ S2 ) ) )
              @ ^ [Uu3: state] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ B2 ) )
                | ( packag4152378406607043437_state @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_state @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag7096211993527611489_state @ Plus @ W @ A5 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_715_package__logic_Owf__assertion_Oelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( packag7298478212905792770_state @ Plus @ Core @ X2 )
          = Y )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( Y
                = ( ~ ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                      & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( Y
                  = ( ~ ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                        & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( Y
                    = ( ~ ( packag2456304381420842418_state @ Plus @ A11 ) ) ) ) ) ) ) ) ).

% package_logic.wf_assertion.elims(1)
thf(fact_716_package__logic_Owf__assertion_Oelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ X2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ~ ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                  & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ~ ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                    & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ~ ( packag2456304381420842418_state @ Plus @ A11 ) ) ) ) ) ) ).

% package_logic.wf_assertion.elims(2)
thf(fact_717_package__logic_Owf__assertion_Oelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( packag7298478212905792770_state @ Plus @ Core @ X2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                  & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( packag2456304381420842418_state @ Plus @ A11 ) ) ) ) ) ) ).

% package_logic.wf_assertion.elims(3)
thf(fact_718_package__logic_Ogeneral__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F: state,A5: aassertion_state,Phi: state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ! [X4: state] :
                ( ( ( some_state @ X4 )
                  = ( Plus @ A6 @ ( T3 @ F ) ) )
               => ( package_sat_state @ Plus @ A5 @ X4 ) ) )
       => ( ( ( some_state @ Phi )
            = ( Plus @ Phi2 @ F ) )
         => ( ( Stable @ F )
           => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ Unit )
             => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
               => ? [X_1: set_Pr4231240917283394591_state] :
                    ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ Unit @ S2
                    @ ^ [Uu3: state] : $true
                    @ A5
                    @ Phi2
                    @ F
                    @ X_1 ) ) ) ) ) ) ) ).

% package_logic.general_completeness
thf(fact_719_accp__subset__induct,axiom,
    ! [D2: produc2756261550280135502_state > $o,R4: produc2756261550280135502_state > produc2756261550280135502_state > $o,X2: produc2756261550280135502_state,P: produc2756261550280135502_state > $o] :
      ( ( ord_le5566168861750573711tate_o @ D2 @ ( accp_P1644577663358802135_state @ R4 ) )
     => ( ! [X4: produc2756261550280135502_state,Z4: produc2756261550280135502_state] :
            ( ( D2 @ X4 )
           => ( ( R4 @ Z4 @ X4 )
             => ( D2 @ Z4 ) ) )
       => ( ( D2 @ X2 )
         => ( ! [X4: produc2756261550280135502_state] :
                ( ( D2 @ X4 )
               => ( ! [Z5: produc2756261550280135502_state] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_720_accp__subset__induct,axiom,
    ! [D2: produc1317171644805857433on_val > $o,R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X2: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( ord_le8528861841627966596_val_o @ D2 @ ( accp_P6797916352740768290on_val @ R4 ) )
     => ( ! [X4: produc1317171644805857433on_val,Z4: produc1317171644805857433on_val] :
            ( ( D2 @ X4 )
           => ( ( R4 @ Z4 @ X4 )
             => ( D2 @ Z4 ) ) )
       => ( ( D2 @ X2 )
         => ( ! [X4: produc1317171644805857433on_val] :
                ( ( D2 @ X4 )
               => ( ! [Z5: produc1317171644805857433on_val] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_721_accp__subset__induct,axiom,
    ! [D2: produc3142500478612311029_state > $o,R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,X2: produc3142500478612311029_state,P: produc3142500478612311029_state > $o] :
      ( ( ord_le5366583112655343912tate_o @ D2 @ ( accp_P7183347322223220350_state @ R4 ) )
     => ( ! [X4: produc3142500478612311029_state,Z4: produc3142500478612311029_state] :
            ( ( D2 @ X4 )
           => ( ( R4 @ Z4 @ X4 )
             => ( D2 @ Z4 ) ) )
       => ( ( D2 @ X2 )
         => ( ! [X4: produc3142500478612311029_state] :
                ( ( D2 @ X4 )
               => ( ! [Z5: produc3142500478612311029_state] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_722_accp__subset__induct,axiom,
    ! [D2: produc4844402304972151915_state > $o,R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,X2: produc4844402304972151915_state,P: produc4844402304972151915_state > $o] :
      ( ( ord_le5681876185599785522tate_o @ D2 @ ( accp_P6233459008282737652_state @ R4 ) )
     => ( ! [X4: produc4844402304972151915_state,Z4: produc4844402304972151915_state] :
            ( ( D2 @ X4 )
           => ( ( R4 @ Z4 @ X4 )
             => ( D2 @ Z4 ) ) )
       => ( ( D2 @ X2 )
         => ( ! [X4: produc4844402304972151915_state] :
                ( ( D2 @ X4 )
               => ( ! [Z5: produc4844402304972151915_state] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_723_accp__subset__induct,axiom,
    ! [D2: produc6251377953716854561_state > $o,R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,X2: produc6251377953716854561_state,P: produc6251377953716854561_state > $o] :
      ( ( ord_le8699832056682887676tate_o @ D2 @ ( accp_P6554740477754186410_state @ R4 ) )
     => ( ! [X4: produc6251377953716854561_state,Z4: produc6251377953716854561_state] :
            ( ( D2 @ X4 )
           => ( ( R4 @ Z4 @ X4 )
             => ( D2 @ Z4 ) ) )
       => ( ( D2 @ X2 )
         => ( ! [X4: produc6251377953716854561_state] :
                ( ( D2 @ X4 )
               => ( ! [Z5: produc6251377953716854561_state] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_724_package__logic_Ocompleteness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,W: state,R4: state > state > state,A5: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag6192661653625646554_state @ Plus @ W @ R4 @ A5 @ B2 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ( ! [A6: state] :
                  ( ( package_sat_state @ Plus @ A5 @ A6 )
                 => ( packag4817583915586052229_state @ Plus @ Unit @ ( R4 @ A6 ) ) )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ ( R4 @ A2 ) ) ) )
                        & ( package_sat_state @ Plus @ A5 @ A2 ) ) )
                  @ ^ [Uu3: state] : $true
                  @ B2
                  @ Sigma2
                  @ W
                  @ X_1 ) ) ) ) ) ) ).

% package_logic.completeness
thf(fact_725_PartialSA_Opackage__rhs_Ocases,axiom,
    ! [A1: state,A22: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
     => ( ! [A11: aassertion_state,Phi6: state,F5: state,S6: set_Pr4231240917283394591_state,B8: aassertion_state] :
            ( ( A52
              = ( aStar_state @ A11 @ B8 ) )
           => ( ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ A42 @ A11 @ Phi6 @ F5 @ S6 )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F5 @ S6 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
       => ( ! [B7: state > $o,A11: aassertion_state] :
              ( ( A52
                = ( aImp_state @ B7 @ A11 ) )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ ( packag1330488657391168505_state @ A42 @ B7 ) @ A11 @ A62 @ A72 @ A82 ) )
         => ( ! [Witness: produc7272280185086996969_state > state,B8: state > $o] :
                ( ( A52
                  = ( aSem_state @ B8 ) )
               => ( ( A62 = A1 )
                 => ( ( A72 = A22 )
                   => ( ! [A3: state,U7: state,T5: state > state] :
                          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) @ A32 )
                         => ( ( A42 @ A3 )
                           => ( ( greater @ A3 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) )
                              & ( B8 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) ) ) ) )
                     => ( A82
                       != ( sup_su3837536599209510219_state
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A2: state,U6: state,T4: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                & ~ ( A42 @ A2 ) ) )
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A2: state,U6: state,T4: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ ( minus @ A2 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) @ T4 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                & ( A42 @ A2 ) ) ) ) ) ) ) ) )
           => ~ ! [Phi6: state,M3: state] :
                  ( ( ( some_state @ A1 )
                    = ( plus @ Phi6 @ M3 ) )
                 => ! [F5: state,S6: set_Pr4231240917283394591_state] :
                      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F5 @ S6 @ A42 @ A52 @ A62 @ A72 @ A82 )
                     => ( ( stable @ M3 )
                       => ( ( ( some_state @ F5 )
                            = ( plus @ A22 @ M3 ) )
                         => ( S6
                           != ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T4: state > state,R2: state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ( ( some_state @ R2 )
                                    = ( plus @ A2 @ ( minus @ ( T4 @ F5 ) @ ( T4 @ A22 ) ) ) )
                                  & ( defined @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.cases
thf(fact_726_PartialSA_Opackage__rhs_Osimps,axiom,
    ! [A1: state,A22: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
      = ( ? [Phi4: state,F6: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,A4: aassertion_state,Phi3: state,F7: state,S8: set_Pr4231240917283394591_state,B5: aassertion_state,Phi8: state,F8: state,S9: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi4 )
            & ( A22 = F6 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aStar_state @ A4 @ B5 ) )
            & ( A62 = Phi8 )
            & ( A72 = F8 )
            & ( A82 = S9 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi4 @ F6 @ S7 @ Pc2 @ A4 @ Phi3 @ F7 @ S8 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F7 @ S8 @ Pc2 @ B5 @ Phi8 @ F8 @ S9 ) )
        | ? [Phi4: state,F6: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,B: state > $o,A4: aassertion_state,Phi3: state,F7: state,S8: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi4 )
            & ( A22 = F6 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aImp_state @ B @ A4 ) )
            & ( A62 = Phi3 )
            & ( A72 = F7 )
            & ( A82 = S8 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi4 @ F6 @ S7 @ ( packag1330488657391168505_state @ Pc2 @ B ) @ A4 @ Phi3 @ F7 @ S8 ) )
        | ? [S7: set_Pr4231240917283394591_state,Pc2: state > $o,Witness2: produc7272280185086996969_state > state,B5: state > $o,S8: set_Pr4231240917283394591_state,Phi4: state,F6: state] :
            ( ( A1 = Phi4 )
            & ( A22 = F6 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aSem_state @ B5 ) )
            & ( A62 = Phi4 )
            & ( A72 = F6 )
            & ( A82 = S8 )
            & ! [X: state,Y3: state,Z6: state > state,Aa2: state] :
                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z6 ) ) @ S7 )
               => ( ( Pc2 @ X )
                 => ( ( Aa2
                      = ( Witness2 @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z6 ) ) ) )
                   => ( ( greater @ X @ Aa2 )
                      & ( B5 @ Aa2 ) ) ) ) )
            & ( S8
              = ( sup_su3837536599209510219_state
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T4: state > state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                      & ~ ( Pc2 @ A2 ) ) )
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T4: state > state,B: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ ( minus @ A2 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B ) ) @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                      & ( Pc2 @ A2 )
                      & ( B
                        = ( Witness2 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) ) )
        | ? [Phi4: state,Phi3: state,M4: state,F7: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,A4: aassertion_state,Phi8: state,F8: state,S9: set_Pr4231240917283394591_state,F6: state,S7: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi4 )
            & ( A22 = F6 )
            & ( A32 = S7 )
            & ( A42 = Pc2 )
            & ( A52 = A4 )
            & ( A62 = Phi8 )
            & ( A72 = F8 )
            & ( A82 = S9 )
            & ( ( some_state @ Phi4 )
              = ( plus @ Phi3 @ M4 ) )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F7 @ S8 @ Pc2 @ A4 @ Phi8 @ F8 @ S9 )
            & ( stable @ M4 )
            & ( ( some_state @ F7 )
              = ( plus @ F6 @ M4 ) )
            & ( S8
              = ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T4: state > state,R2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                    & ( ( some_state @ R2 )
                      = ( plus @ A2 @ ( minus @ ( T4 @ F7 ) @ ( T4 @ F6 ) ) ) )
                    & ( defined @ R2 @ U6 ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.simps
thf(fact_727_PartialSA_OASem,axiom,
    ! [S2: set_Pr4231240917283394591_state,Pc: state > $o,Witness3: produc7272280185086996969_state > state,B2: state > $o,S3: set_Pr4231240917283394591_state,Phi: state,F: state] :
      ( ! [A6: state,U: state,T3: state > state,B7: state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( Pc @ A6 )
           => ( ( B7
                = ( Witness3 @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) ) )
             => ( ( greater @ A6 @ B7 )
                & ( B2 @ B7 ) ) ) ) )
     => ( ( S3
          = ( sup_su3837536599209510219_state
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state,U6: state,T4: state > state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                  & ~ ( Pc @ A2 ) ) )
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A2: state,U6: state,T4: state > state,B: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ ( minus @ A2 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B ) ) @ T4 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                  & ( Pc @ A2 )
                  & ( B
                    = ( Witness3 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aSem_state @ B2 ) @ Phi @ F @ S3 ) ) ) ).

% PartialSA.ASem
thf(fact_728_package__logic_Owf__assertion_Opelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( packag7298478212905792770_state @ Plus @ Core @ X2 )
          = Y )
       => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( Y
                    = ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                      & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) )
                 => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( Y
                      = ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                        & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) )
                   => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( Y
                        = ( packag2456304381420842418_state @ Plus @ A11 ) )
                     => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) ) ) ) ) ) ) ) ) ).

% package_logic.wf_assertion.pelims(1)
thf(fact_729_UnCI,axiom,
    ! [C2: state,B2: set_state,A5: set_state] :
      ( ( ~ ( member_state @ C2 @ B2 )
       => ( member_state @ C2 @ A5 ) )
     => ( member_state @ C2 @ ( sup_sup_set_state @ A5 @ B2 ) ) ) ).

% UnCI
thf(fact_730_UnCI,axiom,
    ! [C2: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( ~ ( member2879030624002546611on_val @ C2 @ B2 )
       => ( member2879030624002546611on_val @ C2 @ A5 ) )
     => ( member2879030624002546611on_val @ C2 @ ( sup_su7506060154676209150on_val @ A5 @ B2 ) ) ) ).

% UnCI
thf(fact_731_UnCI,axiom,
    ! [C2: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,A5: set_Pr2391701600042252046_state] :
      ( ( ~ ( member6715010751053357271_state @ C2 @ B2 )
       => ( member6715010751053357271_state @ C2 @ A5 ) )
     => ( member6715010751053357271_state @ C2 @ ( sup_su1984960714304006370_state @ A5 @ B2 ) ) ) ).

% UnCI
thf(fact_732_UnCI,axiom,
    ! [C2: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,A5: set_Pr4231240917283394591_state] :
      ( ( ~ ( member7461423308196221312_state @ C2 @ B2 )
       => ( member7461423308196221312_state @ C2 @ A5 ) )
     => ( member7461423308196221312_state @ C2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ) ).

% UnCI
thf(fact_733_Un__iff,axiom,
    ! [C2: state,A5: set_state,B2: set_state] :
      ( ( member_state @ C2 @ ( sup_sup_set_state @ A5 @ B2 ) )
      = ( ( member_state @ C2 @ A5 )
        | ( member_state @ C2 @ B2 ) ) ) ).

% Un_iff
thf(fact_734_Un__iff,axiom,
    ! [C2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C2 @ ( sup_su7506060154676209150on_val @ A5 @ B2 ) )
      = ( ( member2879030624002546611on_val @ C2 @ A5 )
        | ( member2879030624002546611on_val @ C2 @ B2 ) ) ) ).

% Un_iff
thf(fact_735_Un__iff,axiom,
    ! [C2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C2 @ ( sup_su1984960714304006370_state @ A5 @ B2 ) )
      = ( ( member6715010751053357271_state @ C2 @ A5 )
        | ( member6715010751053357271_state @ C2 @ B2 ) ) ) ).

% Un_iff
thf(fact_736_Un__iff,axiom,
    ! [C2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
      = ( ( member7461423308196221312_state @ C2 @ A5 )
        | ( member7461423308196221312_state @ C2 @ B2 ) ) ) ).

% Un_iff
thf(fact_737_Un__subset__iff,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A5 @ B2 ) @ C3 )
      = ( ( ord_le4032880777312351359_state @ A5 @ C3 )
        & ( ord_le4032880777312351359_state @ B2 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_738_Un__subset__iff,axiom,
    ! [A5: set_state,B2: set_state,C3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A5 @ B2 ) @ C3 )
      = ( ( ord_le2494988322063910608_state @ A5 @ C3 )
        & ( ord_le2494988322063910608_state @ B2 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_739_Un__insert__left,axiom,
    ! [A: state,B2: set_state,C3: set_state] :
      ( ( sup_sup_set_state @ ( insert_state @ A @ B2 ) @ C3 )
      = ( insert_state @ A @ ( sup_sup_set_state @ B2 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_740_Un__insert__left,axiom,
    ! [A: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ ( insert175804457004744729_state @ A @ B2 ) @ C3 )
      = ( insert175804457004744729_state @ A @ ( sup_su3837536599209510219_state @ B2 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_741_Un__insert__right,axiom,
    ! [A5: set_state,A: state,B2: set_state] :
      ( ( sup_sup_set_state @ A5 @ ( insert_state @ A @ B2 ) )
      = ( insert_state @ A @ ( sup_sup_set_state @ A5 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_742_Un__insert__right,axiom,
    ! [A5: set_Pr4231240917283394591_state,A: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ ( insert175804457004744729_state @ A @ B2 ) )
      = ( insert175804457004744729_state @ A @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_743_UnE,axiom,
    ! [C2: state,A5: set_state,B2: set_state] :
      ( ( member_state @ C2 @ ( sup_sup_set_state @ A5 @ B2 ) )
     => ( ~ ( member_state @ C2 @ A5 )
       => ( member_state @ C2 @ B2 ) ) ) ).

% UnE
thf(fact_744_UnE,axiom,
    ! [C2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C2 @ ( sup_su7506060154676209150on_val @ A5 @ B2 ) )
     => ( ~ ( member2879030624002546611on_val @ C2 @ A5 )
       => ( member2879030624002546611on_val @ C2 @ B2 ) ) ) ).

% UnE
thf(fact_745_UnE,axiom,
    ! [C2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C2 @ ( sup_su1984960714304006370_state @ A5 @ B2 ) )
     => ( ~ ( member6715010751053357271_state @ C2 @ A5 )
       => ( member6715010751053357271_state @ C2 @ B2 ) ) ) ).

% UnE
thf(fact_746_UnE,axiom,
    ! [C2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
     => ( ~ ( member7461423308196221312_state @ C2 @ A5 )
       => ( member7461423308196221312_state @ C2 @ B2 ) ) ) ).

% UnE
thf(fact_747_UnI1,axiom,
    ! [C2: state,A5: set_state,B2: set_state] :
      ( ( member_state @ C2 @ A5 )
     => ( member_state @ C2 @ ( sup_sup_set_state @ A5 @ B2 ) ) ) ).

% UnI1
thf(fact_748_UnI1,axiom,
    ! [C2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C2 @ A5 )
     => ( member2879030624002546611on_val @ C2 @ ( sup_su7506060154676209150on_val @ A5 @ B2 ) ) ) ).

% UnI1
thf(fact_749_UnI1,axiom,
    ! [C2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C2 @ A5 )
     => ( member6715010751053357271_state @ C2 @ ( sup_su1984960714304006370_state @ A5 @ B2 ) ) ) ).

% UnI1
thf(fact_750_UnI1,axiom,
    ! [C2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C2 @ A5 )
     => ( member7461423308196221312_state @ C2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ) ).

% UnI1
thf(fact_751_UnI2,axiom,
    ! [C2: state,B2: set_state,A5: set_state] :
      ( ( member_state @ C2 @ B2 )
     => ( member_state @ C2 @ ( sup_sup_set_state @ A5 @ B2 ) ) ) ).

% UnI2
thf(fact_752_UnI2,axiom,
    ! [C2: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C2 @ B2 )
     => ( member2879030624002546611on_val @ C2 @ ( sup_su7506060154676209150on_val @ A5 @ B2 ) ) ) ).

% UnI2
thf(fact_753_UnI2,axiom,
    ! [C2: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,A5: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C2 @ B2 )
     => ( member6715010751053357271_state @ C2 @ ( sup_su1984960714304006370_state @ A5 @ B2 ) ) ) ).

% UnI2
thf(fact_754_UnI2,axiom,
    ! [C2: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,A5: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C2 @ B2 )
     => ( member7461423308196221312_state @ C2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ) ).

% UnI2
thf(fact_755_bex__Un,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( ? [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
            & ( P @ X ) ) )
      = ( ? [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ A5 )
            & ( P @ X ) )
        | ? [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ B2 )
            & ( P @ X ) ) ) ) ).

% bex_Un
thf(fact_756_ball__Un,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
           => ( P @ X ) ) )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ A5 )
           => ( P @ X ) )
        & ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ B2 )
           => ( P @ X ) ) ) ) ).

% ball_Un
thf(fact_757_Un__assoc,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ ( sup_su3837536599209510219_state @ A5 @ B2 ) @ C3 )
      = ( sup_su3837536599209510219_state @ A5 @ ( sup_su3837536599209510219_state @ B2 @ C3 ) ) ) ).

% Un_assoc
thf(fact_758_Un__absorb,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ A5 )
      = A5 ) ).

% Un_absorb
thf(fact_759_Un__commute,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] : ( sup_su3837536599209510219_state @ B5 @ A4 ) ) ) ).

% Un_commute
thf(fact_760_Un__left__absorb,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
      = ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ).

% Un_left_absorb
thf(fact_761_Un__left__commute,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ ( sup_su3837536599209510219_state @ B2 @ C3 ) )
      = ( sup_su3837536599209510219_state @ B2 @ ( sup_su3837536599209510219_state @ A5 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_762_Un__mono,axiom,
    ! [A5: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,D2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A5 @ C3 )
     => ( ( ord_le4032880777312351359_state @ B2 @ D2 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A5 @ B2 ) @ ( sup_su3837536599209510219_state @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_763_Un__mono,axiom,
    ! [A5: set_state,C3: set_state,B2: set_state,D2: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ C3 )
     => ( ( ord_le2494988322063910608_state @ B2 @ D2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A5 @ B2 ) @ ( sup_sup_set_state @ C3 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_764_Un__least,axiom,
    ! [A5: set_Pr4231240917283394591_state,C3: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A5 @ C3 )
     => ( ( ord_le4032880777312351359_state @ B2 @ C3 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A5 @ B2 ) @ C3 ) ) ) ).

