TPTP Problem File: SLH0060^1.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_00666_032821__8490772_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1902 ( 483 unt; 621 typ;   0 def)
%            Number of atoms       : 5056 (1863 equ;   0 cnn)
%            Maximal formula atoms :   23 (   3 avg)
%            Number of connectives : 17733 ( 424   ~;  54   |; 839   &;14165   @)
%                                         (   0 <=>;2251  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   8 avg)
%            Number of types       :   98 (  97 usr)
%            Number of type conns  : 3896 (3896   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  527 ( 524 usr;  26 con; 0-11 aty)
%            Number of variables   : 5949 ( 690   ^;4723   !; 536   ?;5949   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:09:26.926
%------------------------------------------------------------------------------
% Could-be-implicit typings (97)
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__PartialHeapSA__Ostate_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr4913868530397957098_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > option_state ) > ( produc3882902684156697774_state > option_state ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    bNF_Gr5413576055295359779_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > state ) > ( produc3882902684156697774_state > state > state ) > set_Pr2391701600042252046_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    bNF_Gr18303635036277918_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > state ) > ( produc3882902684156697774_state > produc3882902684156697774_state ) > set_Pr4231240917283394591_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
    bNF_Gr5432878666621363808_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > prat ) > ( produc3882902684156697774_state > state ) > set_Pr6039890582625207499_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr6818451474134737942_state: set_Pr2391701600042252046_state > ( produc3882902684156697774_state > prat ) > ( produc3882902684156697774_state > set_state ) > set_Pr6847263425858009217_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_001t__PosRat__Oprat_001t__PartialHeapSA__Ostate,type,
    bNF_Gr2250442185837678927_state: set_Pr4231240917283394591_state > ( produc7272280185086996969_state > prat ) > ( produc7272280185086996969_state > state ) > set_Pr6039890582625207499_state ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_001t__PosRat__Oprat_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr7237560867856044805_state: set_Pr4231240917283394591_state > ( produc7272280185086996969_state > prat ) > ( produc7272280185086996969_state > set_state ) > set_Pr6847263425858009217_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr63754628658600314_state: set_Pr1688445902015331925_state > ( option_state > option_state ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_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,
    bNF_Gr4325946583917791628on_val: set_Pr1688445902015331925_state > ( option_state > produc9003781588162623004on_val ) > set_Pr4305388979797279737on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    bNF_Gr6805951496242791002_state: set_Pr1785066336555260981_state > ( state > state ) > set_Pr1785066336555260981_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_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__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr6895291047975414284_state: set_Pr4305388979797279737on_val > ( produc9003781588162623004on_val > option_state ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_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__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,
    bNF_Gr4118820077530253598on_val: set_Pr4305388979797279737on_val > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > set_Pr4305388979797279737on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr621135371174781015_state: set_option_state > set_Pr1688445902015331925_state > ( option_state > option_state ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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,
    bNF_Gr1795481779025939945on_val: set_option_state > set_Pr4305388979797279737on_val > ( option_state > produc9003781588162623004on_val ) > set_Pr1688445902015331925_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__PartialHeapSA__Ostate_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr8759033791985395207_state: set_state > set_Pr1688445902015331925_state > ( state > option_state ) > set_Pr1785066336555260981_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    bNF_Gr4114592414835599415_state: set_state > set_Pr1785066336555260981_state > ( state > state ) > set_Pr1785066336555260981_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__PartialHeapSA__Ostate_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,
    bNF_Gr4428767212045630361on_val: set_state > set_Pr4305388979797279737on_val > ( state > produc9003781588162623004on_val ) > set_Pr1785066336555260981_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr4364826243083562601_state: set_Pr7649655576376376530on_val > set_Pr1688445902015331925_state > ( produc9003781588162623004on_val > option_state ) > set_Pr4305388979797279737on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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__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,
    bNF_Gr6662968685559237371on_val: set_Pr7649655576376376530on_val > set_Pr4305388979797279737on_val > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > set_Pr4305388979797279737on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr919661598790814775_state: set_Pr2391701600042252046_state > set_Pr1688445902015331925_state > ( produc3882902684156697774_state > option_state ) > set_Pr4846723933414538759_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_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,
    bNF_Gr1139690717726542281on_val: set_Pr2391701600042252046_state > set_Pr4305388979797279737on_val > ( produc3882902684156697774_state > produc9003781588162623004on_val ) > set_Pr4846723933414538759_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    bNF_Gr5630821380644201398_state: set_Pr4231240917283394591_state > set_Pr1688445902015331925_state > ( produc7272280185086996969_state > option_state ) > set_Pr7975217055296730899_state ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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_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,
    bNF_Gr7039482972660942280on_val: set_Pr4231240917283394591_state > set_Pr4305388979797279737on_val > ( produc7272280185086996969_state > produc9003781588162623004on_val ) > set_Pr7975217055296730899_state ).

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__Mask__Oval,type,
    id_val: val > val ).

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__Mask__Oval,type,
    fun_THE_default_val: val > ( val > $o ) > 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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    packag6843733491101478048on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > $o ) > produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr1565700067222966883on_val > ( produc9003781588162623004on_val > $o ) > aasser1163550960519286933on_val > produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr1565700067222966883on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag4061699384143886122_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > ( produc3882902684156697774_state > produc3882902684156697774_state ) > ( produc3882902684156697774_state > $o ) > produc3882902684156697774_state > produc3882902684156697774_state > set_Pr5308428015838290133_state > ( produc3882902684156697774_state > $o ) > aasser3179708585289803787_state > produc3882902684156697774_state > produc3882902684156697774_state > set_Pr5308428015838290133_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs_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,
    packag3630742828623380589_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > ( produc7272280185086996969_state > produc7272280185086996969_state ) > ( produc7272280185086996969_state > $o ) > produc7272280185086996969_state > produc7272280185086996969_state > set_Pr3289333603297600445_state > ( produc7272280185086996969_state > $o ) > aasser7019605857572776418_state > produc7272280185086996969_state > produc7272280185086996969_state > set_Pr3289333603297600445_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs__connection_001t__Mask__Oval,type,
    packag1944724486178758143on_val: ( val > val > option_val ) > ( val > val ) > ( val > $o ) > val > val > set_Pr4723104092121316667al_val > ( val > $o ) > aassertion_val > val > val > set_Pr4723104092121316667al_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__rhs__connection_001t__PartialHeapSA__Ostate,type,
    packag2054414653763826108_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > state > state > set_Pr4231240917283394591_state > ( state > $o ) > aassertion_state > state > state > set_Pr4231240917283394591_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__sat_001t__Mask__Oval,type,
    packag5057539306968662900at_val: ( val > val > option_val ) > ( val > val ) > ( val > $o ) > aassertion_val > val > val > val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opackage__sat_001t__PartialHeapSA__Ostate,type,
    packag8070727096299822599_state: ( state > state > option_state ) > ( state > state ) > ( state > $o ) > aassertion_state > state > state > state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_001t__Mask__Oval_001t__Mask__Oval_001_062_It__Mask__Oval_Mt__Mask__Oval_J,type,
    packag3202507263771882441al_val: ( val > val > option_val ) > set_Pr4723104092121316667al_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    packag4152378406607043437_state: ( state > state > option_state ) > set_Pr4231240917283394591_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag9013619817833129428_state: ( state > state > option_state ) > set_Pr523392378426063460_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_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__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_001_062_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,
    packag161099568566171125on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > set_Pr1565700067222966883on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J_001_062_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,
    packag1386629450763126729_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > set_Pr5308428015838290133_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Opure__remains_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_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_001_062_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_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,
    packag3875795242785447337_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > set_Pr3289333603297600445_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__Mask__Oval,type,
    package_sat_val: ( val > val > option_val ) > aassertion_val > val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__PartialHeapSA__Ostate,type,
    package_sat_state: ( state > state > option_state ) > aassertion_state > state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_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,
    packag3677137384183703351on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > aasser1163550960519286933on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag2518615827549365843_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > aasser3179708585289803787_state > produc3882902684156697774_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat_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,
    packag4865960244449834244_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > aasser7019605857572776418_state > produc7272280185086996969_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat__rel_001t__Mask__Oval,type,
    package_sat_rel_val: produc1707155752992360906al_val > produc1707155752992360906al_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Osat__rel_001t__PartialHeapSA__Ostate,type,
    packag1918864394588125018_state: produc2756261550280135502_state > produc2756261550280135502_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Ovalid__package__set_001t__Mask__Oval,type,
    packag7517340229427718612et_val: ( val > val > option_val ) > ( val > val ) > val > set_Pr4723104092121316667al_val > val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Ovalid__package__set_001t__PartialHeapSA__Ostate,type,
    packag3877571396724326311_state: ( state > state > option_state ) > ( state > state ) > state > set_Pr4231240917283394591_state > state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Mask__Oval,type,
    packag5732365195153136185on_val: ( val > val > option_val ) > ( val > val ) > aassertion_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__PartialHeapSA__Ostate,type,
    packag7298478212905792770_state: ( state > state > option_state ) > ( state > state ) > aassertion_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_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,
    packag5541956234368469604on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > aasser1163550960519286933on_val > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_001t__Product____Type__Oprod_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J_J,type,
    packag8517968027831007846_state: ( produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state ) > ( produc3882902684156697774_state > produc3882902684156697774_state ) > aasser3179708585289803787_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion_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,
    packag6254535061403973169_state: ( produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state ) > ( produc7272280185086996969_state > produc7272280185086996969_state ) > aasser7019605857572776418_state > $o ).

thf(sy_c_PackageLogic_Opackage__logic_Owf__assertion__rel_001t__PartialHeapSA__Ostate,type,
    packag4834522500328176237_state: aassertion_state > aassertion_state > $o ).