% Un_least
thf(fact_765_Un__least,axiom,
    ! [A5: set_state,C3: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ C3 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A5 @ B2 ) @ C3 ) ) ) ).

% Un_least
thf(fact_766_Un__upper1,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ A5 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ).

% Un_upper1
thf(fact_767_Un__upper1,axiom,
    ! [A5: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ A5 @ ( sup_sup_set_state @ A5 @ B2 ) ) ).

% Un_upper1
thf(fact_768_Un__upper2,axiom,
    ! [B2: set_Pr4231240917283394591_state,A5: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ B2 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) ) ).

% Un_upper2
thf(fact_769_Un__upper2,axiom,
    ! [B2: set_state,A5: set_state] : ( ord_le2494988322063910608_state @ B2 @ ( sup_sup_set_state @ A5 @ B2 ) ) ).

% Un_upper2
thf(fact_770_Un__absorb1,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A5 @ B2 )
     => ( ( sup_su3837536599209510219_state @ A5 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_771_Un__absorb1,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ B2 )
     => ( ( sup_sup_set_state @ A5 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_772_Un__absorb2,axiom,
    ! [B2: set_Pr4231240917283394591_state,A5: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B2 @ A5 )
     => ( ( sup_su3837536599209510219_state @ A5 @ B2 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_773_Un__absorb2,axiom,
    ! [B2: set_state,A5: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A5 )
     => ( ( sup_sup_set_state @ A5 @ B2 )
        = A5 ) ) ).

% Un_absorb2
thf(fact_774_subset__UnE,axiom,
    ! [C3: set_Pr4231240917283394591_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C3 @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
     => ~ ! [A12: set_Pr4231240917283394591_state] :
            ( ( ord_le4032880777312351359_state @ A12 @ A5 )
           => ! [B10: set_Pr4231240917283394591_state] :
                ( ( ord_le4032880777312351359_state @ B10 @ B2 )
               => ( C3
                 != ( sup_su3837536599209510219_state @ A12 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_775_subset__UnE,axiom,
    ! [C3: set_state,A5: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ C3 @ ( sup_sup_set_state @ A5 @ B2 ) )
     => ~ ! [A12: set_state] :
            ( ( ord_le2494988322063910608_state @ A12 @ A5 )
           => ! [B10: set_state] :
                ( ( ord_le2494988322063910608_state @ B10 @ B2 )
               => ( C3
                 != ( sup_sup_set_state @ A12 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_776_subset__Un__eq,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A4 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_777_subset__Un__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A4: set_state,B5: set_state] :
          ( ( sup_sup_set_state @ A4 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_778_Un__UNIV__right,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ top_to174962084833059535_state )
      = top_to174962084833059535_state ) ).

% Un_UNIV_right
thf(fact_779_Un__UNIV__right,axiom,
    ! [A5: set_state] :
      ( ( sup_sup_set_state @ A5 @ top_top_set_state )
      = top_top_set_state ) ).

% Un_UNIV_right
thf(fact_780_Un__UNIV__left,axiom,
    ! [B2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ top_to174962084833059535_state @ B2 )
      = top_to174962084833059535_state ) ).

% Un_UNIV_left
thf(fact_781_Un__UNIV__left,axiom,
    ! [B2: set_state] :
      ( ( sup_sup_set_state @ top_top_set_state @ B2 )
      = top_top_set_state ) ).

% Un_UNIV_left
thf(fact_782_image__Un,axiom,
    ! [F: state > state,A5: set_state,B2: set_state] :
      ( ( image_state_state @ F @ ( sup_sup_set_state @ A5 @ B2 ) )
      = ( sup_sup_set_state @ ( image_state_state @ F @ A5 ) @ ( image_state_state @ F @ B2 ) ) ) ).

% image_Un
thf(fact_783_image__Un,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( image_3064086290059306969_state @ F @ ( sup_su3837536599209510219_state @ A5 @ B2 ) )
      = ( sup_su3837536599209510219_state @ ( image_3064086290059306969_state @ F @ A5 ) @ ( image_3064086290059306969_state @ F @ B2 ) ) ) ).

% image_Un
thf(fact_784_Un__def,axiom,
    ( sup_su1984960714304006370_state
    = ( ^ [A4: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [X: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X @ A4 )
              | ( member6715010751053357271_state @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_785_Un__def,axiom,
    ( sup_sup_set_state
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ^ [X: state] :
              ( ( member_state @ X @ A4 )
              | ( member_state @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_786_Un__def,axiom,
    ( sup_su7506060154676209150on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [X: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X @ A4 )
              | ( member2879030624002546611on_val @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_787_Un__def,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ A4 )
              | ( member7461423308196221312_state @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_788_Collect__disj__eq,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ( collect_state
        @ ^ [X: state] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_state @ ( collect_state @ P ) @ ( collect_state @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_789_Collect__disj__eq,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su7506060154676209150on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_790_Collect__disj__eq,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su3837536599209510219_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_791_insert__def,axiom,
    ( insert_state
    = ( ^ [A2: state] :
          ( sup_sup_set_state
          @ ( collect_state
            @ ^ [X: state] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_792_insert__def,axiom,
    ( insert3030518898131756108on_val
    = ( ^ [A2: produc9003781588162623004on_val] :
          ( sup_su7506060154676209150on_val
          @ ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_793_insert__def,axiom,
    ( insert175804457004744729_state
    = ( ^ [A2: produc7272280185086996969_state] :
          ( sup_su3837536599209510219_state
          @ ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_794_PartialSA_Owf__assertion_Opelims_I3_J,axiom,
    ! [X2: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) )
               => ( ( packag7298478212905792770_state @ plus @ core @ A11 )
                  & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) )
                 => ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                    & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) )
                   => ( packag2456304381420842418_state @ plus @ A11 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(3)
thf(fact_795_PartialSA_Owf__assertion_Opelims_I2_J,axiom,
    ! [X2: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X2 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) )
               => ~ ( ( packag7298478212905792770_state @ plus @ core @ A11 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) )
                 => ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) )
                   => ~ ( packag2456304381420842418_state @ plus @ A11 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(2)
thf(fact_796_PartialSA_Owf__assertion_Opelims_I1_J,axiom,
    ! [X2: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X2 )
        = Y )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( Y
                  = ( ( packag7298478212905792770_state @ plus @ core @ A11 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
               => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( Y
                    = ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A11 ) ) )
                 => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( Y
                      = ( packag2456304381420842418_state @ plus @ A11 ) )
                   => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(1)
thf(fact_797_package__logic_Owf__assertion_Opelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( packag7298478212905792770_state @ Plus @ Core @ X2 )
       => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) )
                 => ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                    & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) )
                   => ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                      & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) )
                     => ( packag2456304381420842418_state @ Plus @ A11 ) ) ) ) ) ) ) ) ).

% package_logic.wf_assertion.pelims(3)
thf(fact_798_package__logic_Owf__assertion_Opelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ X2 )
       => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X2 )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A11 @ B8 ) )
                 => ~ ( ( packag7298478212905792770_state @ Plus @ Core @ A11 )
                      & ( packag7298478212905792770_state @ Plus @ Core @ B8 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A11 ) )
                   => ~ ( ( packag6595153354952283300_state @ Plus @ Core @ B7 )
                        & ( packag7298478212905792770_state @ Plus @ Core @ A11 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A11 ) )
                     => ~ ( packag2456304381420842418_state @ Plus @ A11 ) ) ) ) ) ) ) ) ).

% package_logic.wf_assertion.pelims(2)
thf(fact_799_sup_Obounded__iff,axiom,
    ! [B3: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B3 @ C2 ) @ A )
      = ( ( ord_le4032880777312351359_state @ B3 @ A )
        & ( ord_le4032880777312351359_state @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_800_sup_Obounded__iff,axiom,
    ! [B3: set_state,C2: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B3 @ C2 ) @ A )
      = ( ( ord_le2494988322063910608_state @ B3 @ A )
        & ( ord_le2494988322063910608_state @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_801_le__sup__iff,axiom,
    ! [X2: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state,Z: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ X2 @ Y ) @ Z )
      = ( ( ord_le4032880777312351359_state @ X2 @ Z )
        & ( ord_le4032880777312351359_state @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_802_le__sup__iff,axiom,
    ! [X2: set_state,Y: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ X2 @ Y ) @ Z )
      = ( ( ord_le2494988322063910608_state @ X2 @ Z )
        & ( ord_le2494988322063910608_state @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_803_package__logic_Osat_Opelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X2 @ Xa2 )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) )
                 => ? [A3: state,B9: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( Plus @ A3 @ B9 ) )
                      & ( package_sat_state @ Plus @ A11 @ A3 )
                      & ( package_sat_state @ Plus @ B8 @ B9 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) )
                   => ( ( B7 @ Xa2 )
                     => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) )
                     => ( A11 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(3)
thf(fact_804_package__logic_Osat_Opelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X2 @ Xa2 )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) )
                 => ~ ? [A6: state,B7: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( Plus @ A6 @ B7 ) )
                        & ( package_sat_state @ Plus @ A11 @ A6 )
                        & ( package_sat_state @ Plus @ B8 @ B7 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) )
                   => ~ ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) )
                     => ~ ( A11 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(2)
thf(fact_805_sup__set__def,axiom,
    ( sup_su1984960714304006370_state
    = ( ^ [A4: set_Pr2391701600042252046_state,B5: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ( sup_su1130956725435522171tate_o
            @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ A4 )
            @ ^ [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_806_sup__set__def,axiom,
    ( sup_sup_set_state
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ( sup_sup_state_o
            @ ^ [X: state] : ( member_state @ X @ A4 )
            @ ^ [X: state] : ( member_state @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_807_sup__set__def,axiom,
    ( sup_su7506060154676209150on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ( sup_su3200866414790943751_val_o
            @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ A4 )
            @ ^ [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_808_sup__set__def,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B5: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ( sup_su4345871077856951546tate_o
            @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ A4 )
            @ ^ [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_809_sup__Un__eq2,axiom,
    ! [R4: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( sup_su1108609720783396685tate_o
        @ ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ R4 )
        @ ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ ( sup_su1984960714304006370_state @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_810_sup__Un__eq2,axiom,
    ! [R4: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( sup_su177996417187360546_val_o
        @ ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ R4 )
        @ ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ ( sup_su3890347010539237325on_val @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_811_sup__Un__eq2,axiom,
    ! [R4: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( sup_su2989194110892726342tate_o
        @ ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 )
        @ ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ ( sup_su5006661989123481129_state @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_812_sup__Un__eq2,axiom,
    ! [R4: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( sup_sup_prat_state_o
        @ ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ R4 )
        @ ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ ( sup_su1826719071296284831_state @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_813_sup__Un__eq2,axiom,
    ! [R4: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( sup_su7614437069348798106tate_o
        @ ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ R4 )
        @ ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ ( sup_su5611131606495579221_state @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_814_sup__Un__eq2,axiom,
    ! [R4: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( sup_su4175861339346331443_val_o
        @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ R4 )
        @ ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ ( sup_su7506060154676209150on_val @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_815_sup__Un__eq2,axiom,
    ! [R4: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( sup_su1162340662190093652tate_o
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ R4 )
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ S2 ) )
      = ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ ( sup_su3837536599209510219_state @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_816_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C2 @ B3 )
     => ( ord_le4032880777312351359_state @ C2 @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_817_sup_OcoboundedI2,axiom,
    ! [C2: set_state,B3: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ B3 )
     => ( ord_le2494988322063910608_state @ C2 @ ( sup_sup_set_state @ A @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_818_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C2 @ A )
     => ( ord_le4032880777312351359_state @ C2 @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_819_sup_OcoboundedI1,axiom,
    ! [C2: set_state,A: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ A )
     => ( ord_le2494988322063910608_state @ C2 @ ( sup_sup_set_state @ A @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_820_sup_Oabsorb__iff2,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A2: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A2 @ B )
          = B ) ) ) ).

% sup.absorb_iff2
thf(fact_821_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_822_sup_Oabsorb__iff1,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [B: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A2 @ B )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_823_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_824_sup_Ocobounded2,axiom,
    ! [B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ B3 @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ).

% sup.cobounded2
thf(fact_825_sup_Ocobounded2,axiom,
    ! [B3: set_state,A: set_state] : ( ord_le2494988322063910608_state @ B3 @ ( sup_sup_set_state @ A @ B3 ) ) ).

% sup.cobounded2
thf(fact_826_sup_Ocobounded1,axiom,
    ! [A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ A @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ).

% sup.cobounded1
thf(fact_827_sup_Ocobounded1,axiom,
    ! [A: set_state,B3: set_state] : ( ord_le2494988322063910608_state @ A @ ( sup_sup_set_state @ A @ B3 ) ) ).

% sup.cobounded1
thf(fact_828_sup_Oorder__iff,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [B: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] :
          ( A2
          = ( sup_su3837536599209510219_state @ A2 @ B ) ) ) ) ).

% sup.order_iff
thf(fact_829_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_830_sup_OboundedI,axiom,
    ! [B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B3 @ A )
     => ( ( ord_le4032880777312351359_state @ C2 @ A )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B3 @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_831_sup_OboundedI,axiom,
    ! [B3: set_state,A: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A )
     => ( ( ord_le2494988322063910608_state @ C2 @ A )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B3 @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_832_sup_OboundedE,axiom,
    ! [B3: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B3 @ C2 ) @ A )
     => ~ ( ( ord_le4032880777312351359_state @ B3 @ A )
         => ~ ( ord_le4032880777312351359_state @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_833_sup_OboundedE,axiom,
    ! [B3: set_state,C2: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B3 @ C2 ) @ A )
     => ~ ( ( ord_le2494988322063910608_state @ B3 @ A )
         => ~ ( ord_le2494988322063910608_state @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_834_sup__absorb2,axiom,
    ! [X2: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X2 @ Y )
     => ( ( sup_su3837536599209510219_state @ X2 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_835_sup__absorb2,axiom,
    ! [X2: set_state,Y: set_state] :
      ( ( ord_le2494988322063910608_state @ X2 @ Y )
     => ( ( sup_sup_set_state @ X2 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_836_sup__absorb1,axiom,
    ! [Y: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ Y @ X2 )
     => ( ( sup_su3837536599209510219_state @ X2 @ Y )
        = X2 ) ) ).

% sup_absorb1
thf(fact_837_sup__absorb1,axiom,
    ! [Y: set_state,X2: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X2 )
     => ( ( sup_sup_set_state @ X2 @ Y )
        = X2 ) ) ).

% sup_absorb1
thf(fact_838_sup_Oabsorb2,axiom,
    ! [A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ B3 )
     => ( ( sup_su3837536599209510219_state @ A @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_839_sup_Oabsorb2,axiom,
    ! [A: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B3 )
     => ( ( sup_sup_set_state @ A @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_840_sup_Oabsorb1,axiom,
    ! [B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B3 @ A )
     => ( ( sup_su3837536599209510219_state @ A @ B3 )
        = A ) ) ).

% sup.absorb1
thf(fact_841_sup_Oabsorb1,axiom,
    ! [B3: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A )
     => ( ( sup_sup_set_state @ A @ B3 )
        = A ) ) ).

% sup.absorb1
thf(fact_842_sup__unique,axiom,
    ! [F: set_Pr4231240917283394591_state > set_Pr4231240917283394591_state > set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] :
      ( ! [X4: set_Pr4231240917283394591_state,Y4: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X4 @ ( F @ X4 @ Y4 ) )
     => ( ! [X4: set_Pr4231240917283394591_state,Y4: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y4 @ ( F @ X4 @ Y4 ) )
       => ( ! [X4: set_Pr4231240917283394591_state,Y4: set_Pr4231240917283394591_state,Z4: set_Pr4231240917283394591_state] :
              ( ( ord_le4032880777312351359_state @ Y4 @ X4 )
             => ( ( ord_le4032880777312351359_state @ Z4 @ X4 )
               => ( ord_le4032880777312351359_state @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
         => ( ( sup_su3837536599209510219_state @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_843_sup__unique,axiom,
    ! [F: set_state > set_state > set_state,X2: set_state,Y: set_state] :
      ( ! [X4: set_state,Y4: set_state] : ( ord_le2494988322063910608_state @ X4 @ ( F @ X4 @ Y4 ) )
     => ( ! [X4: set_state,Y4: set_state] : ( ord_le2494988322063910608_state @ Y4 @ ( F @ X4 @ Y4 ) )
       => ( ! [X4: set_state,Y4: set_state,Z4: set_state] :
              ( ( ord_le2494988322063910608_state @ Y4 @ X4 )
             => ( ( ord_le2494988322063910608_state @ Z4 @ X4 )
               => ( ord_le2494988322063910608_state @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
         => ( ( sup_sup_set_state @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_844_sup_OorderI,axiom,
    ! [A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( A
        = ( sup_su3837536599209510219_state @ A @ B3 ) )
     => ( ord_le4032880777312351359_state @ B3 @ A ) ) ).

% sup.orderI
thf(fact_845_sup_OorderI,axiom,
    ! [A: set_state,B3: set_state] :
      ( ( A
        = ( sup_sup_set_state @ A @ B3 ) )
     => ( ord_le2494988322063910608_state @ B3 @ A ) ) ).

% sup.orderI
thf(fact_846_sup_OorderE,axiom,
    ! [B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B3 @ A )
     => ( A
        = ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ).

% sup.orderE
thf(fact_847_sup_OorderE,axiom,
    ! [B3: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B3 @ A )
     => ( A
        = ( sup_sup_set_state @ A @ B3 ) ) ) ).

% sup.orderE
thf(fact_848_le__iff__sup,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [X: set_Pr4231240917283394591_state,Y3: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ X @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_849_le__iff__sup,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [X: set_state,Y3: set_state] :
          ( ( sup_sup_set_state @ X @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_850_sup__least,axiom,
    ! [Y: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state,Z: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ Y @ X2 )
     => ( ( ord_le4032880777312351359_state @ Z @ X2 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ Y @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_851_sup__least,axiom,
    ! [Y: set_state,X2: set_state,Z: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X2 )
     => ( ( ord_le2494988322063910608_state @ Z @ X2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ Y @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_852_sup__mono,axiom,
    ! [A: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state,D: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ C2 )
     => ( ( ord_le4032880777312351359_state @ B3 @ D )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B3 ) @ ( sup_su3837536599209510219_state @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_853_sup__mono,axiom,
    ! [A: set_state,C2: set_state,B3: set_state,D: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ C2 )
     => ( ( ord_le2494988322063910608_state @ B3 @ D )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B3 ) @ ( sup_sup_set_state @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_854_sup_Omono,axiom,
    ! [C2: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,D: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C2 @ A )
     => ( ( ord_le4032880777312351359_state @ D @ B3 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ C2 @ D ) @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ) ).

% sup.mono
thf(fact_855_sup_Omono,axiom,
    ! [C2: set_state,A: set_state,D: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ A )
     => ( ( ord_le2494988322063910608_state @ D @ B3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ C2 @ D ) @ ( sup_sup_set_state @ A @ B3 ) ) ) ) ).

% sup.mono
thf(fact_856_le__supI2,axiom,
    ! [X2: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X2 @ B3 )
     => ( ord_le4032880777312351359_state @ X2 @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ).

% le_supI2
thf(fact_857_le__supI2,axiom,
    ! [X2: set_state,B3: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ X2 @ B3 )
     => ( ord_le2494988322063910608_state @ X2 @ ( sup_sup_set_state @ A @ B3 ) ) ) ).

% le_supI2
thf(fact_858_le__supI1,axiom,
    ! [X2: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X2 @ A )
     => ( ord_le4032880777312351359_state @ X2 @ ( sup_su3837536599209510219_state @ A @ B3 ) ) ) ).

% le_supI1
thf(fact_859_le__supI1,axiom,
    ! [X2: set_state,A: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ X2 @ A )
     => ( ord_le2494988322063910608_state @ X2 @ ( sup_sup_set_state @ A @ B3 ) ) ) ).

% le_supI1
thf(fact_860_sup__ge2,axiom,
    ! [Y: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y @ ( sup_su3837536599209510219_state @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_861_sup__ge2,axiom,
    ! [Y: set_state,X2: set_state] : ( ord_le2494988322063910608_state @ Y @ ( sup_sup_set_state @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_862_sup__ge1,axiom,
    ! [X2: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X2 @ ( sup_su3837536599209510219_state @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_863_sup__ge1,axiom,
    ! [X2: set_state,Y: set_state] : ( ord_le2494988322063910608_state @ X2 @ ( sup_sup_set_state @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_864_le__supI,axiom,
    ! [A: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ X2 )
     => ( ( ord_le4032880777312351359_state @ B3 @ X2 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_865_le__supI,axiom,
    ! [A: set_state,X2: set_state,B3: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ X2 )
     => ( ( ord_le2494988322063910608_state @ B3 @ X2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_866_le__supE,axiom,
    ! [A: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B3 ) @ X2 )
     => ~ ( ( ord_le4032880777312351359_state @ A @ X2 )
         => ~ ( ord_le4032880777312351359_state @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_867_le__supE,axiom,
    ! [A: set_state,B3: set_state,X2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B3 ) @ X2 )
     => ~ ( ( ord_le2494988322063910608_state @ A @ X2 )
         => ~ ( ord_le2494988322063910608_state @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_868_inf__sup__ord_I3_J,axiom,
    ! [X2: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X2 @ ( sup_su3837536599209510219_state @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_869_inf__sup__ord_I3_J,axiom,
    ! [X2: set_state,Y: set_state] : ( ord_le2494988322063910608_state @ X2 @ ( sup_sup_set_state @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_870_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4231240917283394591_state,X2: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y @ ( sup_su3837536599209510219_state @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_871_inf__sup__ord_I4_J,axiom,
    ! [Y: set_state,X2: set_state] : ( ord_le2494988322063910608_state @ Y @ ( sup_sup_set_state @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_872_PartialSA_Osat_Opelims_I3_J,axiom,
    ! [X2: aassertion_state,Xa2: state] :
      ( ~ ( package_sat_state @ plus @ X2 @ Xa2 )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) )
               => ? [A3: state,B9: state] :
                    ( ( ( some_state @ Xa2 )
                      = ( plus @ A3 @ B9 ) )
                    & ( package_sat_state @ plus @ A11 @ A3 )
                    & ( package_sat_state @ plus @ B8 @ B9 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) )
                 => ( ( B7 @ Xa2 )
                   => ( package_sat_state @ plus @ A11 @ Xa2 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) )
                   => ( A11 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(3)
thf(fact_873_PartialSA_Osat_Opelims_I2_J,axiom,
    ! [X2: aassertion_state,Xa2: state] :
      ( ( package_sat_state @ plus @ X2 @ Xa2 )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) )
               => ~ ? [A6: state,B7: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( plus @ A6 @ B7 ) )
                      & ( package_sat_state @ plus @ A11 @ A6 )
                      & ( package_sat_state @ plus @ B8 @ B7 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) )
                 => ~ ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A11 @ Xa2 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) )
                   => ~ ( A11 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(2)
thf(fact_874_PartialSA_Osat_Opelims_I1_J,axiom,
    ! [X2: aassertion_state,Xa2: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
       => ( ! [A11: aassertion_state,B8: aassertion_state] :
              ( ( X2
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( Y
                  = ( ? [A2: state,B: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( plus @ A2 @ B ) )
                        & ( package_sat_state @ plus @ A11 @ A2 )
                        & ( package_sat_state @ plus @ B8 @ B ) ) ) )
               => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( X2
                  = ( aImp_state @ B7 @ A11 ) )
               => ( ( Y
                    = ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A11 @ Xa2 ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) ) ) )
           => ~ ! [A11: state > $o] :
                  ( ( X2
                    = ( aSem_state @ A11 ) )
                 => ( ( Y
                      = ( A11 @ Xa2 ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(1)
thf(fact_875_package__logic_Osat_Opelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X2: aassertion_state,Xa2: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X2 @ Xa2 )
          = Y )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X2 @ Xa2 ) )
         => ( ! [A11: aassertion_state,B8: aassertion_state] :
                ( ( X2
                  = ( aStar_state @ A11 @ B8 ) )
               => ( ( Y
                    = ( ? [A2: state,B: state] :
                          ( ( ( some_state @ Xa2 )
                            = ( Plus @ A2 @ B ) )
                          & ( package_sat_state @ Plus @ A11 @ A2 )
                          & ( package_sat_state @ Plus @ B8 @ B ) ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A11 @ B8 ) @ Xa2 ) ) ) )
           => ( ! [B7: state > $o,A11: aassertion_state] :
                  ( ( X2
                    = ( aImp_state @ B7 @ A11 ) )
                 => ( ( Y
                      = ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A11 @ Xa2 ) ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A11 ) @ Xa2 ) ) ) )
             => ~ ! [A11: state > $o] :
                    ( ( X2
                      = ( aSem_state @ A11 ) )
                   => ( ( Y
                        = ( A11 @ Xa2 ) )
                     => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A11 ) @ Xa2 ) ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(1)
thf(fact_876_PartialSA_Osplus_Opelims,axiom,
    ! [X2: option_state,Xa2: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X2 @ Xa2 ) )
       => ( ( ( X2 = none_state )
           => ( ( Y = none_state )
             => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
         => ( ! [V: state] :
                ( ( X2
                  = ( some_state @ V ) )
               => ( ( Xa2 = none_state )
                 => ( ( Y = none_state )
                   => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) ) ) ) )
           => ~ ! [A6: state] :
                  ( ( X2
                    = ( some_state @ A6 ) )
                 => ! [B7: state] :
                      ( ( Xa2
                        = ( some_state @ B7 ) )
                     => ( ( Y
                          = ( plus @ A6 @ B7 ) )
                       => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.splus.pelims
thf(fact_877_package__logic_OAddFromOutside,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,Phi2: state,M2: state,F3: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi7: state,F4: state,S5: set_Pr4231240917283394591_state,F: state,S2: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi )
          = ( Plus @ Phi2 @ M2 ) )
       => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F3 @ S3 @ Pc @ A5 @ Phi7 @ F4 @ S5 )
         => ( ( Stable @ M2 )
           => ( ( ( some_state @ F3 )
                = ( Plus @ F @ M2 ) )
             => ( ( S3
                  = ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T4: state > state,R2: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                        & ( ( some_state @ R2 )
                          = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) ) )
                        & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) )
               => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi7 @ F4 @ S5 ) ) ) ) ) ) ) ).

% package_logic.AddFromOutside
thf(fact_878_package__logic_OASem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,Pc: state > $o,Witness3: produc7272280185086996969_state > state,B2: state > $o,S3: set_Pr4231240917283394591_state,Phi: state,F: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,U: state,T3: state > state,B7: state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( Pc @ A6 )
             => ( ( B7
                  = ( Witness3 @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) ) )
               => ( ( sep_greater_state @ Plus @ A6 @ B7 )
                  & ( B2 @ B7 ) ) ) ) )
       => ( ( S3
            = ( sup_su3837536599209510219_state
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T4: state > state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                    & ~ ( Pc @ A2 ) ) )
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A2: state,U6: state,T4: state > state,B: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B ) ) @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                    & ( Pc @ A2 )
                    & ( B
                      = ( Witness3 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) )
         => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ ( aSem_state @ B2 ) @ Phi @ F @ S3 ) ) ) ) ).

% package_logic.ASem
thf(fact_879_PartialSA_Ogeneral__soundness,axiom,
    ! [Phi: state,S2: set_Pr2391701600042252046_state,A5: aassertion_state,Phi2: state,F: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ unit
        @ ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A2: state,T4: state > state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ unit @ T4 ) ) )
              & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A2 @ T4 ) @ S2 ) ) )
        @ ^ [Uu3: state] : $true
        @ A5
        @ Phi2
        @ F
        @ S3 )
     => ( ! [A6: state,T3: state > state] :
            ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A6 @ T3 ) @ S2 )
           => ( packag4817583915586052229_state @ plus @ unit @ T3 ) )
       => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
         => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ A5 ) )
              | ( packag4152378406607043437_state @ plus @ S3 ) )
           => ( ( ( some_state @ Phi )
                = ( plus @ Phi2 @ F ) )
              & ( stable @ F )
              & ! [X3: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X3 @ S2 )
                 => ( produc3645877548210629341tate_o
                    @ ^ [A2: state,T4: state > state] :
                        ( ( defined @ A2 @ ( T4 @ F ) )
                       => ( package_sat_state @ plus @ A5 @ ( the_state @ ( plus @ A2 @ ( T4 @ F ) ) ) ) )
                    @ X3 ) ) ) ) ) ) ) ).

% PartialSA.general_soundness
thf(fact_880_split__part,axiom,
    ! [P: $o,Q: state > ( state > state ) > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [A2: state,B: state > state] :
            ( P
            & ( Q @ A2 @ B ) ) )
      = ( ^ [Ab2: produc3882902684156697774_state] :
            ( P
            & ( produc3645877548210629341tate_o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_881_split__part,axiom,
    ! [P: $o,Q: state > produc3882902684156697774_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [A2: state,B: produc3882902684156697774_state] :
            ( P
            & ( Q @ A2 @ B ) ) )
      = ( ^ [Ab2: produc7272280185086996969_state] :
            ( P
            & ( produc1745419677418529872tate_o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_882_case__prod__conv,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B3: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B3 ) )
      = ( F @ A @ B3 ) ) ).

% case_prod_conv
thf(fact_883_case__prod__conv,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B3: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B3 ) )
      = ( F @ A @ B3 ) ) ).

% case_prod_conv
thf(fact_884_pair__imageI,axiom,
    ! [A: prat,B3: state,A5: set_Pr6039890582625207499_state,F: prat > state > state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B3 ) @ A5 )
     => ( member_state @ ( F @ A @ B3 ) @ ( image_6883929567913436896_state @ ( produc5191658985856023922_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_885_pair__imageI,axiom,
    ! [A: prat,B3: set_state,A5: set_Pr6847263425858009217_state,F: prat > set_state > state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B3 ) @ A5 )
     => ( member_state @ ( F @ A @ B3 ) @ ( image_1091725855630477206_state @ ( produc2626907587099862312_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_886_pair__imageI,axiom,
    ! [A: state,B3: state > state,A5: set_Pr2391701600042252046_state,F: state > ( state > state ) > state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B3 ) @ A5 )
     => ( member_state @ ( F @ A @ B3 ) @ ( image_6759111139246981411_state @ ( produc8575939086779714485_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_887_pair__imageI,axiom,
    ! [A: option_state,B3: option_state,A5: set_Pr1688445902015331925_state,F: option_state > option_state > state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B3 ) @ A5 )
     => ( member_state @ ( F @ A @ B3 ) @ ( image_393491784373993066_state @ ( produc7211634496914305532_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_888_pair__imageI,axiom,
    ! [A: state,B3: state > state,A5: set_Pr2391701600042252046_state,F: state > ( state > state ) > $o] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B3 ) @ A5 )
     => ( member_o @ ( F @ A @ B3 ) @ ( image_528283217102427119tate_o @ ( produc3645877548210629341tate_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_889_pair__imageI,axiom,
    ! [A: state,B3: produc3882902684156697774_state,A5: set_Pr4231240917283394591_state,F: state > produc3882902684156697774_state > state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B3 ) @ A5 )
     => ( member_state @ ( F @ A @ B3 ) @ ( image_2710133271811360426_state @ ( produc8258182183023037954_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_890_pair__imageI,axiom,
    ! [A: prat,B3: state,A5: set_Pr6039890582625207499_state,F: prat > state > produc3882902684156697774_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B3 ) @ A5 )
     => ( member6715010751053357271_state @ ( F @ A @ B3 ) @ ( image_1979208616201753608_state @ ( produc6727336724318121462_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_891_pair__imageI,axiom,
    ! [A: state,B3: produc3882902684156697774_state,A5: set_Pr4231240917283394591_state,F: state > produc3882902684156697774_state > $o] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B3 ) @ A5 )
     => ( member_o @ ( F @ A @ B3 ) @ ( image_6262870063963634088tate_o @ ( produc1745419677418529872tate_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_892_pair__imageI,axiom,
    ! [A: prat,B3: set_state,A5: set_Pr6847263425858009217_state,F: prat > set_state > produc3882902684156697774_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B3 ) @ A5 )
     => ( member6715010751053357271_state @ ( F @ A @ B3 ) @ ( image_7417905060788143314_state @ ( produc8936804542704588736_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_893_pair__imageI,axiom,
    ! [A: state,B3: state > state,A5: set_Pr2391701600042252046_state,F: state > ( state > state ) > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B3 ) @ A5 )
     => ( member6715010751053357271_state @ ( F @ A @ B3 ) @ ( image_508898238547643525_state @ ( produc5527672907597820531_state @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_894_case__prodI2,axiom,
    ! [P2: produc1317171644805857433on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ! [A6: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] :
          ( ( P2
            = ( produc4958730763592889809on_val @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc1845727980445322802_val_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_895_case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,C2: option_state > option_state > $o] :
      ( ! [A6: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc6082990154846732246tate_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_896_case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,C2: prat > state > $o] :
      ( ! [A6: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc1228703470279370976tate_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_897_case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,C2: prat > set_state > $o] :
      ( ! [A6: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc7925652287191295402tate_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_898_case__prodI2,axiom,
    ! [P2: produc9003781588162623004on_val,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ! [A6: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( ( P2
            = ( produc74620467203698198on_val @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc5248837549992124053_val_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_899_case__prodI2,axiom,
    ! [P2: produc3882902684156697774_state,C2: state > ( state > state ) > $o] :
      ( ! [A6: state,B7: state > state] :
          ( ( P2
            = ( produc8632743443909064678_state @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc3645877548210629341tate_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_900_case__prodI2,axiom,
    ! [P2: produc7272280185086996969_state,C2: state > produc3882902684156697774_state > $o] :
      ( ! [A6: state,B7: produc3882902684156697774_state] :
          ( ( P2
            = ( produc5886875603596089627_state @ A6 @ B7 ) )
         => ( C2 @ A6 @ B7 ) )
     => ( produc1745419677418529872tate_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_901_case__prodI,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
      ( ( F @ A @ B3 )
     => ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_902_case__prodI,axiom,
    ! [F: option_state > option_state > $o,A: option_state,B3: option_state] :
      ( ( F @ A @ B3 )
     => ( produc6082990154846732246tate_o @ F @ ( produc9160152616361873709_state @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_903_case__prodI,axiom,
    ! [F: prat > state > $o,A: prat,B3: state] :
      ( ( F @ A @ B3 )
     => ( produc1228703470279370976tate_o @ F @ ( produc6205719116230455971_state @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_904_case__prodI,axiom,
    ! [F: prat > set_state > $o,A: prat,B3: set_state] :
      ( ( F @ A @ B3 )
     => ( produc7925652287191295402tate_o @ F @ ( produc9080284787853255769_state @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_905_case__prodI,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
      ( ( F @ A @ B3 )
     => ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_906_case__prodI,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B3: state > state] :
      ( ( F @ A @ B3 )
     => ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_907_case__prodI,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B3: produc3882902684156697774_state] :
      ( ( F @ A @ B3 )
     => ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_908_Ball__def,axiom,
    ( ball_state
    = ( ^ [A4: set_state,P5: state > $o] :
        ! [X: state] :
          ( ( member_state @ X @ A4 )
         => ( P5 @ X ) ) ) ) ).

% Ball_def
thf(fact_909_Ball__def,axiom,
    ( ball_P7403102246598991807_state
    = ( ^ [A4: set_Pr4231240917283394591_state,P5: produc7272280185086996969_state > $o] :
        ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A4 )
         => ( P5 @ X ) ) ) ) ).

% Ball_def
thf(fact_910_Ball__def,axiom,
    ( ball_P2871712132005019634on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,P5: produc9003781588162623004on_val > $o] :
        ! [X: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X @ A4 )
         => ( P5 @ X ) ) ) ) ).

% Ball_def
thf(fact_911_Ball__def,axiom,
    ( ball_P1066487999220904024_state
    = ( ^ [A4: set_Pr2391701600042252046_state,P5: produc3882902684156697774_state > $o] :
        ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A4 )
         => ( P5 @ X ) ) ) ) ).

% Ball_def
thf(fact_912_sep__algebra_Odefined_Ocong,axiom,
    sep_defined_state = sep_defined_state ).

% sep_algebra.defined.cong
thf(fact_913_sep__algebra_Ogreater_Ocong,axiom,
    sep_greater_state = sep_greater_state ).

% sep_algebra.greater.cong
thf(fact_914_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F: state > ( state > state ) > $o,Prod: produc3882902684156697774_state] :
      ( ( H3 @ ( produc3645877548210629341tate_o @ F @ Prod ) )
      = ( produc3645877548210629341tate_o
        @ ^ [X13: state,X24: state > state] : ( H3 @ ( F @ X13 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_915_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F: state > produc3882902684156697774_state > $o,Prod: produc7272280185086996969_state] :
      ( ( H3 @ ( produc1745419677418529872tate_o @ F @ Prod ) )
      = ( produc1745419677418529872tate_o
        @ ^ [X13: state,X24: produc3882902684156697774_state] : ( H3 @ ( F @ X13 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_916_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3882902684156697774_state] :
      ( produc3645877548210629341tate_o
      @ ^ [Uu3: state,Uv3: state > state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_917_prod_Odisc__eq__case,axiom,
    ! [Prod: produc7272280185086996969_state] :
      ( produc1745419677418529872tate_o
      @ ^ [Uu3: state,Uv3: produc3882902684156697774_state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_918_case__prodD,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
      ( ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_919_case__prodD,axiom,
    ! [F: option_state > option_state > $o,A: option_state,B3: option_state] :
      ( ( produc6082990154846732246tate_o @ F @ ( produc9160152616361873709_state @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_920_case__prodD,axiom,
    ! [F: prat > state > $o,A: prat,B3: state] :
      ( ( produc1228703470279370976tate_o @ F @ ( produc6205719116230455971_state @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_921_case__prodD,axiom,
    ! [F: prat > set_state > $o,A: prat,B3: set_state] :
      ( ( produc7925652287191295402tate_o @ F @ ( produc9080284787853255769_state @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_922_case__prodD,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
      ( ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_923_case__prodD,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B3: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_924_case__prodD,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B3: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B3 ) )
     => ( F @ A @ B3 ) ) ).

% case_prodD
thf(fact_925_case__prodE,axiom,
    ! [C2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,P2: produc1317171644805857433on_val] :
      ( ( produc1845727980445322802_val_o @ C2 @ P2 )
     => ~ ! [X4: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( P2
              = ( produc4958730763592889809on_val @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_926_case__prodE,axiom,
    ! [C2: option_state > option_state > $o,P2: produc3142500478612311029_state] :
      ( ( produc6082990154846732246tate_o @ C2 @ P2 )
     => ~ ! [X4: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_927_case__prodE,axiom,
    ! [C2: prat > state > $o,P2: produc4844402304972151915_state] :
      ( ( produc1228703470279370976tate_o @ C2 @ P2 )
     => ~ ! [X4: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_928_case__prodE,axiom,
    ! [C2: prat > set_state > $o,P2: produc6251377953716854561_state] :
      ( ( produc7925652287191295402tate_o @ C2 @ P2 )
     => ~ ! [X4: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_929_case__prodE,axiom,
    ! [C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: produc9003781588162623004on_val] :
      ( ( produc5248837549992124053_val_o @ C2 @ P2 )
     => ~ ! [X4: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
            ( ( P2
              = ( produc74620467203698198on_val @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_930_case__prodE,axiom,
    ! [C2: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ( produc3645877548210629341tate_o @ C2 @ P2 )
     => ~ ! [X4: state,Y4: state > state] :
            ( ( P2
              = ( produc8632743443909064678_state @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_931_case__prodE,axiom,
    ! [C2: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ( produc1745419677418529872tate_o @ C2 @ P2 )
     => ~ ! [X4: state,Y4: produc3882902684156697774_state] :
            ( ( P2
              = ( produc5886875603596089627_state @ X4 @ Y4 ) )
           => ~ ( C2 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_932_package__logic_Ocore__mono,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
       => ( sep_greater_state @ Plus @ ( Core @ A ) @ ( Core @ B3 ) ) ) ) ).

% package_logic.core_mono
thf(fact_933_package__logic_Ounit__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( sep_greater_state @ Plus @ Phi @ Unit ) ) ).

% package_logic.unit_smaller
thf(fact_934_old_Oprod_Ocase,axiom,
    ! [F: state > ( state > state ) > $o,X1: state,X22: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_935_old_Oprod_Ocase,axiom,
    ! [F: state > produc3882902684156697774_state > $o,X1: state,X22: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_936_cond__case__prod__eta,axiom,
    ! [F: state > ( state > state ) > $o,G: produc3882902684156697774_state > $o] :
      ( ! [X4: state,Y4: state > state] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( produc8632743443909064678_state @ X4 @ Y4 ) ) )
     => ( ( produc3645877548210629341tate_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_937_cond__case__prod__eta,axiom,
    ! [F: state > produc3882902684156697774_state > $o,G: produc7272280185086996969_state > $o] :
      ( ! [X4: state,Y4: produc3882902684156697774_state] :
          ( ( F @ X4 @ Y4 )
          = ( G @ ( produc5886875603596089627_state @ X4 @ Y4 ) ) )
     => ( ( produc1745419677418529872tate_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_938_case__prod__eta,axiom,
    ! [F: produc3882902684156697774_state > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [X: state,Y3: state > state] : ( F @ ( produc8632743443909064678_state @ X @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_939_case__prod__eta,axiom,
    ! [F: produc7272280185086996969_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [X: state,Y3: produc3882902684156697774_state] : ( F @ ( produc5886875603596089627_state @ X @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_940_case__prodE2,axiom,
    ! [Q: $o > $o,P: state > ( state > state ) > $o,Z: produc3882902684156697774_state] :
      ( ( Q @ ( produc3645877548210629341tate_o @ P @ Z ) )
     => ~ ! [X4: state,Y4: state > state] :
            ( ( Z
              = ( produc8632743443909064678_state @ X4 @ Y4 ) )
           => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_941_case__prodE2,axiom,
    ! [Q: $o > $o,P: state > produc3882902684156697774_state > $o,Z: produc7272280185086996969_state] :
      ( ( Q @ ( produc1745419677418529872tate_o @ P @ Z ) )
     => ~ ! [X4: state,Y4: produc3882902684156697774_state] :
            ( ( Z
              = ( produc5886875603596089627_state @ X4 @ Y4 ) )
           => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_942_greater__def,axiom,
    ( greater
    = ( sep_greater_state @ plus ) ) ).

% greater_def
thf(fact_943_package__logic_Obigger__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,A: state,B3: state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi )
          = ( Plus @ A @ B3 ) )
       => ( ( sep_greater_state @ Plus @ Phi2 @ Phi )
         => ? [B4: state] :
              ( ( sep_greater_state @ Plus @ B4 @ B3 )
              & ( ( some_state @ Phi2 )
                = ( Plus @ A @ B4 ) ) ) ) ) ) ).

% package_logic.bigger_sum
thf(fact_944_package__logic_Obigger__core__sum__defined,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ ( Core @ A ) @ B3 )
       => ( ( some_state @ A )
          = ( Plus @ A @ B3 ) ) ) ) ).

% package_logic.bigger_core_sum_defined
thf(fact_945_package__logic_Ogreater__than__sum__exists,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state,B1: state,B22: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
       => ( ( ( some_state @ B3 )
            = ( Plus @ B1 @ B22 ) )
         => ? [R3: state] :
              ( ( ( some_state @ A )
                = ( Plus @ R3 @ B22 ) )
              & ( sep_greater_state @ Plus @ ( Core @ R3 ) @ ( Core @ A ) )
              & ( sep_greater_state @ Plus @ R3 @ B1 ) ) ) ) ) ).

% package_logic.greater_than_sum_exists
thf(fact_946_Collect__case__prod__mono,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,B2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( ord_le3852884270828607079_val_o @ A5 @ B2 )
     => ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ A5 ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_947_Collect__case__prod__mono,axiom,
    ! [A5: state > ( state > state ) > $o,B2: state > ( state > state ) > $o] :
      ( ( ord_le1106678366802930969tate_o @ A5 @ B2 )
     => ( ord_le6483787752022044334_state @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ A5 ) ) @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_948_Collect__case__prod__mono,axiom,
    ! [A5: state > produc3882902684156697774_state > $o,B2: state > produc3882902684156697774_state > $o] :
      ( ( ord_le6704609105599929480tate_o @ A5 @ B2 )
     => ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ A5 ) ) @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_949_package__logic_Omono__transformer__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,T2: state > state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag4817583915586052229_state @ Plus @ Unit @ T2 )
        = ( ! [Phi4: state,Phi3: state] :
              ( ( sep_greater_state @ Plus @ Phi3 @ Phi4 )
             => ( sep_greater_state @ Plus @ ( T2 @ Phi3 ) @ ( T2 @ Phi4 ) ) )
          & ( ( T2 @ Unit )
            = Unit ) ) ) ) ).