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

thf(sy_c_PartialHeapSA_Oadd__states,type,
    add_states: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Oadd__states__rel,type,
    add_states_rel: produc1317171644805857433on_val > produc1317171644805857433on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible,type,
    compatible: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__heaps,type,
    compatible_heaps: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Mask__Oval,type,
    compat7209469420100717861ns_val: option_val > option_val > $o ).

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

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Mask__Oval,type,
    compat8294034345482142370el_val: produc5122021904355189735on_val > produc5122021904355189735on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__PartialHeapSA__Ostate,type,
    compat1844074344818064729_state: produc3142500478612311029_state > produc3142500478612311029_state > $o ).

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

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

thf(sy_c_PartialHeapSA_Oempty__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
    empty_7512026568161686585ar_val: produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__h,type,
    get_h: state > produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__m,type,
    get_m: state > produc1457211279475724562t_char > prat ).

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

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

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

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

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

thf(sy_c_PartialHeapSA_Ostate_OAbs__state,type,
    abs_state: produc9003781588162623004on_val > state ).

thf(sy_c_PartialHeapSA_Ostate_ORep__state,type,
    rep_state: state > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Ounit,type,
    unit: state ).

thf(sy_c_PartialHeapSA_Ovalid__heap,type,
    valid_heap: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state,type,
    valid_state: produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state__rel,type,
    valid_state_rel: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PosRat_Opadd,type,
    padd: prat > prat > prat ).

thf(sy_c_PosRat_Opgte,type,
    pgte: prat > prat > $o ).

thf(sy_c_PosRat_Opmult,type,
    pmult: prat > prat > prat ).

thf(sy_c_PosRat_Oppos,type,
    ppos: prat > $o ).

thf(sy_c_PosRat_Opwrite,type,
    pwrite: prat ).

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,
    produc5362444129537099455_state: ( state > state ) > ( state > state ) > produc948624807956558599_state ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    produc8008623296994167373on_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > produc1757238238866670357on_val ).

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__Option__Ooption_It__Mask__Oval_J_J,type,
    produc74620467203698198on_val: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ).

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,
    produc1164665353665699927r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc3336119888008106215r_prat ).

thf(sy_c_Product__Type_OPair_001t__Mask__Oval_001_062_It__Mask__Oval_Mt__Mask__Oval_J,type,
    produc303288909735261391al_val: val > ( val > val ) > produc5503543764249810261al_val ).

thf(sy_c_Product__Type_OPair_001t__Mask__Oval_001t__Product____Type__Oprod_It__Mask__Oval_M_062_It__Mask__Oval_Mt__Mask__Oval_J_J,type,
    produc7187740677493149823al_val: val > produc5503543764249810261al_val > produc2072440392815185925al_val ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    produc904852209897547607on_val: option_val > option_val > produc5122021904355189735on_val ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    produc9160152616361873709_state: option_state > option_state > produc3142500478612311029_state ).

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,
    produc6512732258356509631on_val: option_state > produc9003781588162623004on_val > produc5811935988241853191on_val ).

thf(sy_c_Product__Type_OPair_001t__PackageLogic__Oaassertion_It__Mask__Oval_J_001t__Mask__Oval,type,
    produc1969469836606013754al_val: aassertion_val > val > produc1707155752992360906al_val ).

thf(sy_c_Product__Type_OPair_001t__PackageLogic__Oaassertion_It__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate,type,
    produc6654088501289988998_state: aassertion_state > state > produc2756261550280135502_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001_062_It__PartialHeapSA__Ostate_Mt__PartialHeapSA__Ostate_J,type,
    produc8632743443909064678_state: state > ( state > state ) > produc3882902684156697774_state ).

thf(sy_c_Product__Type_OPair_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
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    member_option_val: option_val > set_option_val > $o ).

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    member_set_state: set_state > set_set_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_bb____,type,
    bb: state ).

thf(sy_v_p____,type,
    p: prat ).

thf(sy_v_thesis____,type,
    thesis: $o ).

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 (1274)
thf(fact_0_calculation_I5_J,axiom,
    ( ( some_state @ ba )
    = ( plus @ ( r @ a2 @ wa ) @ a2 ) ) ).

% calculation(5)
thf(fact_1__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_2__092_060open_062_092_060not_062_Ascalable_Awa_Aa_092_060close_062,axiom,
    ~ ( scalable @ wa @ a2 ) ).

% \<open>\<not> scalable wa a\<close>
thf(fact_3_calculation_I7_J,axiom,
    ( ( some_state @ bb )
    = ( plus @ ( r @ a2 @ wb ) @ a2 ) ) ).

% calculation(7)
thf(fact_4__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_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062bb_O_ASome_Abb_A_061_AR_Aa_Awb_A_092_060oplus_062_Aa_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Bb: state] :
        ( ( some_state @ Bb )
       != ( plus @ ( r @ a2 @ wb ) @ a2 ) ) ).

% \<open>\<And>thesis. (\<And>bb. Some bb = R a wb \<oplus> a \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_option_Oinject,axiom,
    ! [X2: state,Y2: state] :
      ( ( ( some_state @ X2 )
        = ( some_state @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_7_option_Oinject,axiom,
    ! [X2: val,Y2: val] :
      ( ( ( some_val @ X2 )
        = ( some_val @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_8_asso1,axiom,
    ! [A: state,B: state,Ab: state,C: state,Bc: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C )
          = ( some_state @ Bc ) ) )
     => ( ( plus @ Ab @ C )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_9_positivity,axiom,
    ! [A: state,B: state,C: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ C ) )
     => ( ( ( some_state @ C )
          = ( plus @ C @ C ) )
       => ( ( some_state @ A )
          = ( plus @ A @ A ) ) ) ) ).

% positivity
thf(fact_10_asm1,axiom,
    ( ( member_state @ a2 @ a )
    & ( ( some_state @ x )
      = ( plus @ ( r @ a2 @ w ) @ a2 ) ) ) ).

% asm1
thf(fact_11_wandI,axiom,
    ! [A2: set_state,W: state,B2: set_state] :
      ( ! [A3: state,X: state] :
          ( ( ( member_state @ A3 @ A2 )
            & ( ( some_state @ X )
              = ( plus @ A3 @ W ) ) )
         => ( member_state @ X @ B2 ) )
     => ( member_state @ W @ ( wand @ A2 @ B2 ) ) ) ).

% wandI
thf(fact_12_in__cwand,axiom,
    ! [A2: set_state,W: state,B2: set_state] :
      ( ! [A3: state,X: state] :
          ( ( ( member_state @ A3 @ A2 )
            & ( ( some_state @ X )
              = ( plus @ ( r @ A3 @ W ) @ A3 ) ) )
         => ( member_state @ X @ B2 ) )
     => ( member_state @ W @ ( cwand @ A2 @ B2 ) ) ) ).

% in_cwand
thf(fact_13_r_I1_J,axiom,
    ! [A: state,X3: state] :
      ( ( ( member_state @ A @ a )
        & ( ( some_state @ X3 )
          = ( plus @ ( r @ A @ wa ) @ A ) ) )
     => ( member_state @ X3 @ b ) ) ).

% r(1)
thf(fact_14_commutative,axiom,
    ( plus
    = ( ^ [A4: state,B3: state] : ( plus @ B3 @ A4 ) ) ) ).

% commutative
thf(fact_15__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_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_assms_I1_J,axiom,
    combinable @ b ).

% assms(1)
thf(fact_18_assms_I2_J,axiom,
    intuitionistic @ b ).

% assms(2)
thf(fact_19__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_20__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_21__092_060open_062_092_060not_062_Ascalable_Awb_Aa_092_060close_062,axiom,
    ~ ( scalable @ wb @ a2 ) ).

% \<open>\<not> scalable wb a\<close>
thf(fact_22__092_060open_062_092_060not_062_Ascalable_Aw_Aa_092_060close_062,axiom,
    ~ ( scalable @ w @ a2 ) ).

% \<open>\<not> scalable w a\<close>
thf(fact_23__092_060open_062R_Aa_Awb_A_124_D_124_Aa_092_060close_062,axiom,
    defined @ ( r @ a2 @ wb ) @ a2 ).

% \<open>R a wb |#| a\<close>
thf(fact_24_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_25_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_26_r_I2_J,axiom,
    ! [A: state,X3: state] :
      ( ( ( member_state @ A @ a )
        & ( ( some_state @ X3 )
          = ( plus @ ( r @ A @ wb ) @ A ) ) )
     => ( member_state @ X3 @ b ) ) ).

% r(2)
thf(fact_27__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,
    ~ ( ! [A5: state,X4: state] :
          ( ( ( member_state @ A5 @ a )
            & ( ( some_state @ X4 )
              = ( plus @ ( r @ A5 @ wa ) @ A5 ) ) )
         => ( member_state @ X4 @ b ) )
     => ~ ! [A5: state,X4: state] :
            ( ( ( member_state @ A5 @ a )
              & ( ( some_state @ X4 )
                = ( plus @ ( r @ A5 @ wb ) @ A5 ) ) )
           => ( member_state @ X4 @ 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_28_state__ext,axiom,
    ! [A: state,B: state] :
      ( ( ( get_h @ A )
        = ( get_h @ B ) )
     => ( ( ( get_m @ A )
          = ( get_m @ B ) )
       => ( A = B ) ) ) ).

% state_ext
thf(fact_29_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_30_PartialSA_Odefined__sum__move,axiom,
    ! [A: state,B: state,X3: state,Y: state,A6: state] :
      ( ( defined @ A @ B )
     => ( ( ( some_state @ B )
          = ( plus @ X3 @ Y ) )
       => ( ( ( some_state @ A6 )
            = ( plus @ A @ X3 ) )
         => ( defined @ A6 @ Y ) ) ) ) ).

% PartialSA.defined_sum_move
thf(fact_31_R__compatible__same,axiom,
    ! [A: state,W: state] :
      ( ( defined @ A @ W )
     => ( ( r @ A @ W )
        = W ) ) ).

% R_compatible_same
thf(fact_32_calculation_I2_J,axiom,
    defined @ a2 @ ( multiply @ p @ w ) ).

% calculation(2)
thf(fact_33__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_34__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_35_scalable__def,axiom,
    ( scalable
    = ( ^ [W2: state,A4: state] :
        ! [X5: state] :
          ( ( member_state @ X5 @ ( scaled @ W2 ) )
         => ~ ( defined @ A4 @ X5 ) ) ) ) ).

% scalable_def
thf(fact_36__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_37_asm,axiom,
    member_state @ w @ ( add_set @ ( multip8064567061438756306ertion @ alpha @ ( cwand @ a @ b ) ) @ ( multip8064567061438756306ertion @ beta @ ( cwand @ a @ b ) ) ) ).

% asm
thf(fact_38_R__def,axiom,
    ( r
    = ( ^ [A4: state,W2: state] : ( if_state @ ( scalable @ W2 @ A4 ) @ W2 @ ( abs_state @ ( produc74620467203698198on_val @ ( comp_min_mask @ ( get_m @ A4 ) @ ( get_m @ W2 ) ) @ ( get_h @ W2 ) ) ) ) ) ) ).

% R_def
thf(fact_39__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_40_plus__charact_I1_J,axiom,
    ! [A: state,B: state,X3: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X3 ) )
     => ( ( get_m @ X3 )
        = ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) ) ) ).

% plus_charact(1)
thf(fact_41_plus__charact_I2_J,axiom,
    ! [A: state,B: state,X3: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X3 ) )
     => ( ( get_h @ X3 )
        = ( map_ad2908576761495157028ar_val @ ( get_h @ A ) @ ( get_h @ B ) ) ) ) ).

% plus_charact(2)
thf(fact_42__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_43__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_44_valid__state_Ocases,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( X3
         != ( produc74620467203698198on_val @ Pi @ H ) ) ).

% valid_state.cases
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,
    ! [A2: set_Pr2391701600042252046_state] :
      ( ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A2: set_Pr4231240917283394591_state] :
      ( ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A2: set_state] :
      ( ( collect_state
        @ ^ [X5: state] : ( member_state @ X5 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A2: set_Pr7649655576376376530on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ A2 ) )
      = A2 ) ).

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

% Collect_cong
thf(fact_54_Collect__cong,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ! [X: state] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_state @ P )
        = ( collect_state @ Q ) ) ) ).

% Collect_cong
thf(fact_55_Collect__cong,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X: produc9003781588162623004on_val] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collec7475294502966882545on_val @ P )
        = ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_cong
thf(fact_56_PartialSA_Osucc__refl,axiom,
    ! [A: state] : ( greater @ A @ A ) ).

% PartialSA.succ_refl
thf(fact_57_PartialSA_Osucc__trans,axiom,
    ! [A: state,B: state,C: state] :
      ( ( greater @ A @ B )
     => ( ( greater @ B @ C )
       => ( greater @ A @ C ) ) ) ).

% PartialSA.succ_trans
thf(fact_58_PartialSA_Oadd__set__asso,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( add_set @ ( add_set @ A2 @ B2 ) @ C2 )
      = ( add_set @ A2 @ ( add_set @ B2 @ C2 ) ) ) ).

% PartialSA.add_set_asso
thf(fact_59_PartialSA_Osucc__antisym,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( ( greater @ B @ A )
       => ( A = B ) ) ) ).

% PartialSA.succ_antisym
thf(fact_60_PartialSA_Oadd__set__commm,axiom,
    ( add_set
    = ( ^ [A7: set_state,B4: set_state] : ( add_set @ B4 @ A7 ) ) ) ).

% PartialSA.add_set_commm
thf(fact_61_intuitionistic__def,axiom,
    ( intuitionistic
    = ( ^ [A7: set_state] :
        ! [A4: state,B3: state] :
          ( ( ( greater @ A4 @ B3 )
            & ( member_state @ B3 @ A7 ) )
         => ( member_state @ A4 @ A7 ) ) ) ) ).

% intuitionistic_def
thf(fact_62_in__multiply__refl,axiom,
    ! [X3: state,A2: set_state,P2: prat] :
      ( ( member_state @ X3 @ A2 )
     => ( member_state @ ( multiply @ P2 @ X3 ) @ ( multip8064567061438756306ertion @ P2 @ A2 ) ) ) ).

% in_multiply_refl
thf(fact_63_in__multiply__sem,axiom,
    ! [X3: state,P2: prat,A2: set_state] :
      ( ( member_state @ X3 @ ( multip8064567061438756306ertion @ P2 @ A2 ) )
     => ? [X: state] :
          ( ( member_state @ X @ A2 )
          & ( greater @ X3 @ ( multiply @ P2 @ X ) ) ) ) ).

% in_multiply_sem
thf(fact_64_w__in__scaled,axiom,
    ! [W: state] : ( member_state @ W @ ( scaled @ W ) ) ).

% w_in_scaled
thf(fact_65_PartialSA_Osmaller__compatible,axiom,
    ! [A6: state,B: state,A: state] :
      ( ( defined @ A6 @ B )
     => ( ( greater @ A6 @ A )
       => ( defined @ A @ B ) ) ) ).

% PartialSA.smaller_compatible
thf(fact_66_R__smaller,axiom,
    ! [W: state,A: state] : ( greater @ W @ ( r @ A @ W ) ) ).

% R_smaller
thf(fact_67_PartialSA_Obigger__sum,axiom,
    ! [Phi: state,A: state,B: state,Phi2: state] :
      ( ( ( some_state @ Phi )
        = ( plus @ A @ B ) )
     => ( ( greater @ Phi2 @ Phi )
       => ? [B5: state] :
            ( ( greater @ B5 @ B )
            & ( ( some_state @ Phi2 )
              = ( plus @ A @ B5 ) ) ) ) ) ).

% PartialSA.bigger_sum
thf(fact_68_PartialSA_Ogreater__def,axiom,
    ( greater
    = ( ^ [A4: state,B3: state] :
        ? [C3: state] :
          ( ( some_state @ A4 )
          = ( plus @ B3 @ C3 ) ) ) ) ).

% PartialSA.greater_def
thf(fact_69_PartialSA_Ogreater__equiv,axiom,
    ( greater
    = ( ^ [A4: state,B3: state] :
        ? [C3: state] :
          ( ( some_state @ A4 )
          = ( plus @ C3 @ B3 ) ) ) ) ).

% PartialSA.greater_equiv
thf(fact_70_PartialSA_Oaddition__bigger,axiom,
    ! [A6: state,A: state,X6: state,B: state,X3: state] :
      ( ( greater @ A6 @ A )
     => ( ( ( some_state @ X6 )
          = ( plus @ A6 @ B ) )
       => ( ( ( some_state @ X3 )
            = ( plus @ A @ B ) )
         => ( greater @ X6 @ X3 ) ) ) ) ).

% PartialSA.addition_bigger
thf(fact_71_PartialSA_Obigger__sum__smaller,axiom,
    ! [C: state,A: state,B: state,A6: state] :
      ( ( ( some_state @ C )
        = ( plus @ A @ B ) )
     => ( ( greater @ A @ A6 )
       => ? [B5: state] :
            ( ( greater @ B5 @ B )
            & ( ( some_state @ C )
              = ( plus @ A6 @ B5 ) ) ) ) ) ).

% PartialSA.bigger_sum_smaller
thf(fact_72_PartialSA_Oadd__set__elem,axiom,
    ! [Phi: state,A2: set_state,B2: set_state] :
      ( ( member_state @ Phi @ ( add_set @ A2 @ B2 ) )
      = ( ? [A4: state,B3: state] :
            ( ( ( some_state @ Phi )
              = ( plus @ A4 @ B3 ) )
            & ( member_state @ A4 @ A2 )
            & ( member_state @ B3 @ B2 ) ) ) ) ).

% PartialSA.add_set_elem
thf(fact_73_PartialSA_Ox__elem__set__product,axiom,
    ! [X3: state,A2: set_state,B2: set_state] :
      ( ( member_state @ X3 @ ( add_set @ A2 @ B2 ) )
      = ( ? [A4: state,B3: state] :
            ( ( member_state @ A4 @ A2 )
            & ( member_state @ B3 @ B2 )
            & ( ( some_state @ X3 )
              = ( plus @ A4 @ B3 ) ) ) ) ) ).

% PartialSA.x_elem_set_product
thf(fact_74_map__add__find__right,axiom,
    ! [N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,Xx: val,M: produc1457211279475724562t_char > option_val] :
      ( ( ( N @ K )
        = ( some_val @ Xx ) )
     => ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ Xx ) ) ) ).

% map_add_find_right
thf(fact_75_add__states_Oelims,axiom,
    ! [X3: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X3 @ Xa2 )
        = Y )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
                ( ( Xa2
                  = ( produc74620467203698198on_val @ Pi2 @ H2 ) )
               => ( Y
                 != ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi2 ) @ ( map_ad2908576761495157028ar_val @ H @ H2 ) ) ) ) ) ) ).