% package_logic.mono_transformer_def
thf(fact_950_package__logic_Ointuitionistic__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8361946002163212404_state @ Plus @ A5 )
        = ( ! [Phi3: state,Phi4: state] :
              ( ( ( sep_greater_state @ Plus @ Phi3 @ Phi4 )
                & ( A5 @ Phi4 ) )
             => ( A5 @ Phi3 ) ) ) ) ) ).

% package_logic.intuitionistic_def
thf(fact_951_internal__case__prod__def,axiom,
    produc1225464857956169863tate_o = produc3645877548210629341tate_o ).

% internal_case_prod_def
thf(fact_952_internal__case__prod__def,axiom,
    produc3899216655821101094tate_o = produc1745419677418529872tate_o ).

% internal_case_prod_def
thf(fact_953_package__logic_Odefined__sum__move,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state,X2: state,Y: state,A7: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_state @ Plus @ A @ B3 )
       => ( ( ( some_state @ B3 )
            = ( Plus @ X2 @ Y ) )
         => ( ( ( some_state @ A7 )
              = ( Plus @ A @ X2 ) )
           => ( sep_defined_state @ Plus @ A7 @ Y ) ) ) ) ) ).

% package_logic.defined_sum_move
thf(fact_954_defined__def,axiom,
    ( defined
    = ( sep_defined_state @ plus ) ) ).

% defined_def
thf(fact_955_package__logic_Obigger__the,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,X5: state,Y: state,X2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ A )
          = ( Plus @ X5 @ Y ) )
       => ( ( sep_greater_state @ Plus @ X5 @ X2 )
         => ( sep_greater_state @ Plus @ ( the_state @ ( Plus @ ( Core @ A ) @ X5 ) ) @ ( the_state @ ( Plus @ ( Core @ A ) @ X2 ) ) ) ) ) ) ).

% package_logic.bigger_the
thf(fact_956_package__logic_Oprove__last__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A7: state,A: state,Nf1: state,F2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A7 @ A )
       => ( ( ( some_state @ A )
            = ( Plus @ Nf1 @ F2 ) )
         => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ A7 @ Nf1 ) @ F2 ) ) ) ) ).

% package_logic.prove_last_completeness
thf(fact_957_package__logic_Ominus__some,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
       => ( ( some_state @ A )
          = ( Plus @ B3 @ ( sep_minus_state @ Plus @ Core @ A @ B3 ) ) ) ) ) ).

% package_logic.minus_some
thf(fact_958_package__logic_Ovalid__package__setI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( sep_defined_state @ Plus @ A6 @ U )
              & ( sep_greater_state @ Plus @ ( Core @ A6 ) @ ( Core @ U ) )
              & ( packag4817583915586052229_state @ Plus @ Unit @ T3 )
              & ( sep_greater_state @ Plus @ A6 @ ( Core @ ( T3 @ F ) ) ) ) )
       => ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F ) ) ) ).

% package_logic.valid_package_setI
thf(fact_959_package__logic_Opackage__rhs__connection__instantiate,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,A: state,U3: state,T2: state > state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
       => ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ ( produc8632743443909064678_state @ U3 @ T2 ) ) @ S2 )
         => ~ ! [Au2: state] :
                ( ( ( some_state @ Au2 )
                  = ( Plus @ A @ U3 ) )
               => ~ ( ( sep_defined_state @ Plus @ Au2 @ ( sep_minus_state @ Plus @ Core @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                   => ? [A10: state,U4: state] :
                        ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A10 @ ( produc8632743443909064678_state @ U4 @ T2 ) ) @ S3 )
                        & ( sep_greater_state @ Plus @ ( Core @ A10 ) @ ( Core @ A ) )
                        & ( ( Plus @ Au2 @ ( sep_minus_state @ Plus @ Core @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                          = ( Plus @ A10 @ U4 ) )
                        & ( sep_greater_state @ Plus @ U4 @ U3 )
                        & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A5 @ A10 @ U4 @ U3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connection_instantiate
thf(fact_960_package__logic_Opackage__rhs__connectionI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,Phi2: state,F3: state,S2: set_Pr4231240917283394591_state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Plus @ Phi @ F )
          = ( Plus @ Phi2 @ F3 ) )
       => ( ( Stable @ F3 )
         => ( ( sep_defined_state @ Plus @ Phi @ F )
           => ( ( sep_greater_state @ Plus @ F3 @ F )
             => ( ! [A6: state,U: state,T3: state > state] :
                    ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
                   => ? [Au: state] :
                        ( ( ( some_state @ Au )
                          = ( Plus @ A6 @ U ) )
                        & ( ( sep_defined_state @ Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                         => ? [A9: state,U2: state] :
                              ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A9 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                              & ( sep_greater_state @ Plus @ ( Core @ A9 ) @ ( Core @ A6 ) )
                              & ( ( Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                                = ( Plus @ A9 @ U2 ) )
                              & ( sep_greater_state @ Plus @ U2 @ U )
                              & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A5 @ A9 @ U2 @ U ) ) ) ) )
               => ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connectionI
thf(fact_961_package__logic_Osat__star__exists__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A5: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A5 @ B2 ) @ Phi )
       => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
           => ? [A6: state,B7: state] :
                ( ( sep_greater_state @ Plus @ ( Core @ A6 ) @ ( Core @ Phi ) )
                & ( sep_greater_state @ Plus @ ( Core @ B7 ) @ ( Core @ Phi ) )
                & ( ( some_state @ Phi )
                  = ( Plus @ A6 @ B7 ) )
                & ( package_sat_state @ Plus @ A5 @ A6 )
                & ( package_sat_state @ Plus @ B2 @ B7 ) ) ) ) ) ) ).

% package_logic.sat_star_exists_bigger
thf(fact_962_package__logic_Ogeneral__soundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,S2: set_Pr2391701600042252046_state,A5: aassertion_state,Phi2: state,F: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ Unit
          @ ( collec906845016635587262_state
            @ ^ [Uu3: produc7272280185086996969_state] :
              ? [A2: state,T4: state > state] :
                ( ( Uu3
                  = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ Unit @ T4 ) ) )
                & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A2 @ T4 ) @ S2 ) ) )
          @ ^ [Uu3: state] : $true
          @ A5
          @ Phi2
          @ F
          @ S3 )
       => ( ! [A6: state,T3: state > state] :
              ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A6 @ T3 ) @ S2 )
             => ( packag4817583915586052229_state @ Plus @ Unit @ T3 ) )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
           => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ A5 ) )
                | ( packag4152378406607043437_state @ Plus @ S3 ) )
             => ( ( ( some_state @ Phi )
                  = ( Plus @ Phi2 @ F ) )
                & ( Stable @ F )
                & ! [X3: produc3882902684156697774_state] :
                    ( ( member6715010751053357271_state @ X3 @ S2 )
                   => ( produc3645877548210629341tate_o
                      @ ^ [A2: state,T4: state > state] :
                          ( ( sep_defined_state @ Plus @ A2 @ ( T4 @ F ) )
                         => ( package_sat_state @ Plus @ A5 @ ( the_state @ ( Plus @ A2 @ ( T4 @ F ) ) ) ) )
                      @ X3 ) ) ) ) ) ) ) ) ).

% package_logic.general_soundness
thf(fact_963_package__logic_Owf__assertion__and__the,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B3: state,A5: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_state @ Plus @ ( Core @ A ) @ B3 )
       => ( ( package_sat_state @ Plus @ A5 @ B3 )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
           => ( package_sat_state @ Plus @ A5 @ ( the_state @ ( Plus @ ( Core @ A ) @ B3 ) ) ) ) ) ) ) ).

% package_logic.wf_assertion_and_the
thf(fact_964_package__logic_Opackage__rhs_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A1: state,A22: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
       => ( ! [A11: aassertion_state,Phi6: state,F5: state,S6: set_Pr4231240917283394591_state,B8: aassertion_state] :
              ( ( A52
                = ( aStar_state @ A11 @ B8 ) )
             => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A11 @ Phi6 @ F5 @ S6 )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F5 @ S6 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
         => ( ! [B7: state > $o,A11: aassertion_state] :
                ( ( A52
                  = ( aImp_state @ B7 @ A11 ) )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ ( packag1330488657391168505_state @ A42 @ B7 ) @ A11 @ A62 @ A72 @ A82 ) )
           => ( ! [Witness: produc7272280185086996969_state > state,B8: state > $o] :
                  ( ( A52
                    = ( aSem_state @ B8 ) )
                 => ( ( A62 = A1 )
                   => ( ( A72 = A22 )
                     => ( ! [A3: state,U7: state,T5: state > state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) @ A32 )
                           => ( ( A42 @ A3 )
                             => ( ( sep_greater_state @ Plus @ A3 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) )
                                & ( B8 @ ( Witness @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) ) ) ) )
                       => ( A82
                         != ( sup_su3837536599209510219_state
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T4: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ~ ( A42 @ A2 ) ) )
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A2: state,U6: state,T4: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ ( Witness @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ( A42 @ A2 ) ) ) ) ) ) ) ) )
             => ~ ! [Phi6: state,M3: state] :
                    ( ( ( some_state @ A1 )
                      = ( Plus @ Phi6 @ M3 ) )
                   => ! [F5: state,S6: set_Pr4231240917283394591_state] :
                        ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F5 @ S6 @ A42 @ A52 @ A62 @ A72 @ A82 )
                       => ( ( Stable @ M3 )
                         => ( ( ( some_state @ F5 )
                              = ( Plus @ A22 @ M3 ) )
                           => ( S6
                             != ( collec906845016635587262_state
                                @ ^ [Uu3: produc7272280185086996969_state] :
                                  ? [A2: state,U6: state,T4: state > state,R2: state] :
                                    ( ( Uu3
                                      = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                    & ( ( some_state @ R2 )
                                      = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F5 ) @ ( T4 @ A22 ) ) ) )
                                    & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.cases
thf(fact_965_package__logic_Opackage__rhs_Osimps,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A1: state,A22: state,A32: set_Pr4231240917283394591_state,A42: state > $o,A52: aassertion_state,A62: state,A72: state,A82: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
        = ( ? [Phi4: state,F6: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,A4: aassertion_state,Phi3: state,F7: state,S8: set_Pr4231240917283394591_state,B5: aassertion_state,Phi8: state,F8: state,S9: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi4 )
              & ( A22 = F6 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aStar_state @ A4 @ B5 ) )
              & ( A62 = Phi8 )
              & ( A72 = F8 )
              & ( A82 = S9 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi4 @ F6 @ S7 @ Pc2 @ A4 @ Phi3 @ F7 @ S8 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi3 @ F7 @ S8 @ Pc2 @ B5 @ Phi8 @ F8 @ S9 ) )
          | ? [Phi4: state,F6: state,S7: set_Pr4231240917283394591_state,Pc2: state > $o,B: state > $o,A4: aassertion_state,Phi3: state,F7: state,S8: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi4 )
              & ( A22 = F6 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aImp_state @ B @ A4 ) )
              & ( A62 = Phi3 )
              & ( A72 = F7 )
              & ( A82 = S8 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi4 @ F6 @ S7 @ ( packag1330488657391168505_state @ Pc2 @ B ) @ A4 @ Phi3 @ F7 @ S8 ) )
          | ? [S7: set_Pr4231240917283394591_state,Pc2: state > $o,Witness2: produc7272280185086996969_state > state,B5: state > $o,S8: set_Pr4231240917283394591_state,Phi4: state,F6: state] :
              ( ( A1 = Phi4 )
              & ( A22 = F6 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aSem_state @ B5 ) )
              & ( A62 = Phi4 )
              & ( A72 = F6 )
              & ( A82 = S8 )
              & ! [X: state,Y3: state,Z6: state > state,Aa2: state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z6 ) ) @ S7 )
                 => ( ( Pc2 @ X )
                   => ( ( Aa2
                        = ( Witness2 @ ( produc5886875603596089627_state @ X @ ( produc8632743443909064678_state @ Y3 @ Z6 ) ) ) )
                     => ( ( sep_greater_state @ Plus @ X @ Aa2 )
                        & ( B5 @ Aa2 ) ) ) ) )
              & ( S8
                = ( sup_su3837536599209510219_state
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T4: state > state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                        & ~ ( Pc2 @ A2 ) ) )
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A2: state,U6: state,T4: state > state,B: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A2 @ B ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B ) ) @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                        & ( Pc2 @ A2 )
                        & ( B
                          = ( Witness2 @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) ) )
          | ? [Phi4: state,Phi3: state,M4: state,F7: state,S8: set_Pr4231240917283394591_state,Pc2: state > $o,A4: aassertion_state,Phi8: state,F8: state,S9: set_Pr4231240917283394591_state,F6: state,S7: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi4 )
              & ( A22 = F6 )
              & ( A32 = S7 )
              & ( A42 = Pc2 )
              & ( A52 = A4 )
              & ( A62 = Phi8 )
              & ( A72 = F8 )
              & ( A82 = S9 )
              & ( ( some_state @ Phi4 )
                = ( Plus @ Phi3 @ M4 ) )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi3 @ F7 @ S8 @ Pc2 @ A4 @ Phi8 @ F8 @ S9 )
              & ( Stable @ M4 )
              & ( ( some_state @ F7 )
                = ( Plus @ F6 @ M4 ) )
              & ( S8
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A2: state,U6: state,T4: state > state,R2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S7 )
                      & ( ( some_state @ R2 )
                        = ( Plus @ A2 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F7 ) @ ( T4 @ F6 ) ) ) )
                      & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.simps
thf(fact_966_package__logic_Opackage__rhs__proof,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
       => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
           => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
             => ( ( Stable @ F )
               => ( ( sep_defined_state @ Plus @ Phi @ F )
                 => ( ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
                    & ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S3 @ F3 ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs_proof
thf(fact_967_compatible__options_Opelims_I1_J,axiom,
    ! [X2: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X2 @ Xa2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa2 ) )
       => ( ! [A6: state] :
              ( ( X2
                = ( some_state @ A6 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( Y
                      = ( A6 = B7 ) )
                   => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) ) ) ) )
         => ( ( ( X2 = none_state )
             => ( Y
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
           => ~ ( ( Xa2 = none_state )
               => ( Y
                 => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ none_state ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_968_compatible__options_Opelims_I2_J,axiom,
    ! [X2: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X2 @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa2 ) )
       => ( ! [A6: state] :
              ( ( X2
                = ( some_state @ A6 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) )
                   => ( A6 != B7 ) ) ) )
         => ( ( ( X2 = none_state )
             => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) )
           => ~ ( ( Xa2 = none_state )
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ none_state ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_969_PartialSA_Ounder__def,axiom,
    ! [A5: set_state,Omega2: state] :
      ( ( sep_under_state @ plus @ A5 @ Omega2 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Omega3: state] :
            ( ( Uu3 = Omega3 )
            & ( member_state @ Omega3 @ A5 )
            & ( greater @ Omega2 @ Omega3 ) ) ) ) ).

% PartialSA.under_def
thf(fact_970_PartialSA_Ocompleteness__aux,axiom,
    ! [S2: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F2: state > state > ( state > state ) > state,Pc: state > $o,A5: aassertion_state,F: state,Phi: state] :
      ( ! [A6: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( greater @ ( core @ ( F1 @ A6 @ U @ T3 ) ) @ ( core @ A6 ) )
            & ( ( some_state @ A6 )
              = ( plus @ ( F1 @ A6 @ U @ T3 ) @ ( F2 @ A6 @ U @ T3 ) ) )
            & ( ( Pc @ ( core @ A6 ) )
             => ( package_sat_state @ plus @ A5 @ ( the_state @ ( plus @ ( core @ A6 ) @ ( F1 @ A6 @ U @ T3 ) ) ) ) ) ) )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
       => ( ( packag7298478212905792770_state @ plus @ core @ A5 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( ( stable @ F )
                & ( defined @ Phi @ F ) )
             => ? [S6: set_Pr4231240917283394591_state] :
                  ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi @ F @ S6 )
                  & ! [X3: produc7272280185086996969_state] :
                      ( ( member7461423308196221312_state @ X3 @ S6 )
                     => ( produc1745419677418529872tate_o
                        @ ^ [A13: state] :
                            ( produc3645877548210629341tate_o
                            @ ^ [U8: state,T4: state > state] :
                              ? [A2: state,U6: state] :
                                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                                & ( greater @ A13 @ ( F2 @ A2 @ U6 @ T4 ) )
                                & ( ( core @ A13 )
                                  = ( core @ A2 ) ) ) )
                        @ X3 ) ) ) ) ) ) ) ) ).

% PartialSA.completeness_aux
thf(fact_971_case__prod__Pair,axiom,
    ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val )
    = id_Pro2088550933252328842on_val ) ).

% case_prod_Pair
thf(fact_972_case__prod__Pair,axiom,
    ( ( produc803252138055838771_state @ produc9160152616361873709_state )
    = id_Pro8715877757508063718_state ) ).

% case_prod_Pair
thf(fact_973_case__prod__Pair,axiom,
    ( ( produc1922103714420728243_state @ produc6205719116230455971_state )
    = id_Pro5916511905602125660_state ) ).

% case_prod_Pair
thf(fact_974_case__prod__Pair,axiom,
    ( ( produc897481912382009139_state @ produc9080284787853255769_state )
    = id_Pro8860473219607035410_state ) ).

% case_prod_Pair
thf(fact_975_case__prod__Pair,axiom,
    ( ( produc4192255817509754207on_val @ produc74620467203698198on_val )
    = id_Pro1402184692240358763on_val ) ).

% case_prod_Pair
thf(fact_976_mem__case__prodI,axiom,
    ! [Z: state,C2: prat > state > set_state,A: prat,B3: state] :
      ( ( member_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member_state @ Z @ ( produc2825319096956678952_state @ C2 @ ( produc6205719116230455971_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_977_mem__case__prodI,axiom,
    ! [Z: state,C2: prat > set_state > set_state,A: prat,B3: set_state] :
      ( ( member_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member_state @ Z @ ( produc5034060725774254814_state @ C2 @ ( produc9080284787853255769_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_978_mem__case__prodI,axiom,
    ! [Z: state,C2: option_state > option_state > set_state,A: option_state,B3: option_state] :
      ( ( member_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member_state @ Z @ ( produc3424155258940669874_state @ C2 @ ( produc9160152616361873709_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_979_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C2: prat > state > set_Pr2391701600042252046_state,A: prat,B3: state] :
      ( ( member6715010751053357271_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C2 @ ( produc6205719116230455971_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_980_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C2: prat > set_state > set_Pr2391701600042252046_state,A: prat,B3: set_state] :
      ( ( member6715010751053357271_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C2 @ ( produc9080284787853255769_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_981_mem__case__prodI,axiom,
    ! [Z: produc3882902684156697774_state,C2: option_state > option_state > set_Pr2391701600042252046_state,A: option_state,B3: option_state] :
      ( ( member6715010751053357271_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C2 @ ( produc9160152616361873709_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_982_mem__case__prodI,axiom,
    ! [Z: produc7272280185086996969_state,C2: prat > state > set_Pr4231240917283394591_state,A: prat,B3: state] :
      ( ( member7461423308196221312_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C2 @ ( produc6205719116230455971_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_983_mem__case__prodI,axiom,
    ! [Z: produc7272280185086996969_state,C2: prat > set_state > set_Pr4231240917283394591_state,A: prat,B3: set_state] :
      ( ( member7461423308196221312_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C2 @ ( produc9080284787853255769_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_984_mem__case__prodI,axiom,
    ! [Z: produc7272280185086996969_state,C2: option_state > option_state > set_Pr4231240917283394591_state,A: option_state,B3: option_state] :
      ( ( member7461423308196221312_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member7461423308196221312_state @ Z @ ( produc6752009811597927777_state @ C2 @ ( produc9160152616361873709_state @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_985_mem__case__prodI,axiom,
    ! [Z: state,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state,A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
      ( ( member_state @ Z @ ( C2 @ A @ B3 ) )
     => ( member_state @ Z @ ( produc7498597735380428723_state @ C2 @ ( produc74620467203698198on_val @ A @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_986_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: state,C2: prat > state > set_state] :
      ( ! [A6: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A6 @ B7 ) )
         => ( member_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member_state @ Z @ ( produc2825319096956678952_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_987_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: state,C2: prat > set_state > set_state] :
      ( ! [A6: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A6 @ B7 ) )
         => ( member_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member_state @ Z @ ( produc5034060725774254814_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_988_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z: state,C2: option_state > option_state > set_state] :
      ( ! [A6: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A6 @ B7 ) )
         => ( member_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member_state @ Z @ ( produc3424155258940669874_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_989_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: produc3882902684156697774_state,C2: prat > state > set_Pr2391701600042252046_state] :
      ( ! [A6: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A6 @ B7 ) )
         => ( member6715010751053357271_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_990_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: produc3882902684156697774_state,C2: prat > set_state > set_Pr2391701600042252046_state] :
      ( ! [A6: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A6 @ B7 ) )
         => ( member6715010751053357271_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_991_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z: produc3882902684156697774_state,C2: option_state > option_state > set_Pr2391701600042252046_state] :
      ( ! [A6: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A6 @ B7 ) )
         => ( member6715010751053357271_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_992_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z: produc7272280185086996969_state,C2: prat > state > set_Pr4231240917283394591_state] :
      ( ! [A6: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A6 @ B7 ) )
         => ( member7461423308196221312_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_993_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z: produc7272280185086996969_state,C2: prat > set_state > set_Pr4231240917283394591_state] :
      ( ! [A6: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A6 @ B7 ) )
         => ( member7461423308196221312_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_994_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z: produc7272280185086996969_state,C2: option_state > option_state > set_Pr4231240917283394591_state] :
      ( ! [A6: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A6 @ B7 ) )
         => ( member7461423308196221312_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z @ ( produc6752009811597927777_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_995_mem__case__prodI2,axiom,
    ! [P2: produc9003781588162623004on_val,Z: state,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state] :
      ( ! [A6: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( ( P2
            = ( produc74620467203698198on_val @ A6 @ B7 ) )
         => ( member_state @ Z @ ( C2 @ A6 @ B7 ) ) )
     => ( member_state @ Z @ ( produc7498597735380428723_state @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_996_mem__case__prodE,axiom,
    ! [Z: state,C2: prat > state > set_state,P2: produc4844402304972151915_state] :
      ( ( member_state @ Z @ ( produc2825319096956678952_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X4 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_997_mem__case__prodE,axiom,
    ! [Z: state,C2: prat > set_state > set_state,P2: produc6251377953716854561_state] :
      ( ( member_state @ Z @ ( produc5034060725774254814_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X4 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_998_mem__case__prodE,axiom,
    ! [Z: state,C2: option_state > option_state > set_state,P2: produc3142500478612311029_state] :
      ( ( member_state @ Z @ ( produc3424155258940669874_state @ C2 @ P2 ) )
     => ~ ! [X4: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X4 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_999_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C2: prat > state > set_Pr2391701600042252046_state,P2: produc4844402304972151915_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc4369308447073708118_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X4 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1000_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C2: prat > set_state > set_Pr2391701600042252046_state,P2: produc6251377953716854561_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc9043967962556017184_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X4 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1001_mem__case__prodE,axiom,
    ! [Z: produc3882902684156697774_state,C2: option_state > option_state > set_Pr2391701600042252046_state,P2: produc3142500478612311029_state] :
      ( ( member6715010751053357271_state @ Z @ ( produc2209043727768083020_state @ C2 @ P2 ) )
     => ~ ! [X4: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X4 @ Y4 ) )
           => ~ ( member6715010751053357271_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1002_mem__case__prodE,axiom,
    ! [Z: produc7272280185086996969_state,C2: prat > state > set_Pr4231240917283394591_state,P2: produc4844402304972151915_state] :
      ( ( member7461423308196221312_state @ Z @ ( produc569869127943504087_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X4 @ Y4 ) )
           => ~ ( member7461423308196221312_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1003_mem__case__prodE,axiom,
    ! [Z: produc7272280185086996969_state,C2: prat > set_state > set_Pr4231240917283394591_state,P2: produc6251377953716854561_state] :
      ( ( member7461423308196221312_state @ Z @ ( produc7793558081207154829_state @ C2 @ P2 ) )
     => ~ ! [X4: prat,Y4: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X4 @ Y4 ) )
           => ~ ( member7461423308196221312_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1004_mem__case__prodE,axiom,
    ! [Z: produc7272280185086996969_state,C2: option_state > option_state > set_Pr4231240917283394591_state,P2: produc3142500478612311029_state] :
      ( ( member7461423308196221312_state @ Z @ ( produc6752009811597927777_state @ C2 @ P2 ) )
     => ~ ! [X4: option_state,Y4: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X4 @ Y4 ) )
           => ~ ( member7461423308196221312_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1005_mem__case__prodE,axiom,
    ! [Z: state,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state,P2: produc9003781588162623004on_val] :
      ( ( member_state @ Z @ ( produc7498597735380428723_state @ C2 @ P2 ) )
     => ~ ! [X4: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] :
            ( ( P2
              = ( produc74620467203698198on_val @ X4 @ Y4 ) )
           => ~ ( member_state @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1006_sep__algebra_Ounder_Ocong,axiom,
    sep_under_state = sep_under_state ).