% add_states.elims
thf(fact_76_add__states_Osimps,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val,Pi4: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
      ( ( add_states @ ( produc74620467203698198on_val @ Pi3 @ H3 ) @ ( produc74620467203698198on_val @ Pi4 @ H4 ) )
      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi3 @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H3 @ H4 ) ) ) ).

% add_states.simps
thf(fact_77_map__add__assoc,axiom,
    ! [M1: produc1457211279475724562t_char > option_val,M2: produc1457211279475724562t_char > option_val,M3: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ M1 @ ( map_ad2908576761495157028ar_val @ M2 @ M3 ) )
      = ( map_ad2908576761495157028ar_val @ ( map_ad2908576761495157028ar_val @ M1 @ M2 ) @ M3 ) ) ).

% map_add_assoc
thf(fact_78_old_Oprod_Oinject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A6: produc9003781588162623004on_val,B6: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_79_old_Oprod_Oinject,axiom,
    ! [A: option_state,B: option_state,A6: option_state,B6: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B )
        = ( produc9160152616361873709_state @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_80_old_Oprod_Oinject,axiom,
    ! [A: prat,B: state,A6: prat,B6: state] :
      ( ( ( produc6205719116230455971_state @ A @ B )
        = ( produc6205719116230455971_state @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_81_old_Oprod_Oinject,axiom,
    ! [A: prat,B: set_state,A6: prat,B6: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B )
        = ( produc9080284787853255769_state @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_82_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A6: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_83_prod_Oinject,axiom,
    ! [X1: produc9003781588162623004on_val,X2: produc9003781588162623004on_val,Y1: produc9003781588162623004on_val,Y2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ X1 @ X2 )
        = ( produc4958730763592889809on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_84_prod_Oinject,axiom,
    ! [X1: option_state,X2: option_state,Y1: option_state,Y2: option_state] :
      ( ( ( produc9160152616361873709_state @ X1 @ X2 )
        = ( produc9160152616361873709_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_85_prod_Oinject,axiom,
    ! [X1: prat,X2: state,Y1: prat,Y2: state] :
      ( ( ( produc6205719116230455971_state @ X1 @ X2 )
        = ( produc6205719116230455971_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_86_prod_Oinject,axiom,
    ! [X1: prat,X2: set_state,Y1: prat,Y2: set_state] :
      ( ( ( produc9080284787853255769_state @ X1 @ X2 )
        = ( produc9080284787853255769_state @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_87_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X2: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X2 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_88_multiply_Osimps,axiom,
    ( multiply
    = ( ^ [P3: prat,Phi3: state] : ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P3 @ ( get_m @ Phi3 ) ) @ ( get_h @ Phi3 ) ) ) ) ) ).

% multiply.simps
thf(fact_89_multiply_Oelims,axiom,
    ! [X3: prat,Xa2: state,Y: state] :
      ( ( ( multiply @ X3 @ Xa2 )
        = Y )
     => ( Y
        = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X3 @ ( get_m @ Xa2 ) ) @ ( get_h @ Xa2 ) ) ) ) ) ).

% multiply.elims
thf(fact_90_binary__def,axiom,
    ( binary
    = ( ^ [A7: set_state] :
        ! [X5: state] :
          ( ( member_state @ X5 @ A7 )
         => ( member_state @ ( abs_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ X5 ) ) @ ( get_h @ X5 ) ) ) @ A7 ) ) ) ) ).

% binary_def
thf(fact_91_empty__heap__neutral,axiom,
    ! [A: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ A @ empty_7512026568161686585ar_val )
      = A ) ).

% empty_heap_neutral
thf(fact_92_get__h__m,axiom,
    ( rep_state
    = ( ^ [X5: state] : ( produc74620467203698198on_val @ ( get_m @ X5 ) @ ( get_h @ X5 ) ) ) ) ).

% get_h_m
thf(fact_93_Rep__state__inject,axiom,
    ! [X3: state,Y: state] :
      ( ( ( rep_state @ X3 )
        = ( rep_state @ Y ) )
      = ( X3 = Y ) ) ).

% Rep_state_inject
thf(fact_94_add__states_Ocases,axiom,
    ! [X3: produc1317171644805857433on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
          ( X3
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi2 @ H2 ) ) ) ).

% add_states.cases
thf(fact_95_Rep__state__inverse,axiom,
    ! [X3: state] :
      ( ( abs_state @ ( rep_state @ X3 ) )
      = X3 ) ).

% Rep_state_inverse
thf(fact_96_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A3: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A3 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_97_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3142500478612311029_state] :
      ~ ! [A3: option_state,B7: option_state] :
          ( Y
         != ( produc9160152616361873709_state @ A3 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_98_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4844402304972151915_state] :
      ~ ! [A3: prat,B7: state] :
          ( Y
         != ( produc6205719116230455971_state @ A3 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_99_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6251377953716854561_state] :
      ~ ! [A3: prat,B7: set_state] :
          ( Y
         != ( produc9080284787853255769_state @ A3 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_100_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ~ ! [A3: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc74620467203698198on_val @ A3 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_101_surj__pair,axiom,
    ! [P2: produc1317171644805857433on_val] :
    ? [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( P2
      = ( produc4958730763592889809on_val @ X @ Y3 ) ) ).

% surj_pair
thf(fact_102_surj__pair,axiom,
    ! [P2: produc3142500478612311029_state] :
    ? [X: option_state,Y3: option_state] :
      ( P2
      = ( produc9160152616361873709_state @ X @ Y3 ) ) ).

% surj_pair
thf(fact_103_surj__pair,axiom,
    ! [P2: produc4844402304972151915_state] :
    ? [X: prat,Y3: state] :
      ( P2
      = ( produc6205719116230455971_state @ X @ Y3 ) ) ).

% surj_pair
thf(fact_104_surj__pair,axiom,
    ! [P2: produc6251377953716854561_state] :
    ? [X: prat,Y3: set_state] :
      ( P2
      = ( produc9080284787853255769_state @ X @ Y3 ) ) ).

% surj_pair
thf(fact_105_surj__pair,axiom,
    ! [P2: produc9003781588162623004on_val] :
    ? [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
      ( P2
      = ( produc74620467203698198on_val @ X @ Y3 ) ) ).

% surj_pair
thf(fact_106_prod__cases,axiom,
    ! [P: produc1317171644805857433on_val > $o,P2: produc1317171644805857433on_val] :
      ( ! [A3: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] : ( P @ ( produc4958730763592889809on_val @ A3 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_107_prod__cases,axiom,
    ! [P: produc3142500478612311029_state > $o,P2: produc3142500478612311029_state] :
      ( ! [A3: option_state,B7: option_state] : ( P @ ( produc9160152616361873709_state @ A3 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_108_prod__cases,axiom,
    ! [P: produc4844402304972151915_state > $o,P2: produc4844402304972151915_state] :
      ( ! [A3: prat,B7: state] : ( P @ ( produc6205719116230455971_state @ A3 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_109_prod__cases,axiom,
    ! [P: produc6251377953716854561_state > $o,P2: produc6251377953716854561_state] :
      ( ! [A3: prat,B7: set_state] : ( P @ ( produc9080284787853255769_state @ A3 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_110_prod__cases,axiom,
    ! [P: produc9003781588162623004on_val > $o,P2: produc9003781588162623004on_val] :
      ( ! [A3: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] : ( P @ ( produc74620467203698198on_val @ A3 @ B7 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_111_Pair__inject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A6: produc9003781588162623004on_val,B6: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_112_Pair__inject,axiom,
    ! [A: option_state,B: option_state,A6: option_state,B6: option_state] :
      ( ( ( produc9160152616361873709_state @ A @ B )
        = ( produc9160152616361873709_state @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_113_Pair__inject,axiom,
    ! [A: prat,B: state,A6: prat,B6: state] :
      ( ( ( produc6205719116230455971_state @ A @ B )
        = ( produc6205719116230455971_state @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_114_Pair__inject,axiom,
    ! [A: prat,B: set_state,A6: prat,B6: set_state] :
      ( ( ( produc9080284787853255769_state @ A @ B )
        = ( produc9080284787853255769_state @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_115_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A6: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_116_prod__cases3,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A3: produc9003781588162623004on_val,B7: produc1457211279475724562t_char > prat,C4: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A3 @ ( produc74620467203698198on_val @ B7 @ C4 ) ) ) ).

% prod_cases3
thf(fact_117_prod__induct3,axiom,
    ! [P: produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ! [A3: produc9003781588162623004on_val,B7: produc1457211279475724562t_char > prat,C4: produc1457211279475724562t_char > option_val] : ( P @ ( produc4958730763592889809on_val @ A3 @ ( produc74620467203698198on_val @ B7 @ C4 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_118_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C ) ) ) ).

% mult_distr_masks
thf(fact_119_binary__same,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( binary @ A2 )
     => ( ( intuitionistic @ B2 )
       => ( ord_le2494988322063910608_state @ ( wand @ A2 @ B2 ) @ ( cwand @ A2 @ B2 ) ) ) ) ).

% binary_same
thf(fact_120_valid__bin,axiom,
    ! [A: state] : ( valid_state @ ( produc74620467203698198on_val @ ( binary_mask @ ( get_m @ A ) ) @ ( get_h @ A ) ) ) ).

% valid_bin
thf(fact_121_multiply_Opelims,axiom,
    ! [X3: prat,Xa2: state,Y: state] :
      ( ( ( multiply @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X3 @ Xa2 ) )
       => ~ ( ( Y
              = ( abs_state @ ( produc74620467203698198on_val @ ( multiply_mask @ X3 @ ( get_m @ Xa2 ) ) @ ( get_h @ Xa2 ) ) ) )
           => ~ ( accp_P6233459008282737652_state @ multiply_rel @ ( produc6205719116230455971_state @ X3 @ Xa2 ) ) ) ) ) ).

% multiply.pelims
thf(fact_122_plus__def,axiom,
    ( plus
    = ( ^ [A4: state,B3: state] : ( if_option_state @ ( compatible @ ( rep_state @ A4 ) @ ( rep_state @ B3 ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A4 ) @ ( rep_state @ B3 ) ) ) ) @ none_state ) ) ) ).

% plus_def
thf(fact_123_add__states_Opelims,axiom,
    ! [X3: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X3 @ Xa2 ) )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
                  ( ( Xa2
                    = ( produc74620467203698198on_val @ Pi2 @ H2 ) )
                 => ( ( Y
                      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi2 ) @ ( map_ad2908576761495157028ar_val @ H @ H2 ) ) )
                   => ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi2 @ H2 ) ) ) ) ) ) ) ) ).

% add_states.pelims
thf(fact_124_PartialSA_Ox__elem__set__product__splus,axiom,
    ! [X3: state,A2: set_state,B2: set_state] :
      ( ( member_state @ X3 @ ( add_set @ A2 @ B2 ) )
      = ( ? [A4: state,B3: state] :
            ( ( member_state @ A4 @ A2 )
            & ( member_state @ B3 @ B2 )
            & ( ( some_state @ X3 )
              = ( sep_splus_state @ plus @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) ) ) ) ) ).

% PartialSA.x_elem_set_product_splus
thf(fact_125_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_126_double__inclusion,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% double_inclusion
thf(fact_127_not__None__eq,axiom,
    ! [X3: option_state] :
      ( ( X3 != none_state )
      = ( ? [Y4: state] :
            ( X3
            = ( some_state @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_128_not__None__eq,axiom,
    ! [X3: option_val] :
      ( ( X3 != none_val )
      = ( ? [Y4: val] :
            ( X3
            = ( some_val @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_129_not__Some__eq,axiom,
    ! [X3: option_state] :
      ( ( ! [Y4: state] :
            ( X3
           != ( some_state @ Y4 ) ) )
      = ( X3 = none_state ) ) ).

% not_Some_eq
thf(fact_130_not__Some__eq,axiom,
    ! [X3: option_val] :
      ( ( ! [Y4: val] :
            ( X3
           != ( some_val @ Y4 ) ) )
      = ( X3 = none_val ) ) ).

% not_Some_eq
thf(fact_131_map__add__eq__empty__iff,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( map_ad2908576761495157028ar_val @ F @ G )
        = ( ^ [X5: produc1457211279475724562t_char] : none_val ) )
      = ( ( F
          = ( ^ [X5: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X5: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% map_add_eq_empty_iff
thf(fact_132_map__add__None,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = none_val )
      = ( ( ( N @ K )
          = none_val )
        & ( ( M @ K )
          = none_val ) ) ) ).

% map_add_None
thf(fact_133_multiply_Ocases,axiom,
    ! [X3: produc4844402304972151915_state] :
      ~ ! [P4: prat,Phi4: state] :
          ( X3
         != ( produc6205719116230455971_state @ P4 @ Phi4 ) ) ).

% multiply.cases
thf(fact_134_PartialSA_Osplus_Osimps_I1_J,axiom,
    ! [Uu: option_state] :
      ( ( sep_splus_state @ plus @ none_state @ Uu )
      = none_state ) ).

% PartialSA.splus.simps(1)
thf(fact_135_PartialSA_Osplus_Osimps_I2_J,axiom,
    ! [V: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ V ) @ none_state )
      = none_state ) ).

% PartialSA.splus.simps(2)
thf(fact_136_PartialSA_Osplus_Oelims,axiom,
    ! [X3: option_state,Xa2: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = none_state )
         => ( Y != none_state ) )
       => ( ( ? [V2: state] :
                ( X3
                = ( some_state @ V2 ) )
           => ( ( Xa2 = none_state )
             => ( Y != none_state ) ) )
         => ~ ! [A3: state] :
                ( ( X3
                  = ( some_state @ A3 ) )
               => ! [B7: state] :
                    ( ( Xa2
                      = ( some_state @ B7 ) )
                   => ( Y
                     != ( plus @ A3 @ B7 ) ) ) ) ) ) ) ).

% PartialSA.splus.elims
thf(fact_137_option_Odistinct_I1_J,axiom,
    ! [X2: state] :
      ( none_state
     != ( some_state @ X2 ) ) ).

% option.distinct(1)
thf(fact_138_option_Odistinct_I1_J,axiom,
    ! [X2: val] :
      ( none_val
     != ( some_val @ X2 ) ) ).

% option.distinct(1)
thf(fact_139_option_OdiscI,axiom,
    ! [Option: option_state,X2: state] :
      ( ( Option
        = ( some_state @ X2 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_140_option_OdiscI,axiom,
    ! [Option: option_val,X2: val] :
      ( ( Option
        = ( some_val @ X2 ) )
     => ( Option != none_val ) ) ).

% option.discI
thf(fact_141_option_Oexhaust,axiom,
    ! [Y: option_state] :
      ( ( Y != none_state )
     => ~ ! [X22: state] :
            ( Y
           != ( some_state @ X22 ) ) ) ).

% option.exhaust
thf(fact_142_option_Oexhaust,axiom,
    ! [Y: option_val] :
      ( ( Y != none_val )
     => ~ ! [X22: val] :
            ( Y
           != ( some_val @ X22 ) ) ) ).

% option.exhaust
thf(fact_143_split__option__ex,axiom,
    ( ( ^ [P5: option_state > $o] :
        ? [X7: option_state] : ( P5 @ X7 ) )
    = ( ^ [P6: option_state > $o] :
          ( ( P6 @ none_state )
          | ? [X5: state] : ( P6 @ ( some_state @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_144_split__option__ex,axiom,
    ( ( ^ [P5: option_val > $o] :
        ? [X7: option_val] : ( P5 @ X7 ) )
    = ( ^ [P6: option_val > $o] :
          ( ( P6 @ none_val )
          | ? [X5: val] : ( P6 @ ( some_val @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_145_split__option__all,axiom,
    ( ( ^ [P5: option_state > $o] :
        ! [X7: option_state] : ( P5 @ X7 ) )
    = ( ^ [P6: option_state > $o] :
          ( ( P6 @ none_state )
          & ! [X5: state] : ( P6 @ ( some_state @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_146_split__option__all,axiom,
    ( ( ^ [P5: option_val > $o] :
        ! [X7: option_val] : ( P5 @ X7 ) )
    = ( ^ [P6: option_val > $o] :
          ( ( P6 @ none_val )
          & ! [X5: val] : ( P6 @ ( some_val @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_147_combine__options__cases,axiom,
    ! [X3: option_state,P: option_state > option_state > $o,Y: option_state] :
      ( ( ( X3 = none_state )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_state )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: state,B7: state] :
              ( ( X3
                = ( some_state @ A3 ) )
             => ( ( Y
                  = ( some_state @ B7 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_148_combine__options__cases,axiom,
    ! [X3: option_state,P: option_state > option_val > $o,Y: option_val] :
      ( ( ( X3 = none_state )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_val )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: state,B7: val] :
              ( ( X3
                = ( some_state @ A3 ) )
             => ( ( Y
                  = ( some_val @ B7 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_149_combine__options__cases,axiom,
    ! [X3: option_val,P: option_val > option_state > $o,Y: option_state] :
      ( ( ( X3 = none_val )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_state )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: val,B7: state] :
              ( ( X3
                = ( some_val @ A3 ) )
             => ( ( Y
                  = ( some_state @ B7 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_150_combine__options__cases,axiom,
    ! [X3: option_val,P: option_val > option_val > $o,Y: option_val] :
      ( ( ( X3 = none_val )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_val )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: val,B7: val] :
              ( ( X3
                = ( some_val @ A3 ) )
             => ( ( Y
                  = ( some_val @ B7 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_151_compatible__options_Ocases,axiom,
    ! [X3: produc3142500478612311029_state] :
      ( ! [A3: state,B7: state] :
          ( X3
         != ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) )
     => ( ! [Uv: option_state] :
            ( X3
           != ( produc9160152616361873709_state @ none_state @ Uv ) )
       => ~ ! [Uu2: option_state] :
              ( X3
             != ( produc9160152616361873709_state @ Uu2 @ none_state ) ) ) ) ).

% compatible_options.cases
thf(fact_152_compatible__options_Ocases,axiom,
    ! [X3: produc5122021904355189735on_val] :
      ( ! [A3: val,B7: val] :
          ( X3
         != ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) )
     => ( ! [Uv: option_val] :
            ( X3
           != ( produc904852209897547607on_val @ none_val @ Uv ) )
       => ~ ! [Uu2: option_val] :
              ( X3
             != ( produc904852209897547607on_val @ Uu2 @ none_val ) ) ) ) ).

% compatible_options.cases
thf(fact_153_PartialSA_Osplus_Ocases,axiom,
    ! [X3: produc3142500478612311029_state] :
      ( ! [Uu2: option_state] :
          ( X3
         != ( produc9160152616361873709_state @ none_state @ Uu2 ) )
     => ( ! [V2: state] :
            ( X3
           != ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) )
       => ~ ! [A3: state,B7: state] :
              ( X3
             != ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) ) ) ) ).

% PartialSA.splus.cases
thf(fact_154_PartialSA_Osplus__asso,axiom,
    ! [A: option_state,B: option_state,C: option_state] :
      ( ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ A @ B ) @ C )
      = ( sep_splus_state @ plus @ A @ ( sep_splus_state @ plus @ B @ C ) ) ) ).