% sep_algebra.under.cong
thf(fact_1007_package__logic_Ovalid__package__set__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F )
        = ( ! [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ S2 )
             => ( produc1745419677418529872tate_o
                @ ^ [A2: state] :
                    ( produc3645877548210629341tate_o
                    @ ^ [U6: state,T4: state > state] :
                        ( ( sep_defined_state @ Plus @ A2 @ U6 )
                        & ( sep_greater_state @ Plus @ ( Core @ A2 ) @ ( Core @ U6 ) )
                        & ( packag4817583915586052229_state @ Plus @ Unit @ T4 )
                        & ( sep_greater_state @ Plus @ A2 @ ( Core @ ( T4 @ F ) ) ) ) )
                @ X ) ) ) ) ) ).

% package_logic.valid_package_set_def
thf(fact_1008_same__fst__def,axiom,
    ( same_f3145481655561538747_state
    = ( ^ [P5: state > $o,R5: state > set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [X8: state,Y8: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [X: state,Y3: state > state] :
                      ( ( X8 = X )
                      & ( P5 @ X )
                      & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ Y8 @ Y3 ) @ ( R5 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_1009_same__fst__def,axiom,
    ( same_f8765775015391913222_state
    = ( ^ [P5: state > $o,R5: state > set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [X8: state,Y8: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [X: state,Y3: produc3882902684156697774_state] :
                      ( ( X8 = X )
                      & ( P5 @ X )
                      & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ Y8 @ Y3 ) @ ( R5 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_1010_PartialSA_Ovalid__package__set__def,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state] :
      ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T4: state > state] :
                      ( ( defined @ A2 @ U6 )
                      & ( greater @ ( core @ A2 ) @ ( core @ U6 ) )
                      & ( packag4817583915586052229_state @ plus @ unit @ T4 )
                      & ( greater @ A2 @ ( core @ ( T4 @ F ) ) ) ) )
              @ X ) ) ) ) ).

% PartialSA.valid_package_set_def
thf(fact_1011_package__logic_Opure__remains__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr523392378426063460_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag9013619817833129428_state @ Plus @ S2 )
        = ( ! [X: produc4357056439811976324_state] :
              ( ( member5752234768399382829_state @ X @ S2 )
             => ( produc8801812494870056711tate_o
                @ ^ [A2: state] :
                    ( produc1745419677418529872tate_o
                    @ ^ [U6: state,P3: produc3882902684156697774_state] : ( sep_pure_state @ Plus @ A2 ) )
                @ X ) ) ) ) ) ).

% package_logic.pure_remains_def
thf(fact_1012_package__logic_Opure__remains__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag4152378406607043437_state @ Plus @ S2 )
        = ( ! [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ S2 )
             => ( produc1745419677418529872tate_o
                @ ^ [A2: state] :
                    ( produc3645877548210629341tate_o
                    @ ^ [U6: state,P3: state > state] : ( sep_pure_state @ Plus @ A2 ) )
                @ X ) ) ) ) ) ).

% package_logic.pure_remains_def
thf(fact_1013_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr523392378426063460_state] :
      ( ( packag9013619817833129428_state @ plus @ S2 )
      = ( ! [X: produc4357056439811976324_state] :
            ( ( member5752234768399382829_state @ X @ S2 )
           => ( produc8801812494870056711tate_o
              @ ^ [A2: state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [U6: state,P3: produc3882902684156697774_state] : ( sep_pure_state @ plus @ A2 ) )
              @ X ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_1014_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr4231240917283394591_state] :
      ( ( packag4152378406607043437_state @ plus @ S2 )
      = ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,P3: state > state] : ( sep_pure_state @ plus @ A2 ) )
              @ X ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_1015_package__logic_Opackage__rhs__connection__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
        = ( ( sep_greater_state @ Plus @ F3 @ F )
          & ( sep_defined_state @ Plus @ Phi @ F )
          & ( ( Plus @ Phi @ F )
            = ( Plus @ Phi2 @ F3 ) )
          & ( Stable @ F3 )
          & ! [X: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X @ S2 )
             => ( produc1745419677418529872tate_o
                @ ^ [A2: state] :
                    ( produc3645877548210629341tate_o
                    @ ^ [U6: state,T4: state > state] :
                      ? [Au3: state] :
                        ( ( ( some_state @ Au3 )
                          = ( Plus @ A2 @ U6 ) )
                        & ( ( sep_defined_state @ Plus @ Au3 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                         => ? [A13: state,U8: state] :
                              ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A13 @ ( produc8632743443909064678_state @ U8 @ T4 ) ) @ S3 )
                              & ( sep_greater_state @ Plus @ ( Core @ A13 ) @ ( Core @ A2 ) )
                              & ( ( Plus @ Au3 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                                = ( Plus @ A13 @ U8 ) )
                              & ( sep_greater_state @ Plus @ U8 @ U6 )
                              & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A5 @ A13 @ U8 @ U6 ) ) ) ) )
                @ X ) ) ) ) ) ).

% package_logic.package_rhs_connection_def
thf(fact_1016_package__logic_Ocompleteness__aux,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F2: state > state > ( state > state ) > state,Pc: state > $o,A5: aassertion_state,F: state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A6: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A6 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( sep_greater_state @ Plus @ ( Core @ ( F1 @ A6 @ U @ T3 ) ) @ ( Core @ A6 ) )
              & ( ( some_state @ A6 )
                = ( Plus @ ( F1 @ A6 @ U @ T3 ) @ ( F2 @ A6 @ U @ T3 ) ) )
              & ( ( Pc @ ( Core @ A6 ) )
               => ( package_sat_state @ Plus @ A5 @ ( the_state @ ( Plus @ ( Core @ A6 ) @ ( F1 @ A6 @ U @ T3 ) ) ) ) ) ) )
       => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A5 )
           => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
             => ( ( ( Stable @ F )
                  & ( sep_defined_state @ Plus @ Phi @ F ) )
               => ? [S6: set_Pr4231240917283394591_state] :
                    ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi @ F @ S6 )
                    & ! [X3: produc7272280185086996969_state] :
                        ( ( member7461423308196221312_state @ X3 @ S6 )
                       => ( produc1745419677418529872tate_o
                          @ ^ [A13: state] :
                              ( produc3645877548210629341tate_o
                              @ ^ [U8: state,T4: state > state] :
                                ? [A2: state,U6: state] :
                                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                                  & ( sep_greater_state @ Plus @ A13 @ ( F2 @ A2 @ U6 @ T4 ) )
                                  & ( ( Core @ A13 )
                                    = ( Core @ A2 ) ) ) )
                          @ X3 ) ) ) ) ) ) ) ) ) ).

% package_logic.completeness_aux
thf(fact_1017_PartialSA_Opackage__rhs__connection__def,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A5: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A5 @ Phi2 @ F3 @ S3 )
      = ( ( greater @ F3 @ F )
        & ( defined @ Phi @ F )
        & ( ( plus @ Phi @ F )
          = ( plus @ Phi2 @ F3 ) )
        & ( stable @ F3 )
        & ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A2: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T4: state > state] :
                    ? [Au3: state] :
                      ( ( ( some_state @ Au3 )
                        = ( plus @ A2 @ U6 ) )
                      & ( ( defined @ Au3 @ ( minus @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                       => ? [A13: state,U8: state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A13 @ ( produc8632743443909064678_state @ U8 @ T4 ) ) @ S3 )
                            & ( greater @ ( core @ A13 ) @ ( core @ A2 ) )
                            & ( ( plus @ Au3 @ ( minus @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                              = ( plus @ A13 @ U8 ) )
                            & ( greater @ U8 @ U6 )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A5 @ A13 @ U8 @ U6 ) ) ) ) )
              @ X ) ) ) ) ).

% PartialSA.package_rhs_connection_def
thf(fact_1018_compatible__options_Opelims_I3_J,axiom,
    ! [X2: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X2 @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X2 @ Xa2 ) )
       => ~ ! [A6: state] :
              ( ( X2
                = ( some_state @ A6 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) )
                   => ( A6 = B7 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_1019_Ball__Collect,axiom,
    ( ball_P7403102246598991807_state
    = ( ^ [A4: set_Pr4231240917283394591_state,P5: produc7272280185086996969_state > $o] : ( ord_le4032880777312351359_state @ A4 @ ( collec906845016635587262_state @ P5 ) ) ) ) ).

% Ball_Collect
thf(fact_1020_Ball__Collect,axiom,
    ( ball_P2871712132005019634on_val
    = ( ^ [A4: set_Pr7649655576376376530on_val,P5: produc9003781588162623004on_val > $o] : ( ord_le6900622908513426738on_val @ A4 @ ( collec7475294502966882545on_val @ P5 ) ) ) ) ).

% Ball_Collect
thf(fact_1021_Ball__Collect,axiom,
    ( ball_state
    = ( ^ [A4: set_state,P5: state > $o] : ( ord_le2494988322063910608_state @ A4 @ ( collect_state @ P5 ) ) ) ) ).

% Ball_Collect
thf(fact_1022_split__cong,axiom,
    ! [Q2: produc3882902684156697774_state,F: state > ( state > state ) > $o,G: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ! [X4: state,Y4: state > state] :
          ( ( ( produc8632743443909064678_state @ X4 @ Y4 )
            = Q2 )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc3645877548210629341tate_o @ F @ P2 )
          = ( produc3645877548210629341tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_1023_split__cong,axiom,
    ! [Q2: produc7272280185086996969_state,F: state > produc3882902684156697774_state > $o,G: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ! [X4: state,Y4: produc3882902684156697774_state] :
          ( ( ( produc5886875603596089627_state @ X4 @ Y4 )
            = Q2 )
         => ( ( F @ X4 @ Y4 )
            = ( G @ X4 @ Y4 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc1745419677418529872tate_o @ F @ P2 )
          = ( produc1745419677418529872tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_1024_sep__algebra_Osplus_Opelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: option_state,Xa2: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X2 @ Xa2 )
          = Y )
       => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X2 @ Xa2 ) )
         => ( ( ( X2 = none_state )
             => ( ( Y = none_state )
               => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
           => ( ! [V: state] :
                  ( ( X2
                    = ( some_state @ V ) )
                 => ( ( Xa2 = none_state )
                   => ( ( Y = none_state )
                     => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) ) ) ) )
             => ~ ! [A6: state] :
                    ( ( X2
                      = ( some_state @ A6 ) )
                   => ! [B7: state] :
                        ( ( Xa2
                          = ( some_state @ B7 ) )
                       => ( ( Y
                            = ( Plus @ A6 @ B7 ) )
                         => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.pelims
thf(fact_1025_sep__algebra_Osucc__refl,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_greater_state @ Plus @ A @ A ) ) ).

% sep_algebra.succ_refl
thf(fact_1026_sep__algebra_Osucc__trans,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
       => ( ( sep_greater_state @ Plus @ B3 @ C2 )
         => ( sep_greater_state @ Plus @ A @ C2 ) ) ) ) ).

% sep_algebra.succ_trans
thf(fact_1027_sep__algebra_Osucc__antisym,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
       => ( ( sep_greater_state @ Plus @ B3 @ A )
         => ( A = B3 ) ) ) ) ).

% sep_algebra.succ_antisym
thf(fact_1028_sep__algebra_Ompp__prop,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( P @ ( F @ X2 ) ) ) ) ).

% sep_algebra.mpp_prop
thf(fact_1029_sep__algebra_Ompp__invo,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( F @ ( F @ X2 ) )
          = ( F @ X2 ) ) ) ) ).

% sep_algebra.mpp_invo
thf(fact_1030_sep__algebra_Osetify__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ A5 )
        = ( ! [X: state] :
              ( ( member_state @ X @ A5 )
             => ( P @ X ) ) ) ) ) ).

% sep_algebra.setify_def
thf(fact_1031_sep__algebra_Ocommutative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Plus @ A @ B3 )
        = ( Plus @ B3 @ A ) ) ) ).

% sep_algebra.commutative
thf(fact_1032_sep__algebra_Oasso1,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,Ab: state,C2: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B3 )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B3 @ C2 )
            = ( some_state @ Bc ) ) )
       => ( ( Plus @ Ab @ C2 )
          = ( Plus @ A @ Bc ) ) ) ) ).

% sep_algebra.asso1
thf(fact_1033_sep__algebra_Ocore__max,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,C2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ X2 @ C2 ) )
       => ? [R3: state] :
            ( ( some_state @ ( Core @ X2 ) )
            = ( Plus @ C2 @ R3 ) ) ) ) ).

% sep_algebra.core_max
thf(fact_1034_sep__algebra_Ocore__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C2: state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C2 )
          = ( Plus @ A @ B3 ) )
       => ( ( some_state @ ( Core @ C2 ) )
          = ( Plus @ ( Core @ A ) @ ( Core @ B3 ) ) ) ) ) ).

% sep_algebra.core_sum
thf(fact_1035_sep__algebra_Opositivity,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A @ B3 )
          = ( some_state @ C2 ) )
       => ( ( ( some_state @ C2 )
            = ( Plus @ C2 @ C2 ) )
         => ( ( some_state @ A )
            = ( Plus @ A @ A ) ) ) ) ) ).

% sep_algebra.positivity
thf(fact_1036_sep__algebra_Ocancellative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,X2: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B3 @ X2 ) )
       => ( ( ( some_state @ A )
            = ( Plus @ B3 @ Y ) )
         => ( ( ( Core @ X2 )
              = ( Core @ Y ) )
           => ( X2 = Y ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_1037_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ ( Core @ X2 ) )
        = ( Plus @ ( Core @ X2 ) @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_1038_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ X2 )
        = ( Plus @ X2 @ ( Core @ X2 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_1039_sep__algebra_Osplus__asso,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: option_state,B3: option_state,C2: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ A @ B3 ) @ C2 )
        = ( sep_splus_state @ Plus @ A @ ( sep_splus_state @ Plus @ B3 @ C2 ) ) ) ) ).

% sep_algebra.splus_asso
thf(fact_1040_sep__algebra_Osplus__comm,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: option_state,B3: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ A @ B3 )
        = ( sep_splus_state @ Plus @ B3 @ A ) ) ) ).

% sep_algebra.splus_comm
thf(fact_1041_sep__algebra_Ominus__core__weaker,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B3 ) )
        = ( sep_minus_state @ Plus @ Core @ ( Core @ A ) @ ( Core @ B3 ) ) ) ) ).

% sep_algebra.minus_core_weaker
thf(fact_1042_sep__algebra_Ominus__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B3 ) )
        = ( Core @ A ) ) ) ).

% sep_algebra.minus_core
thf(fact_1043_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_1044_sep__algebra_Oasso2,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,Ab: state,C2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B3 )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B3 @ C2 )
            = none_state ) )
       => ( ( Plus @ Ab @ C2 )
          = none_state ) ) ) ).

% sep_algebra.asso2
thf(fact_1045_sep__algebra_Oasso3,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A @ B3 )
          = none_state )
       => ( ( ( Plus @ B3 @ C2 )
            = ( some_state @ Bc ) )
         => ( ( Plus @ A @ Bc )
            = none_state ) ) ) ) ).

% sep_algebra.asso3
thf(fact_1046_sep__algebra_Ointro,axiom,
    ! [Plus: state > state > option_state,Core: state > state] :
      ( ! [A6: state,B7: state] :
          ( ( Plus @ A6 @ B7 )
          = ( Plus @ B7 @ A6 ) )
     => ( ! [A6: state,B7: state,Ab3: state,C4: state,Bc2: state] :
            ( ( ( ( Plus @ A6 @ B7 )
                = ( some_state @ Ab3 ) )
              & ( ( Plus @ B7 @ C4 )
                = ( some_state @ Bc2 ) ) )
           => ( ( Plus @ Ab3 @ C4 )
              = ( Plus @ A6 @ Bc2 ) ) )
       => ( ! [A6: state,B7: state,Ab3: state,C4: state] :
              ( ( ( ( Plus @ A6 @ B7 )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus @ B7 @ C4 )
                  = none_state ) )
             => ( ( Plus @ Ab3 @ C4 )
                = none_state ) )
         => ( ! [X4: state] :
                ( ( some_state @ X4 )
                = ( Plus @ X4 @ ( Core @ X4 ) ) )
           => ( ! [X4: state] :
                  ( ( some_state @ ( Core @ X4 ) )
                  = ( Plus @ ( Core @ X4 ) @ ( Core @ X4 ) ) )
             => ( ! [X4: state,C4: state] :
                    ( ( ( some_state @ X4 )
                      = ( Plus @ X4 @ C4 ) )
                   => ? [R6: state] :
                        ( ( some_state @ ( Core @ X4 ) )
                        = ( Plus @ C4 @ R6 ) ) )
               => ( ! [C4: state,A6: state,B7: state] :
                      ( ( ( some_state @ C4 )
                        = ( Plus @ A6 @ B7 ) )
                     => ( ( some_state @ ( Core @ C4 ) )
                        = ( Plus @ ( Core @ A6 ) @ ( Core @ B7 ) ) ) )
                 => ( ! [A6: state,B7: state,C4: state] :
                        ( ( ( Plus @ A6 @ B7 )
                          = ( some_state @ C4 ) )
                       => ( ( ( some_state @ C4 )
                            = ( Plus @ C4 @ C4 ) )
                         => ( ( some_state @ A6 )
                            = ( Plus @ A6 @ A6 ) ) ) )
                   => ( ! [A6: state,B7: state,X4: state,Y4: state] :
                          ( ( ( some_state @ A6 )
                            = ( Plus @ B7 @ X4 ) )
                         => ( ( ( some_state @ A6 )
                              = ( Plus @ B7 @ Y4 ) )
                           => ( ( ( Core @ X4 )
                                = ( Core @ Y4 ) )
                             => ( X4 = Y4 ) ) ) )
                     => ( sep_algebra_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_1047_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,Ab2: state,C: state,Bc3: state] :
              ( ( ( ( Plus2 @ A2 @ B )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus2 @ B @ C )
                  = ( some_state @ Bc3 ) ) )
             => ( ( Plus2 @ Ab2 @ C )
                = ( Plus2 @ A2 @ Bc3 ) ) )
          & ! [A2: state,B: state,Ab2: state,C: state] :
              ( ( ( ( Plus2 @ A2 @ B )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus2 @ B @ C )
                  = none_state ) )
             => ( ( Plus2 @ Ab2 @ C )
                = none_state ) )
          & ! [X: state] :
              ( ( some_state @ X )
              = ( Plus2 @ X @ ( Core2 @ X ) ) )
          & ! [X: state] :
              ( ( some_state @ ( Core2 @ X ) )
              = ( Plus2 @ ( Core2 @ X ) @ ( Core2 @ X ) ) )
          & ! [X: state,C: state] :
              ( ( ( some_state @ X )
                = ( Plus2 @ X @ C ) )
             => ? [R2: state] :
                  ( ( some_state @ ( Core2 @ X ) )
                  = ( Plus2 @ C @ R2 ) ) )
          & ! [C: state,A2: state,B: state] :
              ( ( ( some_state @ C )
                = ( Plus2 @ A2 @ B ) )
             => ( ( some_state @ ( Core2 @ C ) )
                = ( Plus2 @ ( Core2 @ A2 ) @ ( Core2 @ B ) ) ) )
          & ! [A2: state,B: state,C: state] :
              ( ( ( Plus2 @ A2 @ B )
                = ( some_state @ C ) )
             => ( ( ( some_state @ C )
                  = ( Plus2 @ C @ C ) )
               => ( ( some_state @ A2 )
                  = ( Plus2 @ A2 @ A2 ) ) ) )
          & ! [A2: state,B: state,X: state,Y3: state] :
              ( ( ( some_state @ A2 )
                = ( Plus2 @ B @ X ) )
             => ( ( ( some_state @ A2 )
                  = ( Plus2 @ B @ Y3 ) )
               => ( ( ( Core2 @ X )
                    = ( Core2 @ Y3 ) )
                 => ( X = Y3 ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_1048_PartialSA_Osep__algebra__axioms,axiom,
    sep_algebra_state @ plus @ core ).