% PartialSA.splus_asso
thf(fact_155_PartialSA_Osplus__comm,axiom,
    ! [A: option_state,B: option_state] :
      ( ( sep_splus_state @ plus @ A @ B )
      = ( sep_splus_state @ plus @ B @ A ) ) ).

% PartialSA.splus_comm
thf(fact_156_empty__heap__def,axiom,
    ( empty_7512026568161686585ar_val
    = ( ^ [Hl: produc1457211279475724562t_char] : none_val ) ) ).

% empty_heap_def
thf(fact_157_PartialSA_Oasso3,axiom,
    ! [A: state,B: state,C: state,Bc: state] :
      ( ( ( plus @ A @ B )
        = none_state )
     => ( ( ( plus @ B @ C )
          = ( some_state @ Bc ) )
       => ( ( plus @ A @ Bc )
          = none_state ) ) ) ).

% PartialSA.asso3
thf(fact_158_asso2,axiom,
    ! [A: state,B: state,Ab: state,C: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C )
          = none_state ) )
     => ( ( plus @ Ab @ C )
        = none_state ) ) ).

% asso2
thf(fact_159_map__add__SomeD,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X3 ) )
     => ( ( ( N @ K )
          = ( some_val @ X3 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X3 ) ) ) ) ) ).

% map_add_SomeD
thf(fact_160_map__add__Some__iff,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X3: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X3 ) )
      = ( ( ( N @ K )
          = ( some_val @ X3 ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X3 ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_161_PartialSA_Odefined__def,axiom,
    ( defined
    = ( ^ [A4: state,B3: state] :
          ( ( plus @ A4 @ B3 )
         != none_state ) ) ) ).

% PartialSA.defined_def
thf(fact_162_PartialSA_Osplus_Osimps_I3_J,axiom,
    ! [A: state,B: state] :
      ( ( sep_splus_state @ plus @ ( some_state @ A ) @ ( some_state @ B ) )
      = ( plus @ A @ B ) ) ).

% PartialSA.splus.simps(3)
thf(fact_163_PartialSA_Osplus__develop,axiom,
    ! [A: state,B: state,C: state,D: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ C ) )
     => ( ( plus @ A @ D )
        = ( sep_splus_state @ plus @ ( sep_splus_state @ plus @ ( some_state @ B ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ).

% PartialSA.splus_develop
thf(fact_164_cwand__stronger,axiom,
    ! [A2: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ ( cwand @ A2 @ B2 ) @ ( wand @ A2 @ B2 ) ) ).

% cwand_stronger
thf(fact_165_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B ) @ H3 ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H3 ) ) ) ).

% valid_state_decompose
thf(fact_166_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C = D ) ) ).

% add_masks_cancellative
thf(fact_167_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B3 @ A4 ) ) ) ).

% add_masks_comm
thf(fact_168_subsetI,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A2 )
         => ( member7461423308196221312_state @ X @ B2 ) )
     => ( ord_le4032880777312351359_state @ A2 @ B2 ) ) ).

% subsetI
thf(fact_169_subsetI,axiom,
    ! [A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X @ A2 )
         => ( member2879030624002546611on_val @ X @ B2 ) )
     => ( ord_le6900622908513426738on_val @ A2 @ B2 ) ) ).

% subsetI
thf(fact_170_subsetI,axiom,
    ! [A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A2 )
         => ( member6715010751053357271_state @ X @ B2 ) )
     => ( ord_le6483787752022044334_state @ A2 @ B2 ) ) ).

% subsetI
thf(fact_171_subsetI,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ! [X: state] :
          ( ( member_state @ X @ A2 )
         => ( member_state @ X @ B2 ) )
     => ( ord_le2494988322063910608_state @ A2 @ B2 ) ) ).

% subsetI
thf(fact_172_dual__order_Orefl,axiom,
    ! [A: set_state] : ( ord_le2494988322063910608_state @ A @ A ) ).

% dual_order.refl
thf(fact_173_order__refl,axiom,
    ! [X3: set_state] : ( ord_le2494988322063910608_state @ X3 @ X3 ) ).

% order_refl
thf(fact_174_same__fstI,axiom,
    ! [P: prat > $o,X3: prat,Y5: state,Y: state,R: prat > set_Pr1785066336555260981_state] :
      ( ( P @ X3 )
     => ( ( member753036827967488894_state @ ( produc544653723139640077_state @ Y5 @ Y ) @ ( R @ X3 ) )
       => ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ X3 @ Y5 ) @ ( produc6205719116230455971_state @ X3 @ Y ) ) @ ( same_fst_prat_state @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_175_same__fstI,axiom,
    ! [P: prat > $o,X3: prat,Y5: set_state,Y: set_state,R: prat > set_Pr5618112136499017889_state] :
      ( ( P @ X3 )
     => ( ( member1976142906092959210_state @ ( produc1834047914073684089_state @ Y5 @ Y ) @ ( R @ X3 ) )
       => ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ X3 @ Y5 ) @ ( produc9080284787853255769_state @ X3 @ Y ) ) @ ( same_f5333510722715586990_state @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_176_same__fstI,axiom,
    ! [P: ( produc1457211279475724562t_char > prat ) > $o,X3: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P @ X3 )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y5 @ Y ) @ ( R @ X3 ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X3 @ Y5 ) @ ( produc74620467203698198on_val @ X3 @ Y ) ) @ ( same_f8138205353876899841on_val @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_177_same__fstI,axiom,
    ! [P: produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: produc9003781588162623004on_val > set_Pr4305388979797279737on_val] :
      ( ( P @ X3 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ Y ) @ ( R @ X3 ) )
       => ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ X3 @ Y5 ) @ ( produc4958730763592889809on_val @ X3 @ Y ) ) @ ( same_f7521981028318775590on_val @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_178_same__fstI,axiom,
    ! [P: option_state > $o,X3: option_state,Y5: option_state,Y: option_state,R: option_state > set_Pr1688445902015331925_state] :
      ( ( P @ X3 )
     => ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Y5 @ Y ) @ ( R @ X3 ) )
       => ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ X3 @ Y5 ) @ ( produc9160152616361873709_state @ X3 @ Y ) ) @ ( same_f2704544105346032002_state @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_179_sep__algebra_Osplus_Ocong,axiom,
    sep_splus_state = sep_splus_state ).

% sep_algebra.splus.cong
thf(fact_180_compatible__options_Oelims_I1_J,axiom,
    ! [X3: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: state] :
            ( ( X3
              = ( some_state @ A3 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( Y
                  = ( A3 != B7 ) ) ) )
       => ( ( ( X3 = none_state )
           => ~ Y )
         => ~ ( ( Xa2 = none_state )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_181_compatible__options_Oelims_I1_J,axiom,
    ! [X3: option_val,Xa2: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: val] :
            ( ( X3
              = ( some_val @ A3 ) )
           => ! [B7: val] :
                ( ( Xa2
                  = ( some_val @ B7 ) )
               => ( Y
                  = ( A3 != B7 ) ) ) )
       => ( ( ( X3 = none_val )
           => ~ Y )
         => ~ ( ( Xa2 = none_val )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_182_compatible__options_Oelims_I2_J,axiom,
    ! [X3: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X3 @ Xa2 )
     => ( ! [A3: state] :
            ( ( X3
              = ( some_state @ A3 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( A3 != B7 ) ) )
       => ( ( X3 != none_state )
         => ( Xa2 = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_183_compatible__options_Oelims_I2_J,axiom,
    ! [X3: option_val,Xa2: option_val] :
      ( ( compat7209469420100717861ns_val @ X3 @ Xa2 )
     => ( ! [A3: val] :
            ( ( X3
              = ( some_val @ A3 ) )
           => ! [B7: val] :
                ( ( Xa2
                  = ( some_val @ B7 ) )
               => ( A3 != B7 ) ) )
       => ( ( X3 != none_val )
         => ( Xa2 = none_val ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_184_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_185_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_186_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_187_ssubsetI,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ! [Pi: state,H: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H ) @ A2 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ Pi @ H ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_188_ssubsetI,axiom,
    ! [A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ! [Pi: state,H: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H ) @ A2 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ Pi @ H ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_189_ssubsetI,axiom,
    ! [A2: set_Pr4305388979797279737on_val,B2: set_Pr4305388979797279737on_val] :
      ( ! [Pi: produc9003781588162623004on_val,H: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ A2 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ B2 ) )
     => ( ord_le7224449410216889241on_val @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_190_ssubsetI,axiom,
    ! [A2: set_Pr1688445902015331925_state,B2: set_Pr1688445902015331925_state] :
      ( ! [Pi: option_state,H: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H ) @ A2 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ Pi @ H ) @ B2 ) )
     => ( ord_le6423325748750870005_state @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_191_ssubsetI,axiom,
    ! [A2: set_Pr6039890582625207499_state,B2: set_Pr6039890582625207499_state] :
      ( ! [Pi: prat,H: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H ) @ A2 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ Pi @ H ) @ B2 ) )
     => ( ord_le262197048489888875_state @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_192_ssubsetI,axiom,
    ! [A2: set_Pr6847263425858009217_state,B2: set_Pr6847263425858009217_state] :
      ( ! [Pi: prat,H: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H ) @ A2 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ Pi @ H ) @ B2 ) )
     => ( ord_le5761494056910127137_state @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_193_ssubsetI,axiom,
    ! [A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ A2 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ A2 @ B2 ) ) ).