% PartialSA.sep_algebra_axioms
thf(fact_1049_sep__algebra_Ogreater__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
        = ( ? [C: state] :
              ( ( some_state @ A )
              = ( Plus @ B3 @ C ) ) ) ) ) ).

% sep_algebra.greater_def
thf(fact_1050_sep__algebra_Oextract__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B3 )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_state @ Plus @ X2 @ ( Core @ B3 ) ) )
       => ( ( Core @ X2 )
          = ( Core @ B3 ) ) ) ) ).

% sep_algebra.extract_core
thf(fact_1051_sep__algebra_Ominus__exists,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ B3 @ A )
       => ? [X4: state] :
            ( ( ( some_state @ B3 )
              = ( Plus @ A @ X4 ) )
            & ( sep_greater_state @ Plus @ X4 @ ( Core @ B3 ) ) ) ) ) ).

% sep_algebra.minus_exists
thf(fact_1052_sep__algebra_Ominus__unique,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state,X2: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B3 )
            = ( Plus @ A @ X2 ) )
          & ( sep_greater_state @ Plus @ X2 @ ( Core @ B3 ) ) )
       => ( ( ( ( some_state @ B3 )
              = ( Plus @ A @ Y ) )
            & ( sep_greater_state @ Plus @ Y @ ( Core @ B3 ) ) )
         => ( X2 = Y ) ) ) ) ).

% sep_algebra.minus_unique
thf(fact_1053_sep__algebra_Ogreater__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B3 )
        = ( ? [C: state] :
              ( ( some_state @ A )
              = ( Plus @ C @ B3 ) ) ) ) ) ).

% sep_algebra.greater_equiv
thf(fact_1054_sep__algebra_Oaddition__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A7: state,A: state,X5: state,B3: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A7 @ A )
       => ( ( ( some_state @ X5 )
            = ( Plus @ A7 @ B3 ) )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B3 ) )
           => ( sep_greater_state @ Plus @ X5 @ X2 ) ) ) ) ) ).

% sep_algebra.addition_bigger
thf(fact_1055_sep__algebra_Osmaller__than__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X2: state,Z: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X2 )
       => ( ( ( some_state @ Z )
            = ( Plus @ X2 @ ( Core @ Y ) ) )
         => ( ( Core @ Z )
            = ( Core @ Y ) ) ) ) ) ).

% sep_algebra.smaller_than_core
thf(fact_1056_sep__algebra_Obigger__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C2: state,A: state,B3: state,A7: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C2 )
          = ( Plus @ A @ B3 ) )
       => ( ( sep_greater_state @ Plus @ A @ A7 )
         => ? [B4: state] :
              ( ( sep_greater_state @ Plus @ B4 @ B3 )
              & ( ( some_state @ C2 )
                = ( Plus @ A7 @ B4 ) ) ) ) ) ) ).

% sep_algebra.bigger_sum_smaller
thf(fact_1057_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ A @ B3 ) )
       => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) )
          = ( Plus @ B3 @ ( Core @ X2 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_1058_sep__algebra_Osplus__develop,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state,D: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B3 @ C2 ) )
       => ( ( Plus @ A @ D )
          = ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ ( some_state @ B3 ) @ ( some_state @ C2 ) ) @ ( some_state @ D ) ) ) ) ) ).

% sep_algebra.splus_develop
thf(fact_1059_sep__algebra_Osplus_Osimps_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ A ) @ ( some_state @ B3 ) )
        = ( Plus @ A @ B3 ) ) ) ).

% sep_algebra.splus.simps(3)
thf(fact_1060_sep__algebra_Opure__stable,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A )
       => ( ( sep_pure_state @ Plus @ B3 )
         => ( ( ( some_state @ C2 )
              = ( Plus @ A @ B3 ) )
           => ( sep_pure_state @ Plus @ C2 ) ) ) ) ) ).

% sep_algebra.pure_stable
thf(fact_1061_sep__algebra_Opure__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A )
        = ( ( some_state @ A )
          = ( Plus @ A @ A ) ) ) ) ).

% sep_algebra.pure_def
thf(fact_1062_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Uu2: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ none_state @ Uu2 )
        = none_state ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_1063_sep__algebra_Odefined__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_defined_state @ Plus @ A @ B3 )
        = ( ( Plus @ A @ B3 )
         != none_state ) ) ) ).

% sep_algebra.defined_def
thf(fact_1064_sep__algebra_Ominus__default,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ~ ( sep_greater_state @ Plus @ B3 @ A )
       => ( ( sep_minus_state @ Plus @ Core @ B3 @ A )
          = B3 ) ) ) ).

% sep_algebra.minus_default
thf(fact_1065_sep__algebra_Ominus__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ X2 @ A )
       => ( sep_greater_state @ Plus @ X2 @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) ) ) ) ).

% sep_algebra.minus_smaller
thf(fact_1066_sep__algebra_Ogreater__minus__trans,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X2: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X2 )
       => ( ( sep_greater_state @ Plus @ X2 @ A )
         => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ Y @ A ) @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) ) ) ) ) ).

% sep_algebra.greater_minus_trans
thf(fact_1067_sep__algebra_Opure__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A )
       => ( ( sep_greater_state @ Plus @ A @ B3 )
         => ( sep_pure_state @ Plus @ B3 ) ) ) ) ).

% sep_algebra.pure_smaller
thf(fact_1068_sep__algebra_Osmaller__compatible,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A7: state,B3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_defined_state @ Plus @ A7 @ B3 )
       => ( ( sep_greater_state @ Plus @ A7 @ A )
         => ( sep_defined_state @ Plus @ A @ B3 ) ) ) ) ).

% sep_algebra.smaller_compatible
thf(fact_1069_sep__algebra_Osmaller__compatible__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X2 )
       => ( sep_defined_state @ Plus @ X2 @ ( Core @ Y ) ) ) ) ).

% sep_algebra.smaller_compatible_core
thf(fact_1070_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ! [Phi6: produc7272280185086996969_state,Phi5: produc7272280185086996969_state] :
            ( ( ( sep_gr1233117237951779860_state @ Plus @ Phi6 @ Phi5 )
              & ( member7461423308196221312_state @ Phi5 @ A5 ) )
           => ( member7461423308196221312_state @ Phi6 @ A5 ) )
       => ( sep_up5671800011468993857_state @ Plus @ A5 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_1071_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ! [Phi6: produc9003781588162623004on_val,Phi5: produc9003781588162623004on_val] :
            ( ( ( sep_gr8124549141218502215on_val @ Plus @ Phi6 @ Phi5 )
              & ( member2879030624002546611on_val @ Phi5 @ A5 ) )
           => ( member2879030624002546611on_val @ Phi6 @ A5 ) )
       => ( sep_up3467202520194525556on_val @ Plus @ A5 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_1072_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ! [Phi6: produc3882902684156697774_state,Phi5: produc3882902684156697774_state] :
            ( ( ( sep_gr6114829331755084611_state @ Plus @ Phi6 @ Phi5 )
              & ( member6715010751053357271_state @ Phi5 @ A5 ) )
           => ( member6715010751053357271_state @ Phi6 @ A5 ) )
       => ( sep_up2356903840689987414_state @ Plus @ A5 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_1073_sep__algebra_Oup__closedI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [Phi6: state,Phi5: state] :
            ( ( ( sep_greater_state @ Plus @ Phi6 @ Phi5 )
              & ( member_state @ Phi5 @ A5 ) )
           => ( member_state @ Phi6 @ A5 ) )
       => ( sep_up_closed_state @ Plus @ A5 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_1074_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_up5671800011468993857_state @ Plus @ A5 )
        = ( ! [Phi3: produc7272280185086996969_state] :
              ( ? [X: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X @ A5 )
                  & ( sep_gr1233117237951779860_state @ Plus @ Phi3 @ X ) )
             => ( member7461423308196221312_state @ Phi3 @ A5 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_1075_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_up3467202520194525556on_val @ Plus @ A5 )
        = ( ! [Phi3: produc9003781588162623004on_val] :
              ( ? [X: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X @ A5 )
                  & ( sep_gr8124549141218502215on_val @ Plus @ Phi3 @ X ) )
             => ( member2879030624002546611on_val @ Phi3 @ A5 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_1076_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_up2356903840689987414_state @ Plus @ A5 )
        = ( ! [Phi3: produc3882902684156697774_state] :
              ( ? [X: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X @ A5 )
                  & ( sep_gr6114829331755084611_state @ Plus @ Phi3 @ X ) )
             => ( member6715010751053357271_state @ Phi3 @ A5 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_1077_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A5 )
        = ( ! [Phi3: state] :
              ( ? [X: state] :
                  ( ( member_state @ X @ A5 )
                  & ( sep_greater_state @ Plus @ Phi3 @ X ) )
             => ( member_state @ Phi3 @ A5 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_1078_sep__algebra_Omax__projection__propE_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,P2: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( ( P @ P2 )
            & ( sep_greater_state @ Plus @ X2 @ P2 ) )
         => ( sep_greater_state @ Plus @ ( F @ X2 ) @ P2 ) ) ) ) ).

% sep_algebra.max_projection_propE(3)
thf(fact_1079_sep__algebra_OmppI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_greater_state @ Plus @ A @ X2 )
         => ( ( P @ X2 )
           => ( ( sep_greater_state @ Plus @ X2 @ ( F @ A ) )
             => ( X2
                = ( F @ A ) ) ) ) ) ) ) ).

% sep_algebra.mppI
thf(fact_1080_sep__algebra_Ompp__mono,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_greater_state @ Plus @ A @ B3 )
         => ( sep_greater_state @ Plus @ ( F @ A ) @ ( F @ B3 ) ) ) ) ) ).

% sep_algebra.mpp_mono
thf(fact_1081_sep__algebra_Ompp__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( sep_greater_state @ Plus @ X2 @ ( F @ X2 ) ) ) ) ).

% sep_algebra.mpp_smaller
thf(fact_1082_sep__algebra_Omax__projection__propI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,F: state > state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [X4: state] : ( sep_greater_state @ Plus @ X4 @ ( F @ X4 ) )
       => ( ! [X4: state] : ( P @ ( F @ X4 ) )
         => ( ! [X4: state,P4: state] :
                ( ( ( P @ P4 )
                  & ( sep_greater_state @ Plus @ X4 @ P4 ) )
               => ( sep_greater_state @ Plus @ ( F @ X4 ) @ P4 ) )
           => ( sep_ma8214210560313151521_state @ Plus @ P @ F ) ) ) ) ) ).

% sep_algebra.max_projection_propI
thf(fact_1083_sep__algebra_Omax__projection__prop__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
        = ( ! [X: state] :
              ( ( sep_greater_state @ Plus @ X @ ( F @ X ) )
              & ( P @ ( F @ X ) )
              & ! [P3: state] :
                  ( ( ( P @ P3 )
                    & ( sep_greater_state @ Plus @ X @ P3 ) )
                 => ( sep_greater_state @ Plus @ ( F @ X ) @ P3 ) ) ) ) ) ) ).

% sep_algebra.max_projection_prop_def
thf(fact_1084_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_1085_sep__algebra_Ompp__compatible,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_defined_state @ Plus @ A @ B3 )
         => ( sep_defined_state @ Plus @ ( F @ A ) @ ( F @ B3 ) ) ) ) ) ).

% sep_algebra.mpp_compatible
thf(fact_1086_sep__algebra_Omono__prop__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_mono_prop_state @ Plus @ P )
        = ( ! [X: state,Y3: state] :
              ( ( ( sep_greater_state @ Plus @ Y3 @ X )
                & ( P @ X ) )
             => ( P @ Y3 ) ) ) ) ) ).

% sep_algebra.mono_prop_def
thf(fact_1087_sep__algebra_Omono__propI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [X4: state,Y4: state] :
            ( ( ( sep_greater_state @ Plus @ Y4 @ X4 )
              & ( P @ X4 ) )
           => ( P @ Y4 ) )
       => ( sep_mono_prop_state @ Plus @ P ) ) ) ).

% sep_algebra.mono_propI
thf(fact_1088_sep__algebra_Oup__close__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A5 )
       => ( ( sep_up_closed_state @ Plus @ B2 )
         => ( ( sep_equiv_state @ Plus @ A5 @ B2 )
            = ( A5 = B2 ) ) ) ) ) ).

% sep_algebra.up_close_equiv
thf(fact_1089_sep__algebra_Oupper__closure__up__closed,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_up_closed_state @ Plus @ ( sep_up1246176804924251236_state @ Plus @ A5 ) ) ) ).

% sep_algebra.upper_closure_up_closed
thf(fact_1090_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: option_state,Xa2: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X2 @ Xa2 )
          = Y )
       => ( ( ( X2 = none_state )
           => ( Y != none_state ) )
         => ( ( ? [V: state] :
                  ( X2
                  = ( some_state @ V ) )
             => ( ( Xa2 = none_state )
               => ( Y != none_state ) ) )
           => ~ ! [A6: state] :
                  ( ( X2
                    = ( some_state @ A6 ) )
                 => ! [B7: state] :
                      ( ( Xa2
                        = ( some_state @ B7 ) )
                     => ( Y
                       != ( Plus @ A6 @ B7 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_1091_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_1092_sep__algebra_Osplus_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: produc3142500478612311029_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [Uu: option_state] :
            ( X2
           != ( produc9160152616361873709_state @ none_state @ Uu ) )
       => ( ! [V: state] :
              ( X2
             != ( produc9160152616361873709_state @ ( some_state @ V ) @ none_state ) )
         => ~ ! [A6: state,B7: state] :
                ( X2
               != ( produc9160152616361873709_state @ ( some_state @ A6 ) @ ( some_state @ B7 ) ) ) ) ) ) ).

% sep_algebra.splus.cases
thf(fact_1093_sep__algebra_Ominus__equiv__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ B3 @ A )
       => ( ( ( some_state @ B3 )
            = ( Plus @ A @ ( sep_minus_state @ Plus @ Core @ B3 @ A ) ) )
          & ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ B3 @ A ) @ ( Core @ B3 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def
thf(fact_1094_sep__algebra_Ominus__and__plus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Omega: state,Omega2: state,R: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ Omega )
          = ( Plus @ Omega2 @ R ) )
       => ( ( sep_greater_state @ Plus @ Omega2 @ A )
         => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ Omega @ A ) )
            = ( Plus @ ( sep_minus_state @ Plus @ Core @ Omega2 @ A ) @ R ) ) ) ) ) ).

% sep_algebra.minus_and_plus
thf(fact_1095_sep__algebra_Ominus__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A: state,B3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X2 )
          = ( Plus @ A @ B3 ) )
       => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ X2 @ A ) @ B3 ) ) ) ).

% sep_algebra.minus_bigger
thf(fact_1096_sep__algebra_Ominus__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B3: state,C2: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B3 @ C2 ) )
       => ( ( sep_greater_state @ Plus @ X2 @ A )
         => ( ( sep_minus_state @ Plus @ Core @ X2 @ A )
            = ( sep_minus_state @ Plus @ Core @ ( sep_minus_state @ Plus @ Core @ X2 @ B3 ) @ C2 ) ) ) ) ) ).

% sep_algebra.minus_sum
thf(fact_1097_sep__algebra_OminusI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ B3 )
          = ( Plus @ A @ X2 ) )
       => ( ( sep_greater_state @ Plus @ X2 @ ( Core @ B3 ) )
         => ( X2
            = ( sep_minus_state @ Plus @ Core @ B3 @ A ) ) ) ) ) ).

% sep_algebra.minusI
thf(fact_1098_sep__algebra_Osmaller__pure__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,A: state,B3: state,X2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ A )
       => ( ( sep_greater_state @ Plus @ Y @ B3 )
         => ( ( ( some_state @ X2 )
              = ( Plus @ A @ B3 ) )
           => ( ( sep_pure_state @ Plus @ B3 )
             => ( sep_greater_state @ Plus @ Y @ X2 ) ) ) ) ) ) ).

% sep_algebra.smaller_pure_sum_smaller
thf(fact_1099_sep__algebra_Oequiv__up__closed__subset,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state,B2: set_state,C3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A5 )
       => ( ( sep_equiv_state @ Plus @ B2 @ C3 )
         => ( ( ord_le2494988322063910608_state @ B2 @ A5 )
            = ( ord_le2494988322063910608_state @ C3 @ A5 ) ) ) ) ) ).

% sep_algebra.equiv_up_closed_subset
thf(fact_1100_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_up6452048470549419908_state @ Plus @ A5 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Phi3: produc3882902684156697774_state,Phi4: produc3882902684156697774_state] :
              ( ( Uu3 = Phi3 )
              & ( sep_gr6114829331755084611_state @ Plus @ Phi3 @ Phi4 )
              & ( member6715010751053357271_state @ Phi4 @ A5 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_1101_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_up6054414863798424979_state @ Plus @ A5 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Phi3: produc7272280185086996969_state,Phi4: produc7272280185086996969_state] :
              ( ( Uu3 = Phi3 )
              & ( sep_gr1233117237951779860_state @ Plus @ Phi3 @ Phi4 )
              & ( member7461423308196221312_state @ Phi4 @ A5 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_1102_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_up7144661806627585478on_val @ Plus @ A5 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi3: produc9003781588162623004on_val,Phi4: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi3 )
              & ( sep_gr8124549141218502215on_val @ Plus @ Phi3 @ Phi4 )
              & ( member2879030624002546611on_val @ Phi4 @ A5 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_1103_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up1246176804924251236_state @ Plus @ A5 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi3: state,Phi4: state] :
              ( ( Uu3 = Phi3 )
              & ( sep_greater_state @ Plus @ Phi3 @ Phi4 )
              & ( member_state @ Phi4 @ A5 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_1104_sep__algebra_Omono__prop__set__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_mono_prop_state @ Plus @ P )
       => ( ( sep_equiv_state @ Plus @ A5 @ B2 )
         => ( ( sep_setify_state @ P @ A5 )
            = ( sep_setify_state @ P @ B2 ) ) ) ) ) ).

% sep_algebra.mono_prop_set_equiv
thf(fact_1105_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,Omega2: produc3882902684156697774_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_un1732284790778891013_state @ Plus @ A5 @ Omega2 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Omega3: produc3882902684156697774_state] :
              ( ( Uu3 = Omega3 )
              & ( member6715010751053357271_state @ Omega3 @ A5 )
              & ( sep_gr6114829331755084611_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_1106_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,Omega2: produc7272280185086996969_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_un1299216510214111186_state @ Plus @ A5 @ Omega2 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Omega3: produc7272280185086996969_state] :
              ( ( Uu3 = Omega3 )
              & ( member7461423308196221312_state @ Omega3 @ A5 )
              & ( sep_gr1233117237951779860_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_1107_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,Omega2: produc9003781588162623004on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_un3765025543224000005on_val @ Plus @ A5 @ Omega2 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Omega3: produc9003781588162623004on_val] :
              ( ( Uu3 = Omega3 )
              & ( member2879030624002546611on_val @ Omega3 @ A5 )
              & ( sep_gr8124549141218502215on_val @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_1108_sep__algebra_Ounder__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state,Omega2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_under_state @ Plus @ A5 @ Omega2 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Omega3: state] :
              ( ( Uu3 = Omega3 )
              & ( member_state @ Omega3 @ A5 )
              & ( sep_greater_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_1109_sep__algebra_Ominus__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_minus_state @ Plus @ Core @ B3 @ A )
        = ( fun_TH8221542053827138557_state @ B3
          @ ^ [X: state] :
              ( ( ( some_state @ B3 )
                = ( Plus @ A @ X ) )
              & ( sep_greater_state @ Plus @ X @ ( Core @ B3 ) ) ) ) ) ) ).

% sep_algebra.minus_def
thf(fact_1110_lex__prod__def,axiom,
    ( lex_pr8992908938815505897on_val
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr4305388979797279737on_val] :
          ( collec4536212101028033490on_val
          @ ( produc3489267648137303980_val_o
            @ ( produc7816547785737266186_val_o
              @ ^ [A2: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
                  ( produc1845727980445322802_val_o
                  @ ^ [A13: produc9003781588162623004on_val,B11: produc9003781588162623004on_val] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1111_lex__prod__def,axiom,
    ( lex_pr849482514101377111_state
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr1688445902015331925_state] :
          ( collec3985216178052149586_state
          @ ( produc6773883055741934892tate_o
            @ ( produc6001348999437208714tate_o
              @ ^ [A2: produc9003781588162623004on_val,B: option_state] :
                  ( produc3589443509783526980tate_o
                  @ ^ [A13: produc9003781588162623004on_val,B11: option_state] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1112_lex__prod__def,axiom,
    ( lex_pr7503510086898530263on_val
    = ( ^ [Ra: set_Pr1688445902015331925_state,Rb: set_Pr4305388979797279737on_val] :
          ( collec5286658484364120402on_val
          @ ( produc6745459475987627308_val_o
            @ ( produc5489745010067198858_val_o
              @ ^ [A2: option_state,B: produc9003781588162623004on_val] :
                  ( produc8282195775912345284_val_o
                  @ ^ [A13: option_state,B11: produc9003781588162623004on_val] :
                      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1113_lex__prod__def,axiom,
    ( lex_pr8336311929016760645_state
    = ( ^ [Ra: set_Pr1688445902015331925_state,Rb: set_Pr1688445902015331925_state] :
          ( collec6784228404298526930_state
          @ ( produc1935756327278076076tate_o
            @ ( produc5315882979514849802tate_o
              @ ^ [A2: option_state,B: option_state] :
                  ( produc6082990154846732246tate_o
                  @ ^ [A13: option_state,B11: option_state] :
                      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1114_lex__prod__def,axiom,
    ( lex_pr4041356198452583422_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [A2: state,B: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [A13: state,B11: state > state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1115_lex__prod__def,axiom,
    ( lex_pr7268379037978345219_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [A2: state,B: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [A13: state,B11: produc3882902684156697774_state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A2 @ A13 ) @ Ra )
                      | ( ( A2 = A13 )
                        & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ B @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_1116_case__prod__Pair__iden,axiom,
    ! [P2: produc1317171644805857433on_val] :
      ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1117_case__prod__Pair__iden,axiom,
    ! [P2: produc3142500478612311029_state] :
      ( ( produc803252138055838771_state @ produc9160152616361873709_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1118_case__prod__Pair__iden,axiom,
    ! [P2: produc4844402304972151915_state] :
      ( ( produc1922103714420728243_state @ produc6205719116230455971_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1119_case__prod__Pair__iden,axiom,
    ! [P2: produc6251377953716854561_state] :
      ( ( produc897481912382009139_state @ produc9080284787853255769_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1120_case__prod__Pair__iden,axiom,
    ! [P2: produc9003781588162623004on_val] :
      ( ( produc4192255817509754207on_val @ produc74620467203698198on_val @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1121_wand__equiv__def,axiom,
    ( wand
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi4: state] :
              ( ( Uu3 = Phi4 )
              & ( ord_le2494988322063910608_state @ ( add_set @ A4 @ ( insert_state @ Phi4 @ bot_bot_set_state ) ) @ B5 ) ) ) ) ) ).

% wand_equiv_def
thf(fact_1122_empty__Collect__eq,axiom,
    ! [P: produc7272280185086996969_state > $o] :
      ( ( bot_bo6430881413863817907_state
        = ( collec906845016635587262_state @ P ) )
      = ( ! [X: produc7272280185086996969_state] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1123_empty__Collect__eq,axiom,
    ! [P: produc9003781588162623004on_val > $o] :
      ( ( bot_bo6722054719108133734on_val
        = ( collec7475294502966882545on_val @ P ) )
      = ( ! [X: produc9003781588162623004on_val] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1124_empty__Collect__eq,axiom,
    ! [P: state > $o] :
      ( ( bot_bot_set_state
        = ( collect_state @ P ) )
      = ( ! [X: state] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1125_Collect__empty__eq,axiom,
    ! [P: produc7272280185086996969_state > $o] :
      ( ( ( collec906845016635587262_state @ P )
        = bot_bo6430881413863817907_state )
      = ( ! [X: produc7272280185086996969_state] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1126_Collect__empty__eq,axiom,
    ! [P: produc9003781588162623004on_val > $o] :
      ( ( ( collec7475294502966882545on_val @ P )
        = bot_bo6722054719108133734on_val )
      = ( ! [X: produc9003781588162623004on_val] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1127_Collect__empty__eq,axiom,
    ! [P: state > $o] :
      ( ( ( collect_state @ P )
        = bot_bot_set_state )
      = ( ! [X: state] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1128_all__not__in__conv,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( ! [X: produc7272280185086996969_state] :
            ~ ( member7461423308196221312_state @ X @ A5 ) )
      = ( A5 = bot_bo6430881413863817907_state ) ) ).

% all_not_in_conv
thf(fact_1129_all__not__in__conv,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( ! [X: produc9003781588162623004on_val] :
            ~ ( member2879030624002546611on_val @ X @ A5 ) )
      = ( A5 = bot_bo6722054719108133734on_val ) ) ).

% all_not_in_conv
thf(fact_1130_all__not__in__conv,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ( ! [X: produc3882902684156697774_state] :
            ~ ( member6715010751053357271_state @ X @ A5 ) )
      = ( A5 = bot_bo3524941057552913786_state ) ) ).

% all_not_in_conv
thf(fact_1131_all__not__in__conv,axiom,
    ! [A5: set_state] :
      ( ( ! [X: state] :
            ~ ( member_state @ X @ A5 ) )
      = ( A5 = bot_bot_set_state ) ) ).

% all_not_in_conv
thf(fact_1132_empty__iff,axiom,
    ! [C2: produc7272280185086996969_state] :
      ~ ( member7461423308196221312_state @ C2 @ bot_bo6430881413863817907_state ) ).

% empty_iff
thf(fact_1133_empty__iff,axiom,
    ! [C2: produc9003781588162623004on_val] :
      ~ ( member2879030624002546611on_val @ C2 @ bot_bo6722054719108133734on_val ) ).

% empty_iff
thf(fact_1134_empty__iff,axiom,
    ! [C2: produc3882902684156697774_state] :
      ~ ( member6715010751053357271_state @ C2 @ bot_bo3524941057552913786_state ) ).

% empty_iff
thf(fact_1135_empty__iff,axiom,
    ! [C2: state] :
      ~ ( member_state @ C2 @ bot_bot_set_state ) ).

% empty_iff
thf(fact_1136_image__is__empty,axiom,
    ! [F: state > state,A5: set_state] :
      ( ( ( image_state_state @ F @ A5 )
        = bot_bot_set_state )
      = ( A5 = bot_bot_set_state ) ) ).

% image_is_empty
thf(fact_1137_empty__is__image,axiom,
    ! [F: state > state,A5: set_state] :
      ( ( bot_bot_set_state
        = ( image_state_state @ F @ A5 ) )
      = ( A5 = bot_bot_set_state ) ) ).

% empty_is_image
thf(fact_1138_image__empty,axiom,
    ! [F: state > state] :
      ( ( image_state_state @ F @ bot_bot_set_state )
      = bot_bot_set_state ) ).

% image_empty
thf(fact_1139_empty__subsetI,axiom,
    ! [A5: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A5 ) ).

% empty_subsetI
thf(fact_1140_subset__empty,axiom,
    ! [A5: set_state] :
      ( ( ord_le2494988322063910608_state @ A5 @ bot_bot_set_state )
      = ( A5 = bot_bot_set_state ) ) ).

% subset_empty
thf(fact_1141_singletonI,axiom,
    ! [A: produc7272280185086996969_state] : ( member7461423308196221312_state @ A @ ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) ) ).

% singletonI
thf(fact_1142_singletonI,axiom,
    ! [A: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ A @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) ).

% singletonI
thf(fact_1143_singletonI,axiom,
    ! [A: produc3882902684156697774_state] : ( member6715010751053357271_state @ A @ ( insert94804854012721790_state @ A @ bot_bo3524941057552913786_state ) ) ).

% singletonI
thf(fact_1144_singletonI,axiom,
    ! [A: state] : ( member_state @ A @ ( insert_state @ A @ bot_bot_set_state ) ) ).

% singletonI
thf(fact_1145_ball__empty,axiom,
    ! [P: state > $o,X3: state] :
      ( ( member_state @ X3 @ bot_bot_set_state )
     => ( P @ X3 ) ) ).

% ball_empty
thf(fact_1146_Un__empty,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ( sup_su3837536599209510219_state @ A5 @ B2 )
        = bot_bo6430881413863817907_state )
      = ( ( A5 = bot_bo6430881413863817907_state )
        & ( B2 = bot_bo6430881413863817907_state ) ) ) ).

% Un_empty
thf(fact_1147_Un__empty,axiom,
    ! [A5: set_state,B2: set_state] :
      ( ( ( sup_sup_set_state @ A5 @ B2 )
        = bot_bot_set_state )
      = ( ( A5 = bot_bot_set_state )
        & ( B2 = bot_bot_set_state ) ) ) ).

% Un_empty
thf(fact_1148_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec906845016635587262_state
            @ ^ [S10: produc7272280185086996969_state] : P )
          = top_to174962084833059535_state ) )
      & ( ~ P
       => ( ( collec906845016635587262_state
            @ ^ [S10: produc7272280185086996969_state] : P )
          = bot_bo6430881413863817907_state ) ) ) ).

% Collect_const
thf(fact_1149_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec7475294502966882545on_val
            @ ^ [S10: produc9003781588162623004on_val] : P )
          = top_to1561438921946393474on_val ) )
      & ( ~ P
       => ( ( collec7475294502966882545on_val
            @ ^ [S10: produc9003781588162623004on_val] : P )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_const
thf(fact_1150_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_state
            @ ^ [S10: state] : P )
          = top_top_set_state ) )
      & ( ~ P
       => ( ( collect_state
            @ ^ [S10: state] : P )
          = bot_bot_set_state ) ) ) ).

% Collect_const
thf(fact_1151_singleton__conv2,axiom,
    ! [A: produc7272280185086996969_state] :
      ( ( collec906845016635587262_state
        @ ( ^ [Y5: produc7272280185086996969_state,Z2: produc7272280185086996969_state] : ( Y5 = Z2 )
          @ A ) )
      = ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) ) ).

% singleton_conv2
thf(fact_1152_singleton__conv2,axiom,
    ! [A: produc9003781588162623004on_val] :
      ( ( collec7475294502966882545on_val
        @ ( ^ [Y5: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] : ( Y5 = Z2 )
          @ A ) )
      = ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) ).

% singleton_conv2
thf(fact_1153_singleton__conv2,axiom,
    ! [A: state] :
      ( ( collect_state
        @ ( ^ [Y5: state,Z2: state] : ( Y5 = Z2 )
          @ A ) )
      = ( insert_state @ A @ bot_bot_set_state ) ) ).

% singleton_conv2
thf(fact_1154_singleton__conv,axiom,
    ! [A: produc7272280185086996969_state] :
      ( ( collec906845016635587262_state
        @ ^ [X: produc7272280185086996969_state] : ( X = A ) )
      = ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) ) ).

% singleton_conv
thf(fact_1155_singleton__conv,axiom,
    ! [A: produc9003781588162623004on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X: produc9003781588162623004on_val] : ( X = A ) )
      = ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) ).

% singleton_conv
thf(fact_1156_singleton__conv,axiom,
    ! [A: state] :
      ( ( collect_state
        @ ^ [X: state] : ( X = A ) )
      = ( insert_state @ A @ bot_bot_set_state ) ) ).

% singleton_conv
thf(fact_1157_these__empty,axiom,
    ( ( these_state @ bot_bo710180891245420500_state )
    = bot_bot_set_state ) ).

% these_empty
thf(fact_1158_singleton__insert__inj__eq_H,axiom,
    ! [A: state,A5: set_state,B3: state] :
      ( ( ( insert_state @ A @ A5 )
        = ( insert_state @ B3 @ bot_bot_set_state ) )
      = ( ( A = B3 )
        & ( ord_le2494988322063910608_state @ A5 @ ( insert_state @ B3 @ bot_bot_set_state ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1159_singleton__insert__inj__eq,axiom,
    ! [B3: state,A: state,A5: set_state] :
      ( ( ( insert_state @ B3 @ bot_bot_set_state )
        = ( insert_state @ A @ A5 ) )
      = ( ( A = B3 )
        & ( ord_le2494988322063910608_state @ A5 @ ( insert_state @ B3 @ bot_bot_set_state ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1160_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B3: option_state,A7: produc9003781588162623004on_val,B6: option_state,R: set_Pr4305388979797279737on_val,S: set_Pr1688445902015331925_state] :
      ( ( member7415082542495011600_state @ ( produc1288379658824033111_state @ ( produc9082076722414132287_state @ A @ B3 ) @ ( produc9082076722414132287_state @ A7 @ B6 ) ) @ ( lex_pr849482514101377111_state @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1161_in__lex__prod,axiom,
    ! [A: option_state,B3: produc9003781588162623004on_val,A7: option_state,B6: produc9003781588162623004on_val,R: set_Pr1688445902015331925_state,S: set_Pr4305388979797279737on_val] :
      ( ( member8716524848806982416on_val @ ( produc3994388588539396951on_val @ ( produc6512732258356509631on_val @ A @ B3 ) @ ( produc6512732258356509631on_val @ A7 @ B6 ) ) @ ( lex_pr7503510086898530263on_val @ R @ S ) )
      = ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1162_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,A7: produc9003781588162623004on_val,B6: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A @ B3 ) @ ( produc4958730763592889809on_val @ A7 @ B6 ) ) @ ( lex_pr8992908938815505897on_val @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1163_in__lex__prod,axiom,
    ! [A: option_state,B3: option_state,A7: option_state,B6: option_state,R: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
      ( ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ A @ B3 ) @ ( produc9160152616361873709_state @ A7 @ B6 ) ) @ ( lex_pr8336311929016760645_state @ R @ S ) )
      = ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1164_in__lex__prod,axiom,
    ! [A: prat,B3: state,A7: prat,B6: state,R: set_Pr3298805672376727137t_prat,S: set_Pr1785066336555260981_state] :
      ( ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ A @ B3 ) @ ( produc6205719116230455971_state @ A7 @ B6 ) ) @ ( lex_prod_prat_state @ R @ S ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member753036827967488894_state @ ( produc544653723139640077_state @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1165_in__lex__prod,axiom,
    ! [A: prat,B3: set_state,A7: prat,B6: set_state,R: set_Pr3298805672376727137t_prat,S: set_Pr5618112136499017889_state] :
      ( ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ A @ B3 ) @ ( produc9080284787853255769_state @ A7 @ B6 ) ) @ ( lex_pr1787351029354341489_state @ R @ S ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member1976142906092959210_state @ ( produc1834047914073684089_state @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1166_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,A7: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr4698884584343881589on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A @ B3 ) @ ( produc74620467203698198on_val @ A7 @ B6 ) ) @ ( lex_pr8780192377291449342on_val @ R @ S ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A7 ) @ R )
        | ( ( A = A7 )
          & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B3 @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1167_range__constant,axiom,
    ! [X2: state] :
      ( ( image_state_state
        @ ^ [Uu3: state] : X2
        @ top_top_set_state )
      = ( insert_state @ X2 @ bot_bot_set_state ) ) ).

% range_constant
thf(fact_1168_Un__empty__left,axiom,
    ! [B2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ bot_bo6430881413863817907_state @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_1169_Un__empty__left,axiom,
    ! [B2: set_state] :
      ( ( sup_sup_set_state @ bot_bot_set_state @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_1170_Un__empty__right,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A5 @ bot_bo6430881413863817907_state )
      = A5 ) ).

% Un_empty_right
thf(fact_1171_Un__empty__right,axiom,
    ! [A5: set_state] :
      ( ( sup_sup_set_state @ A5 @ bot_bot_set_state )
      = A5 ) ).

% Un_empty_right
thf(fact_1172_Collect__conv__if,axiom,
    ! [P: produc7272280185086996969_state > $o,A: produc7272280185086996969_state] :
      ( ( ( P @ A )
       => ( ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) ) )
      & ( ~ ( P @ A )
       => ( ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo6430881413863817907_state ) ) ) ).

% Collect_conv_if
thf(fact_1173_Collect__conv__if,axiom,
    ! [P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( ( P @ A )
       => ( ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) )
      & ( ~ ( P @ A )
       => ( ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_conv_if
thf(fact_1174_Collect__conv__if,axiom,
    ! [P: state > $o,A: state] :
      ( ( ( P @ A )
       => ( ( collect_state
            @ ^ [X: state] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_state @ A @ bot_bot_set_state ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_state
            @ ^ [X: state] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_state ) ) ) ).

% Collect_conv_if
thf(fact_1175_Collect__conv__if2,axiom,
    ! [P: produc7272280185086996969_state > $o,A: produc7272280185086996969_state] :
      ( ( ( P @ A )
       => ( ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) ) )
      & ( ~ ( P @ A )
       => ( ( collec906845016635587262_state
            @ ^ [X: produc7272280185086996969_state] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo6430881413863817907_state ) ) ) ).

% Collect_conv_if2
thf(fact_1176_Collect__conv__if2,axiom,
    ! [P: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( ( P @ A )
       => ( ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) ) )
      & ( ~ ( P @ A )
       => ( ( collec7475294502966882545on_val
            @ ^ [X: produc9003781588162623004on_val] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_conv_if2
thf(fact_1177_Collect__conv__if2,axiom,
    ! [P: state > $o,A: state] :
      ( ( ( P @ A )
       => ( ( collect_state
            @ ^ [X: state] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_state @ A @ bot_bot_set_state ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_state
            @ ^ [X: state] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_state ) ) ) ).

% Collect_conv_if2
thf(fact_1178_empty__def,axiom,
    ( bot_bo6430881413863817907_state
    = ( collec906845016635587262_state
      @ ^ [X: produc7272280185086996969_state] : $false ) ) ).

% empty_def
thf(fact_1179_empty__def,axiom,
    ( bot_bo6722054719108133734on_val
    = ( collec7475294502966882545on_val
      @ ^ [X: produc9003781588162623004on_val] : $false ) ) ).

% empty_def
thf(fact_1180_empty__def,axiom,
    ( bot_bot_set_state
    = ( collect_state
      @ ^ [X: state] : $false ) ) ).

% empty_def
thf(fact_1181_bot_Oextremum,axiom,
    ! [A: set_state] : ( ord_le2494988322063910608_state @ bot_bot_set_state @ A ) ).

% bot.extremum
thf(fact_1182_bot_Oextremum__unique,axiom,
    ! [A: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ bot_bot_set_state )
      = ( A = bot_bot_set_state ) ) ).

% bot.extremum_unique
thf(fact_1183_bot_Oextremum__uniqueI,axiom,
    ! [A: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ bot_bot_set_state )
     => ( A = bot_bot_set_state ) ) ).

% bot.extremum_uniqueI
thf(fact_1184_ex__in__conv,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( ? [X: produc7272280185086996969_state] : ( member7461423308196221312_state @ X @ A5 ) )
      = ( A5 != bot_bo6430881413863817907_state ) ) ).

% ex_in_conv
thf(fact_1185_ex__in__conv,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( ? [X: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X @ A5 ) )
      = ( A5 != bot_bo6722054719108133734on_val ) ) ).

% ex_in_conv
thf(fact_1186_ex__in__conv,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ( ? [X: produc3882902684156697774_state] : ( member6715010751053357271_state @ X @ A5 ) )
      = ( A5 != bot_bo3524941057552913786_state ) ) ).

% ex_in_conv
thf(fact_1187_ex__in__conv,axiom,
    ! [A5: set_state] :
      ( ( ? [X: state] : ( member_state @ X @ A5 ) )
      = ( A5 != bot_bot_set_state ) ) ).

% ex_in_conv
thf(fact_1188_equals0I,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ! [Y4: produc7272280185086996969_state] :
          ~ ( member7461423308196221312_state @ Y4 @ A5 )
     => ( A5 = bot_bo6430881413863817907_state ) ) ).

% equals0I
thf(fact_1189_equals0I,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ! [Y4: produc9003781588162623004on_val] :
          ~ ( member2879030624002546611on_val @ Y4 @ A5 )
     => ( A5 = bot_bo6722054719108133734on_val ) ) ).

% equals0I
thf(fact_1190_equals0I,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ! [Y4: produc3882902684156697774_state] :
          ~ ( member6715010751053357271_state @ Y4 @ A5 )
     => ( A5 = bot_bo3524941057552913786_state ) ) ).

% equals0I
thf(fact_1191_equals0I,axiom,
    ! [A5: set_state] :
      ( ! [Y4: state] :
          ~ ( member_state @ Y4 @ A5 )
     => ( A5 = bot_bot_set_state ) ) ).

% equals0I
thf(fact_1192_equals0D,axiom,
    ! [A5: set_Pr4231240917283394591_state,A: produc7272280185086996969_state] :
      ( ( A5 = bot_bo6430881413863817907_state )
     => ~ ( member7461423308196221312_state @ A @ A5 ) ) ).

% equals0D
thf(fact_1193_equals0D,axiom,
    ! [A5: set_Pr7649655576376376530on_val,A: produc9003781588162623004on_val] :
      ( ( A5 = bot_bo6722054719108133734on_val )
     => ~ ( member2879030624002546611on_val @ A @ A5 ) ) ).

% equals0D
thf(fact_1194_equals0D,axiom,
    ! [A5: set_Pr2391701600042252046_state,A: produc3882902684156697774_state] :
      ( ( A5 = bot_bo3524941057552913786_state )
     => ~ ( member6715010751053357271_state @ A @ A5 ) ) ).

% equals0D
thf(fact_1195_equals0D,axiom,
    ! [A5: set_state,A: state] :
      ( ( A5 = bot_bot_set_state )
     => ~ ( member_state @ A @ A5 ) ) ).

% equals0D
thf(fact_1196_emptyE,axiom,
    ! [A: produc7272280185086996969_state] :
      ~ ( member7461423308196221312_state @ A @ bot_bo6430881413863817907_state ) ).

% emptyE
thf(fact_1197_emptyE,axiom,
    ! [A: produc9003781588162623004on_val] :
      ~ ( member2879030624002546611on_val @ A @ bot_bo6722054719108133734on_val ) ).

% emptyE
thf(fact_1198_emptyE,axiom,
    ! [A: produc3882902684156697774_state] :
      ~ ( member6715010751053357271_state @ A @ bot_bo3524941057552913786_state ) ).

% emptyE
thf(fact_1199_emptyE,axiom,
    ! [A: state] :
      ~ ( member_state @ A @ bot_bot_set_state ) ).

% emptyE
thf(fact_1200_singletonD,axiom,
    ! [B3: produc7272280185086996969_state,A: produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ B3 @ ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_1201_singletonD,axiom,
    ! [B3: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ B3 @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_1202_singletonD,axiom,
    ! [B3: produc3882902684156697774_state,A: produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ B3 @ ( insert94804854012721790_state @ A @ bot_bo3524941057552913786_state ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_1203_singletonD,axiom,
    ! [B3: state,A: state] :
      ( ( member_state @ B3 @ ( insert_state @ A @ bot_bot_set_state ) )
     => ( B3 = A ) ) ).