% ssubsetI
thf(fact_194_compatible__options_Osimps_I1_J,axiom,
    ! [A: state,B: state] :
      ( ( compat2278460363914054422_state @ ( some_state @ A ) @ ( some_state @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_195_compatible__options_Osimps_I1_J,axiom,
    ! [A: val,B: val] :
      ( ( compat7209469420100717861ns_val @ ( some_val @ A ) @ ( some_val @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_196_compatible__options_Oelims_I3_J,axiom,
    ! [X3: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X3 @ Xa2 )
     => ~ ! [A3: state] :
            ( ( X3
              = ( some_state @ A3 ) )
           => ! [B7: state] :
                ( ( Xa2
                  = ( some_state @ B7 ) )
               => ( A3 = B7 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_197_compatible__options_Oelims_I3_J,axiom,
    ! [X3: option_val,Xa2: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X3 @ Xa2 )
     => ~ ! [A3: val] :
            ( ( X3
              = ( some_val @ A3 ) )
           => ! [B7: val] :
                ( ( Xa2
                  = ( some_val @ B7 ) )
               => ( A3 = B7 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_198_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_state] : ( compat2278460363914054422_state @ none_state @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_199_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_200_compatible__options_Osimps_I3_J,axiom,
    ! [Uu: option_state] : ( compat2278460363914054422_state @ Uu @ none_state ) ).

% compatible_options.simps(3)
thf(fact_201_compatible__options_Osimps_I3_J,axiom,
    ! [Uu: option_val] : ( compat7209469420100717861ns_val @ Uu @ none_val ) ).

% compatible_options.simps(3)
thf(fact_202_minus__empty,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat] :
      ( Pi3
      = ( add_ma6586698345353345811t_char @ Pi3 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_203_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_state,Z: set_state] : ( Y6 = Z ) )
    = ( ^ [X5: set_state,Y4: set_state] :
          ( ( ord_le2494988322063910608_state @ X5 @ Y4 )
          & ( ord_le2494988322063910608_state @ Y4 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_204_ord__eq__le__trans,axiom,
    ! [A: set_state,B: set_state,C: set_state] :
      ( ( A = B )
     => ( ( ord_le2494988322063910608_state @ B @ C )
       => ( ord_le2494988322063910608_state @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_205_ord__le__eq__trans,axiom,
    ! [A: set_state,B: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( B = C )
       => ( ord_le2494988322063910608_state @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_206_order__antisym,axiom,
    ! [X3: set_state,Y: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y )
     => ( ( ord_le2494988322063910608_state @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_207_order_Otrans,axiom,
    ! [A: set_state,B: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( ord_le2494988322063910608_state @ B @ C )
       => ( ord_le2494988322063910608_state @ A @ C ) ) ) ).

% order.trans
thf(fact_208_order__trans,axiom,
    ! [X3: set_state,Y: set_state,Z2: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y )
     => ( ( ord_le2494988322063910608_state @ Y @ Z2 )
       => ( ord_le2494988322063910608_state @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_209_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_state,Z: set_state] : ( Y6 = Z ) )
    = ( ^ [A4: set_state,B3: set_state] :
          ( ( ord_le2494988322063910608_state @ B3 @ A4 )
          & ( ord_le2494988322063910608_state @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_210_dual__order_Oantisym,axiom,
    ! [B: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B @ A )
     => ( ( ord_le2494988322063910608_state @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_211_dual__order_Otrans,axiom,
    ! [B: set_state,A: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ B @ A )
     => ( ( ord_le2494988322063910608_state @ C @ B )
       => ( ord_le2494988322063910608_state @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_212_antisym,axiom,
    ! [A: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( ord_le2494988322063910608_state @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_213_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_state,Z: set_state] : ( Y6 = Z ) )
    = ( ^ [A4: set_state,B3: set_state] :
          ( ( ord_le2494988322063910608_state @ A4 @ B3 )
          & ( ord_le2494988322063910608_state @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_214_order__subst1,axiom,
    ! [A: set_state,F: set_state > set_state,B: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ ( F @ B ) )
     => ( ( ord_le2494988322063910608_state @ B @ C )
       => ( ! [X: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_215_order__subst2,axiom,
    ! [A: set_state,B: set_state,F: set_state > set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( ord_le2494988322063910608_state @ ( F @ B ) @ C )
       => ( ! [X: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_216_order__eq__refl,axiom,
    ! [X3: set_state,Y: set_state] :
      ( ( X3 = Y )
     => ( ord_le2494988322063910608_state @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_217_ord__eq__le__subst,axiom,
    ! [A: set_state,F: set_state > set_state,B: set_state,C: set_state] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2494988322063910608_state @ B @ C )
       => ( ! [X: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_218_ord__le__eq__subst,axiom,
    ! [A: set_state,B: set_state,F: set_state > set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_state,Y3: set_state] :
              ( ( ord_le2494988322063910608_state @ X @ Y3 )
             => ( ord_le2494988322063910608_state @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2494988322063910608_state @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_219_order__antisym__conv,axiom,
    ! [Y: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X3 )
     => ( ( ord_le2494988322063910608_state @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_220_in__mono,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,X3: produc7272280185086996969_state] :
      ( ( ord_le4032880777312351359_state @ A2 @ B2 )
     => ( ( member7461423308196221312_state @ X3 @ A2 )
       => ( member7461423308196221312_state @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_221_in__mono,axiom,
    ! [A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val,X3: produc9003781588162623004on_val] :
      ( ( ord_le6900622908513426738on_val @ A2 @ B2 )
     => ( ( member2879030624002546611on_val @ X3 @ A2 )
       => ( member2879030624002546611on_val @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_222_in__mono,axiom,
    ! [A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state,X3: produc3882902684156697774_state] :
      ( ( ord_le6483787752022044334_state @ A2 @ B2 )
     => ( ( member6715010751053357271_state @ X3 @ A2 )
       => ( member6715010751053357271_state @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_223_in__mono,axiom,
    ! [A2: set_state,B2: set_state,X3: state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ( member_state @ X3 @ A2 )
       => ( member_state @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_224_subsetD,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C: produc7272280185086996969_state] :
      ( ( ord_le4032880777312351359_state @ A2 @ B2 )
     => ( ( member7461423308196221312_state @ C @ A2 )
       => ( member7461423308196221312_state @ C @ B2 ) ) ) ).

% subsetD
thf(fact_225_subsetD,axiom,
    ! [A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val,C: produc9003781588162623004on_val] :
      ( ( ord_le6900622908513426738on_val @ A2 @ B2 )
     => ( ( member2879030624002546611on_val @ C @ A2 )
       => ( member2879030624002546611on_val @ C @ B2 ) ) ) ).

% subsetD
thf(fact_226_subsetD,axiom,
    ! [A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state,C: produc3882902684156697774_state] :
      ( ( ord_le6483787752022044334_state @ A2 @ B2 )
     => ( ( member6715010751053357271_state @ C @ A2 )
       => ( member6715010751053357271_state @ C @ B2 ) ) ) ).

% subsetD
thf(fact_227_subsetD,axiom,
    ! [A2: set_state,B2: set_state,C: state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ( member_state @ C @ A2 )
       => ( member_state @ C @ B2 ) ) ) ).

% subsetD
thf(fact_228_equalityE,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( A2 = B2 )
     => ~ ( ( ord_le2494988322063910608_state @ A2 @ B2 )
         => ~ ( ord_le2494988322063910608_state @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_229_subset__eq,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
        ! [X5: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X5 @ A7 )
         => ( member7461423308196221312_state @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_230_subset__eq,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,B4: set_Pr7649655576376376530on_val] :
        ! [X5: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X5 @ A7 )
         => ( member2879030624002546611on_val @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_231_subset__eq,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A7: set_Pr2391701600042252046_state,B4: set_Pr2391701600042252046_state] :
        ! [X5: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X5 @ A7 )
         => ( member6715010751053357271_state @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_232_subset__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A7: set_state,B4: set_state] :
        ! [X5: state] :
          ( ( member_state @ X5 @ A7 )
         => ( member_state @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_233_equalityD1,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( A2 = B2 )
     => ( ord_le2494988322063910608_state @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_234_equalityD2,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( A2 = B2 )
     => ( ord_le2494988322063910608_state @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_235_subset__iff,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
        ! [T: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ T @ A7 )
         => ( member7461423308196221312_state @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_236_subset__iff,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,B4: set_Pr7649655576376376530on_val] :
        ! [T: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ T @ A7 )
         => ( member2879030624002546611on_val @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_237_subset__iff,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A7: set_Pr2391701600042252046_state,B4: set_Pr2391701600042252046_state] :
        ! [T: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ T @ A7 )
         => ( member6715010751053357271_state @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_238_subset__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A7: set_state,B4: set_state] :
        ! [T: state] :
          ( ( member_state @ T @ A7 )
         => ( member_state @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_239_subset__refl,axiom,
    ! [A2: set_state] : ( ord_le2494988322063910608_state @ A2 @ A2 ) ).

% subset_refl
thf(fact_240_Collect__mono,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_mono
thf(fact_241_Collect__mono,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X: produc9003781588162623004on_val] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_mono
thf(fact_242_Collect__mono,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ! [X: state] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) ) ) ).

% Collect_mono
thf(fact_243_subset__trans,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C2 )
       => ( ord_le2494988322063910608_state @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_244_set__eq__subset,axiom,
    ( ( ^ [Y6: set_state,Z: set_state] : ( Y6 = Z ) )
    = ( ^ [A7: set_state,B4: set_state] :
          ( ( ord_le2494988322063910608_state @ A7 @ B4 )
          & ( ord_le2494988322063910608_state @ B4 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_245_Collect__mono__iff,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) )
      = ( ! [X5: produc7272280185086996969_state] :
            ( ( P @ X5 )
           => ( Q @ X5 ) ) ) ) ).