% singletonD
thf(fact_1204_singleton__iff,axiom,
    ! [B3: produc7272280185086996969_state,A: produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ B3 @ ( insert175804457004744729_state @ A @ bot_bo6430881413863817907_state ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_1205_singleton__iff,axiom,
    ! [B3: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ B3 @ ( insert3030518898131756108on_val @ A @ bot_bo6722054719108133734on_val ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_1206_singleton__iff,axiom,
    ! [B3: produc3882902684156697774_state,A: produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ B3 @ ( insert94804854012721790_state @ A @ bot_bo3524941057552913786_state ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_1207_singleton__iff,axiom,
    ! [B3: state,A: state] :
      ( ( member_state @ B3 @ ( insert_state @ A @ bot_bot_set_state ) )
      = ( B3 = A ) ) ).

% singleton_iff
thf(fact_1208_doubleton__eq__iff,axiom,
    ! [A: state,B3: state,C2: state,D: state] :
      ( ( ( insert_state @ A @ ( insert_state @ B3 @ bot_bot_set_state ) )
        = ( insert_state @ C2 @ ( insert_state @ D @ bot_bot_set_state ) ) )
      = ( ( ( A = C2 )
          & ( B3 = D ) )
        | ( ( A = D )
          & ( B3 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1209_insert__not__empty,axiom,
    ! [A: state,A5: set_state] :
      ( ( insert_state @ A @ A5 )
     != bot_bot_set_state ) ).

% insert_not_empty
thf(fact_1210_singleton__inject,axiom,
    ! [A: state,B3: state] :
      ( ( ( insert_state @ A @ bot_bot_set_state )
        = ( insert_state @ B3 @ bot_bot_set_state ) )
     => ( A = B3 ) ) ).

% singleton_inject
thf(fact_1211_subset__emptyI,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ! [X4: produc7272280185086996969_state] :
          ~ ( member7461423308196221312_state @ X4 @ A5 )
     => ( ord_le4032880777312351359_state @ A5 @ bot_bo6430881413863817907_state ) ) ).

% subset_emptyI
thf(fact_1212_subset__emptyI,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ! [X4: produc9003781588162623004on_val] :
          ~ ( member2879030624002546611on_val @ X4 @ A5 )
     => ( ord_le6900622908513426738on_val @ A5 @ bot_bo6722054719108133734on_val ) ) ).

% subset_emptyI
thf(fact_1213_subset__emptyI,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ! [X4: produc3882902684156697774_state] :
          ~ ( member6715010751053357271_state @ X4 @ A5 )
     => ( ord_le6483787752022044334_state @ A5 @ bot_bo3524941057552913786_state ) ) ).

% subset_emptyI
thf(fact_1214_subset__emptyI,axiom,
    ! [A5: set_state] :
      ( ! [X4: state] :
          ~ ( member_state @ X4 @ A5 )
     => ( ord_le2494988322063910608_state @ A5 @ bot_bot_set_state ) ) ).

% subset_emptyI
thf(fact_1215_empty__not__UNIV,axiom,
    bot_bot_set_state != top_top_set_state ).

% empty_not_UNIV
thf(fact_1216_PartialSA_Oempty__set__sum,axiom,
    ! [A5: set_state] :
      ( ( add_set @ bot_bot_set_state @ A5 )
      = bot_bot_set_state ) ).

% PartialSA.empty_set_sum
thf(fact_1217_these__not__empty__eq,axiom,
    ! [B2: set_option_state] :
      ( ( ( these_state @ B2 )
       != bot_bot_set_state )
      = ( ( B2 != bot_bo710180891245420500_state )
        & ( B2
         != ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_not_empty_eq
thf(fact_1218_these__empty__eq,axiom,
    ! [B2: set_option_state] :
      ( ( ( these_state @ B2 )
        = bot_bot_set_state )
      = ( ( B2 = bot_bo710180891245420500_state )
        | ( B2
          = ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_empty_eq
thf(fact_1219_subset__singleton__iff,axiom,
    ! [X7: set_state,A: state] :
      ( ( ord_le2494988322063910608_state @ X7 @ ( insert_state @ A @ bot_bot_set_state ) )
      = ( ( X7 = bot_bot_set_state )
        | ( X7
          = ( insert_state @ A @ bot_bot_set_state ) ) ) ) ).

% subset_singleton_iff
thf(fact_1220_subset__singletonD,axiom,
    ! [A5: set_state,X2: state] :
      ( ( ord_le2494988322063910608_state @ A5 @ ( insert_state @ X2 @ bot_bot_set_state ) )
     => ( ( A5 = bot_bot_set_state )
        | ( A5
          = ( insert_state @ X2 @ bot_bot_set_state ) ) ) ) ).

% subset_singletonD
thf(fact_1221_singleton__Un__iff,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state )
        = ( sup_su3837536599209510219_state @ A5 @ B2 ) )
      = ( ( ( A5 = bot_bo6430881413863817907_state )
          & ( B2
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) ) )
        | ( ( A5
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) )
          & ( B2 = bot_bo6430881413863817907_state ) )
        | ( ( A5
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) )
          & ( B2
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1222_singleton__Un__iff,axiom,
    ! [X2: state,A5: set_state,B2: set_state] :
      ( ( ( insert_state @ X2 @ bot_bot_set_state )
        = ( sup_sup_set_state @ A5 @ B2 ) )
      = ( ( ( A5 = bot_bot_set_state )
          & ( B2
            = ( insert_state @ X2 @ bot_bot_set_state ) ) )
        | ( ( A5
            = ( insert_state @ X2 @ bot_bot_set_state ) )
          & ( B2 = bot_bot_set_state ) )
        | ( ( A5
            = ( insert_state @ X2 @ bot_bot_set_state ) )
          & ( B2
            = ( insert_state @ X2 @ bot_bot_set_state ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1223_Un__singleton__iff,axiom,
    ! [A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,X2: produc7272280185086996969_state] :
      ( ( ( sup_su3837536599209510219_state @ A5 @ B2 )
        = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) )
      = ( ( ( A5 = bot_bo6430881413863817907_state )
          & ( B2
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) ) )
        | ( ( A5
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) )
          & ( B2 = bot_bo6430881413863817907_state ) )
        | ( ( A5
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) )
          & ( B2
            = ( insert175804457004744729_state @ X2 @ bot_bo6430881413863817907_state ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1224_Un__singleton__iff,axiom,
    ! [A5: set_state,B2: set_state,X2: state] :
      ( ( ( sup_sup_set_state @ A5 @ B2 )
        = ( insert_state @ X2 @ bot_bot_set_state ) )
      = ( ( ( A5 = bot_bot_set_state )
          & ( B2
            = ( insert_state @ X2 @ bot_bot_set_state ) ) )
        | ( ( A5
            = ( insert_state @ X2 @ bot_bot_set_state ) )
          & ( B2 = bot_bot_set_state ) )
        | ( ( A5
            = ( insert_state @ X2 @ bot_bot_set_state ) )
          & ( B2
            = ( insert_state @ X2 @ bot_bot_set_state ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1225_insert__is__Un,axiom,
    ( insert175804457004744729_state
    = ( ^ [A2: produc7272280185086996969_state] : ( sup_su3837536599209510219_state @ ( insert175804457004744729_state @ A2 @ bot_bo6430881413863817907_state ) ) ) ) ).

% insert_is_Un
thf(fact_1226_insert__is__Un,axiom,
    ( insert_state
    = ( ^ [A2: state] : ( sup_sup_set_state @ ( insert_state @ A2 @ bot_bot_set_state ) ) ) ) ).

% insert_is_Un
thf(fact_1227_image__constant,axiom,
    ! [X2: state,A5: set_state,C2: state] :
      ( ( member_state @ X2 @ A5 )
     => ( ( image_state_state
          @ ^ [X: state] : C2
          @ A5 )
        = ( insert_state @ C2 @ bot_bot_set_state ) ) ) ).

% image_constant
thf(fact_1228_image__constant,axiom,
    ! [X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,C2: state] :
      ( ( member7461423308196221312_state @ X2 @ A5 )
     => ( ( image_2710133271811360426_state
          @ ^ [X: produc7272280185086996969_state] : C2
          @ A5 )
        = ( insert_state @ C2 @ bot_bot_set_state ) ) ) ).

% image_constant
thf(fact_1229_image__constant,axiom,
    ! [X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,C2: state] :
      ( ( member2879030624002546611on_val @ X2 @ A5 )
     => ( ( image_1205529093420515805_state
          @ ^ [X: produc9003781588162623004on_val] : C2
          @ A5 )
        = ( insert_state @ C2 @ bot_bot_set_state ) ) ) ).

% image_constant
thf(fact_1230_image__constant,axiom,
    ! [X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,C2: state] :
      ( ( member6715010751053357271_state @ X2 @ A5 )
     => ( ( image_6759111139246981411_state
          @ ^ [X: produc3882902684156697774_state] : C2
          @ A5 )
        = ( insert_state @ C2 @ bot_bot_set_state ) ) ) ).

% image_constant
thf(fact_1231_image__constant__conv,axiom,
    ! [A5: set_state,C2: state] :
      ( ( ( A5 = bot_bot_set_state )
       => ( ( image_state_state
            @ ^ [X: state] : C2
            @ A5 )
          = bot_bot_set_state ) )
      & ( ( A5 != bot_bot_set_state )
       => ( ( image_state_state
            @ ^ [X: state] : C2
            @ A5 )
          = ( insert_state @ C2 @ bot_bot_set_state ) ) ) ) ).

% image_constant_conv
thf(fact_1232_PartialSA_Obigger__singleton,axiom,
    ! [Phi2: state,Phi: state] :
      ( ( greater @ Phi2 @ Phi )
     => ( greater_set @ ( insert_state @ Phi2 @ bot_bot_set_state ) @ ( insert_state @ Phi @ bot_bot_set_state ) ) ) ).

% PartialSA.bigger_singleton
thf(fact_1233_PartialSA_Osetify__sum,axiom,
    ! [P: state > $o,A5: set_state,B2: set_state] :
      ( ( sep_setify_state @ P @ ( add_set @ A5 @ B2 ) )
      = ( ! [X: state] :
            ( ( member_state @ X @ A5 )
           => ( sep_setify_state @ P @ ( add_set @ ( insert_state @ X @ bot_bot_set_state ) @ B2 ) ) ) ) ) ).

% PartialSA.setify_sum
thf(fact_1234_range__eq__singletonD,axiom,
    ! [F: state > state,A: state,X2: state] :
      ( ( ( image_state_state @ F @ top_top_set_state )
        = ( insert_state @ A @ bot_bot_set_state ) )
     => ( ( F @ X2 )
        = A ) ) ).

% range_eq_singletonD
thf(fact_1235_PartialSA_Osum__then__singleton,axiom,
    ! [A: state,B3: state,C2: state] :
      ( ( ( some_state @ A )
        = ( plus @ B3 @ C2 ) )
      = ( ( insert_state @ A @ bot_bot_set_state )
        = ( add_set @ ( insert_state @ B3 @ bot_bot_set_state ) @ ( insert_state @ C2 @ bot_bot_set_state ) ) ) ) ).

% PartialSA.sum_then_singleton
thf(fact_1236_PartialSA_Osetify__sum__image,axiom,
    ! [P: state > $o,F: state > state,A5: set_state,B2: set_state] :
      ( ( sep_setify_state @ P @ ( add_set @ ( image_state_state @ F @ A5 ) @ B2 ) )
      = ( ! [X: state] :
            ( ( member_state @ X @ A5 )
           => ( sep_setify_state @ P @ ( add_set @ ( insert_state @ ( F @ X ) @ bot_bot_set_state ) @ B2 ) ) ) ) ) ).

% PartialSA.setify_sum_image
thf(fact_1237_the__elem__eq,axiom,
    ! [X2: state] :
      ( ( the_elem_state @ ( insert_state @ X2 @ bot_bot_set_state ) )
      = X2 ) ).

% the_elem_eq
thf(fact_1238_is__singletonI,axiom,
    ! [X2: state] : ( is_singleton_state @ ( insert_state @ X2 @ bot_bot_set_state ) ) ).

% is_singletonI
thf(fact_1239_sep__algebra_Osetify__sum__image,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( image_state_state @ F @ A5 ) @ B2 ) )
        = ( ! [X: state] :
              ( ( member_state @ X @ A5 )
             => ( sep_setify_state @ P @ ( sep_add_set_state @ Plus @ ( insert_state @ ( F @ X ) @ bot_bot_set_state ) @ B2 ) ) ) ) ) ) ).

% sep_algebra.setify_sum_image
thf(fact_1240_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] : P ) )
          = top_to1561438921946393474on_val ) )
      & ( ~ P
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] : P ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_const_case_prod
thf(fact_1241_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec394671971739345689_state
            @ ( produc3645877548210629341tate_o
              @ ^ [A2: state,B: state > state] : P ) )
          = top_to1744471583079920734_state ) )
      & ( ~ P
       => ( ( collec394671971739345689_state
            @ ( produc3645877548210629341tate_o
              @ ^ [A2: state,B: state > state] : P ) )
          = bot_bo3524941057552913786_state ) ) ) ).

% Collect_const_case_prod
thf(fact_1242_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec906845016635587262_state
            @ ( produc1745419677418529872tate_o
              @ ^ [A2: state,B: produc3882902684156697774_state] : P ) )
          = top_to174962084833059535_state ) )
      & ( ~ P
       => ( ( collec906845016635587262_state
            @ ( produc1745419677418529872tate_o
              @ ^ [A2: state,B: produc3882902684156697774_state] : P ) )
          = bot_bo6430881413863817907_state ) ) ) ).

% Collect_const_case_prod
thf(fact_1243_sep__algebra_Oadd__set__asso,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state,B2: set_state,C3: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ ( sep_add_set_state @ Plus @ A5 @ B2 ) @ C3 )
        = ( sep_add_set_state @ Plus @ A5 @ ( sep_add_set_state @ Plus @ B2 @ C3 ) ) ) ) ).

% sep_algebra.add_set_asso
thf(fact_1244_sep__algebra_Oadd__set__commm,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ A5 @ B2 )
        = ( sep_add_set_state @ Plus @ B2 @ A5 ) ) ) ).

% sep_algebra.add_set_commm
thf(fact_1245_bot__empty__eq2,axiom,
    ( bot_bo932482984042685116tate_o
    = ( ^ [X: state,Y3: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ bot_bo6430881413863817907_state ) ) ) ).

% bot_empty_eq2
thf(fact_1246_bot__empty__eq2,axiom,
    ( bot_bo641312475164433381tate_o
    = ( ^ [X: state,Y3: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ bot_bo3524941057552913786_state ) ) ) ).

% bot_empty_eq2
thf(fact_1247_bot__empty__eq2,axiom,
    ( bot_bo5848341091568474042_val_o
    = ( ^ [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ bot_bo402267660617762917on_val ) ) ) ).

% bot_empty_eq2
thf(fact_1248_bot__empty__eq2,axiom,
    ( bot_bo5101768366538119390tate_o
    = ( ^ [X: option_state,Y3: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ bot_bo1080640394036989633_state ) ) ) ).

% bot_empty_eq2
thf(fact_1249_bot__empty__eq2,axiom,
    ( bot_bot_prat_state_o
    = ( ^ [X: prat,Y3: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ bot_bo1267616847652589879_state ) ) ) ).

% bot_empty_eq2
thf(fact_1250_bot__empty__eq2,axiom,
    ( bot_bo4139562324058396466tate_o
    = ( ^ [X: prat,Y3: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ bot_bo6311991824895087853_state ) ) ) ).

% bot_empty_eq2
thf(fact_1251_bot__empty__eq2,axiom,
    ( bot_bo3626896261903426715_val_o
    = ( ^ [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ bot_bo6722054719108133734on_val ) ) ) ).

% bot_empty_eq2
thf(fact_1252_bot__set__def,axiom,
    ( bot_bo6430881413863817907_state
    = ( collec906845016635587262_state @ bot_bo4916705512949510546tate_o ) ) ).

% bot_set_def
thf(fact_1253_bot__set__def,axiom,
    ( bot_bo6722054719108133734on_val
    = ( collec7475294502966882545on_val @ bot_bo6370234372831922335_val_o ) ) ).

% bot_set_def
thf(fact_1254_bot__set__def,axiom,
    ( bot_bot_set_state
    = ( collect_state @ bot_bot_state_o ) ) ).

% bot_set_def
thf(fact_1255_sep__algebra_Oadd__set_Ocong,axiom,
    sep_add_set_state = sep_add_set_state ).

% sep_algebra.add_set.cong
thf(fact_1256_is__singleton__the__elem,axiom,
    ( is_singleton_state
    = ( ^ [A4: set_state] :
          ( A4
          = ( insert_state @ ( the_elem_state @ A4 ) @ bot_bot_set_state ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1257_is__singletonI_H,axiom,
    ! [A5: set_Pr4231240917283394591_state] :
      ( ( A5 != bot_bo6430881413863817907_state )
     => ( ! [X4: produc7272280185086996969_state,Y4: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X4 @ A5 )
           => ( ( member7461423308196221312_state @ Y4 @ A5 )
             => ( X4 = Y4 ) ) )
       => ( is_sin5735034780885272053_state @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1258_is__singletonI_H,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( A5 != bot_bo6722054719108133734on_val )
     => ( ! [X4: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X4 @ A5 )
           => ( ( member2879030624002546611on_val @ Y4 @ A5 )
             => ( X4 = Y4 ) ) )
       => ( is_sin1022367449962840616on_val @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1259_is__singletonI_H,axiom,
    ! [A5: set_Pr2391701600042252046_state] :
      ( ( A5 != bot_bo3524941057552913786_state )
     => ( ! [X4: produc3882902684156697774_state,Y4: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X4 @ A5 )
           => ( ( member6715010751053357271_state @ Y4 @ A5 )
             => ( X4 = Y4 ) ) )
       => ( is_sin8914544035436372898_state @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1260_is__singletonI_H,axiom,
    ! [A5: set_state] :
      ( ( A5 != bot_bot_set_state )
     => ( ! [X4: state,Y4: state] :
            ( ( member_state @ X4 @ A5 )
           => ( ( member_state @ Y4 @ A5 )
             => ( X4 = Y4 ) ) )
       => ( is_singleton_state @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1261_sep__algebra_Oempty__set__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A5: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ bot_bot_set_state @ A5 )
        = bot_bot_set_state ) ) ).

% sep_algebra.empty_set_sum
thf(fact_1262_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Phi: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ Phi @ ( sep_ad3908482799505143387_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc7272280185086996969_state,B: produc7272280185086996969_state] :
              ( ( ( some_P6297086604321900740_state @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( member7461423308196221312_state @ A2 @ A5 )
              & ( member7461423308196221312_state @ B @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_1263_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Phi: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ Phi @ ( sep_ad8320153009520438926on_val @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
              ( ( ( some_P2581650809551911671on_val @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( member2879030624002546611on_val @ A2 @ A5 )
              & ( member2879030624002546611on_val @ B @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_1264_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Phi: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ Phi @ ( sep_ad8203089552910907580_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc3882902684156697774_state,B: produc3882902684156697774_state] :
              ( ( ( some_P5383513176685068947_state @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( member6715010751053357271_state @ A2 @ A5 )
              & ( member6715010751053357271_state @ B @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_1265_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Phi: state,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ Phi @ ( sep_add_set_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: state,B: state] :
              ( ( ( some_state @ Phi )
                = ( Plus @ A2 @ B ) )
              & ( member_state @ A2 @ A5 )
              & ( member_state @ B @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_1266_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,X2: produc7272280185086996969_state,A5: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ X2 @ ( sep_ad3908482799505143387_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc7272280185086996969_state,B: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ A2 @ A5 )
              & ( member7461423308196221312_state @ B @ B2 )
              & ( ( some_P6297086604321900740_state @ X2 )
                = ( Plus @ A2 @ B ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_1267_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,X2: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ X2 @ ( sep_ad8320153009520438926on_val @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ A2 @ A5 )
              & ( member2879030624002546611on_val @ B @ B2 )
              & ( ( some_P2581650809551911671on_val @ X2 )
                = ( Plus @ A2 @ B ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_1268_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,X2: produc3882902684156697774_state,A5: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ X2 @ ( sep_ad8203089552910907580_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: produc3882902684156697774_state,B: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ A2 @ A5 )
              & ( member6715010751053357271_state @ B @ B2 )
              & ( ( some_P5383513176685068947_state @ X2 )
                = ( Plus @ A2 @ B ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_1269_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X2: state,A5: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X2 @ ( sep_add_set_state @ Plus @ A5 @ B2 ) )
        = ( ? [A2: state,B: state] :
              ( ( member_state @ A2 @ A5 )
              & ( member_state @ B @ B2 )
              & ( ( some_state @ X2 )
                = ( Plus @ A2 @ B ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_1270_add__set__def,axiom,
    ( add_set
    = ( sep_add_set_state @ plus ) ) ).

% add_set_def
thf(fact_1271_cwand__def,axiom,
    ( cwand
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A2: state,X: state] :
                  ( ( ( member_state @ A2 @ A4 )
                    & ( ( some_state @ X )
                      = ( plus @ ( r @ A2 @ W2 ) @ A2 ) ) )
                 => ( member_state @ X @ B5 ) ) ) ) ) ) ).

% cwand_def
thf(fact_1272_greater__set__def,axiom,
    ( greater_set
    = ( sep_gr7105985528888466643_state @ plus ) ) ).

% greater_set_def
thf(fact_1273_wand__def,axiom,
    ( wand
    = ( ^ [A4: set_state,B5: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A2: state,X: state] :
                  ( ( ( member_state @ A2 @ A4 )
                    & ( ( some_state @ X )
                      = ( plus @ W2 @ A2 ) ) )
                 => ( member_state @ X @ B5 ) ) ) ) ) ) ).

% wand_def
thf(fact_1274_type__definition__state,axiom,
    ( type_d7385716227342696708on_val @ rep_state @ abs_state
    @ ( collec7475294502966882545on_val
      @ ^ [Uu3: produc9003781588162623004on_val] :
        ? [Phi4: produc9003781588162623004on_val] :
          ( ( Uu3 = Phi4 )
          & ( valid_state @ Phi4 ) ) ) ) ).

% type_definition_state

% Helper facts (5)
thf(help_If_2_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X2: state,Y: state] :
      ( ( if_state @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X2: state,Y: state] :
      ( ( if_state @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( if_option_state @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X2: option_state,Y: option_state] :
      ( ( if_option_state @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    defined @ ( r @ a2 @ wb ) @ a2 ).

%------------------------------------------------------------------------------