% Collect_mono_iff
thf(fact_246_Collect__mono__iff,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) )
      = ( ! [X5: produc9003781588162623004on_val] :
            ( ( P @ X5 )
           => ( Q @ X5 ) ) ) ) ).

% Collect_mono_iff
thf(fact_247_Collect__mono__iff,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ( ord_le2494988322063910608_state @ ( collect_state @ P ) @ ( collect_state @ Q ) )
      = ( ! [X5: state] :
            ( ( P @ X5 )
           => ( Q @ X5 ) ) ) ) ).

% Collect_mono_iff
thf(fact_248_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_249_core__def,axiom,
    ( core
    = ( ^ [Phi3: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi3 ) ) ) ) ) ).

% core_def
thf(fact_250_PartialSA_Osplus_Opelims,axiom,
    ! [X3: option_state,Xa2: option_state,Y: option_state] :
      ( ( ( sep_splus_state @ plus @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X3 @ Xa2 ) )
       => ( ( ( X3 = none_state )
           => ( ( Y = none_state )
             => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
         => ( ! [V2: state] :
                ( ( X3
                  = ( some_state @ V2 ) )
               => ( ( Xa2 = none_state )
                 => ( ( Y = none_state )
                   => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) ) ) ) )
           => ~ ! [A3: state] :
                  ( ( X3
                    = ( some_state @ A3 ) )
                 => ! [B7: state] :
                      ( ( Xa2
                        = ( some_state @ B7 ) )
                     => ( ( Y
                          = ( plus @ A3 @ B7 ) )
                       => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.splus.pelims
thf(fact_251_PartialSA_Osmaller__pure__sum__smaller,axiom,
    ! [Y: state,A: state,B: state,X3: state] :
      ( ( greater @ Y @ A )
     => ( ( greater @ Y @ B )
       => ( ( ( some_state @ X3 )
            = ( plus @ A @ B ) )
         => ( ( sep_pure_state @ plus @ B )
           => ( greater @ Y @ X3 ) ) ) ) ) ).

% PartialSA.smaller_pure_sum_smaller
thf(fact_252_unit__charact_I1_J,axiom,
    ( ( get_h @ unit )
    = empty_7512026568161686585ar_val ) ).

% unit_charact(1)
thf(fact_253_PartialSA_Omono__propI,axiom,
    ! [P: state > $o] :
      ( ! [X: state,Y3: state] :
          ( ( ( greater @ Y3 @ X )
            & ( P @ X ) )
         => ( P @ Y3 ) )
     => ( sep_mono_prop_state @ plus @ P ) ) ).

% PartialSA.mono_propI
thf(fact_254_PartialSA_Omono__prop__def,axiom,
    ! [P: state > $o] :
      ( ( sep_mono_prop_state @ plus @ P )
      = ( ! [X5: state,Y4: state] :
            ( ( ( greater @ Y4 @ X5 )
              & ( P @ X5 ) )
           => ( P @ Y4 ) ) ) ) ).

% PartialSA.mono_prop_def
thf(fact_255_PartialSA_Ointuitionistic__def,axiom,
    ! [A2: state > $o] :
      ( ( packag8361946002163212404_state @ plus @ A2 )
      = ( ! [Phi5: state,Phi3: state] :
            ( ( ( greater @ Phi5 @ Phi3 )
              & ( A2 @ Phi3 ) )
           => ( A2 @ Phi5 ) ) ) ) ).

% PartialSA.intuitionistic_def
thf(fact_256_add__masks__equiv__valid__null,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( valid_null @ ( add_ma6586698345353345811t_char @ A @ B ) )
      = ( ( valid_null @ A )
        & ( valid_null @ B ) ) ) ).

% add_masks_equiv_valid_null
thf(fact_257_sep__algebra_Opure_Ocong,axiom,
    sep_pure_state = sep_pure_state ).

% sep_algebra.pure.cong
thf(fact_258_sep__algebra_Omono__prop_Ocong,axiom,
    sep_mono_prop_state = sep_mono_prop_state ).

% sep_algebra.mono_prop.cong
thf(fact_259_PartialSA_Ounit__core,axiom,
    ( ( core @ unit )
    = unit ) ).

% PartialSA.unit_core
thf(fact_260_PartialSA_Ocore__mono,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( greater @ ( core @ A ) @ ( core @ B ) ) ) ).

% PartialSA.core_mono
thf(fact_261_core__defined_I1_J,axiom,
    ! [Phi: state] :
      ( ( get_h @ ( core @ Phi ) )
      = ( get_h @ Phi ) ) ).

% core_defined(1)
thf(fact_262_PartialSA_Ounit__smaller,axiom,
    ! [Phi: state] : ( greater @ Phi @ unit ) ).

% PartialSA.unit_smaller
thf(fact_263_core__max,axiom,
    ! [X3: state,C: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ X3 @ C ) )
     => ? [R3: state] :
          ( ( some_state @ ( core @ X3 ) )
          = ( plus @ C @ R3 ) ) ) ).

% core_max
thf(fact_264_core__sum,axiom,
    ! [C: state,A: state,B: state] :
      ( ( ( some_state @ C )
        = ( plus @ A @ B ) )
     => ( ( some_state @ ( core @ C ) )
        = ( plus @ ( core @ A ) @ ( core @ B ) ) ) ) ).

% core_sum
thf(fact_265_cancellative,axiom,
    ! [A: state,B: state,X3: state,Y: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ X3 ) )
     => ( ( ( some_state @ A )
          = ( plus @ B @ Y ) )
       => ( ( ( core @ X3 )
            = ( core @ Y ) )
         => ( X3 = Y ) ) ) ) ).

% cancellative
thf(fact_266_core__is__pure,axiom,
    ! [X3: state] :
      ( ( some_state @ ( core @ X3 ) )
      = ( plus @ ( core @ X3 ) @ ( core @ X3 ) ) ) ).

% core_is_pure
thf(fact_267_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X5: state] : ( plus @ X5 @ ( core @ X5 ) ) ) ) ).

% core_is_smaller
thf(fact_268_PartialSA_Osmaller__compatible__core,axiom,
    ! [Y: state,X3: state] :
      ( ( greater @ Y @ X3 )
     => ( defined @ X3 @ ( core @ Y ) ) ) ).

% PartialSA.smaller_compatible_core
thf(fact_269_core__defined_I2_J,axiom,
    ! [Phi: state] :
      ( ( get_m @ ( core @ Phi ) )
      = empty_3446695950879338768t_char ) ).

% core_defined(2)
thf(fact_270_PartialSA_Opure__def,axiom,
    ! [A: state] :
      ( ( sep_pure_state @ plus @ A )
      = ( ( some_state @ A )
        = ( plus @ A @ A ) ) ) ).

% PartialSA.pure_def
thf(fact_271_PartialSA_Opure__stable,axiom,
    ! [A: state,B: state,C: state] :
      ( ( sep_pure_state @ plus @ A )
     => ( ( sep_pure_state @ plus @ B )
       => ( ( ( some_state @ C )
            = ( plus @ A @ B ) )
         => ( sep_pure_state @ plus @ C ) ) ) ) ).

% PartialSA.pure_stable
thf(fact_272_PartialSA_Opure__smaller,axiom,
    ! [A: state,B: state] :
      ( ( sep_pure_state @ plus @ A )
     => ( ( greater @ A @ B )
       => ( sep_pure_state @ plus @ B ) ) ) ).

% PartialSA.pure_smaller
thf(fact_273_unit__neutral,axiom,
    ( some_state
    = ( ^ [A4: state] : ( plus @ A4 @ unit ) ) ) ).

% unit_neutral
thf(fact_274_PartialSA_Oextract__core,axiom,
    ! [B: state,A: state,X3: state] :
      ( ( ( ( some_state @ B )
          = ( plus @ A @ X3 ) )
        & ( greater @ X3 @ ( core @ B ) ) )
     => ( ( core @ X3 )
        = ( core @ B ) ) ) ).

% PartialSA.extract_core
thf(fact_275_PartialSA_Ominus__exists,axiom,
    ! [B: state,A: state] :
      ( ( greater @ B @ A )
     => ? [X: state] :
          ( ( ( some_state @ B )
            = ( plus @ A @ X ) )
          & ( greater @ X @ ( core @ B ) ) ) ) ).

% PartialSA.minus_exists
thf(fact_276_PartialSA_Ominus__unique,axiom,
    ! [B: state,A: state,X3: state,Y: state] :
      ( ( ( ( some_state @ B )
          = ( plus @ A @ X3 ) )
        & ( greater @ X3 @ ( core @ B ) ) )
     => ( ( ( ( some_state @ B )
            = ( plus @ A @ Y ) )
          & ( greater @ Y @ ( core @ B ) ) )
       => ( X3 = Y ) ) ) ).

% PartialSA.minus_unique
thf(fact_277_PartialSA_Osmaller__than__core,axiom,
    ! [Y: state,X3: state,Z2: state] :
      ( ( greater @ Y @ X3 )
     => ( ( ( some_state @ Z2 )
          = ( plus @ X3 @ ( core @ Y ) ) )
       => ( ( core @ Z2 )
          = ( core @ Y ) ) ) ) ).

% PartialSA.smaller_than_core
thf(fact_278_PartialSA_Obigger__core__sum__defined,axiom,
    ! [A: state,B: state] :
      ( ( greater @ ( core @ A ) @ B )
     => ( ( some_state @ A )
        = ( plus @ A @ B ) ) ) ).

% PartialSA.bigger_core_sum_defined
thf(fact_279_PartialSA_Ogreater__than__sum__exists,axiom,
    ! [A: state,B: state,B1: state,B22: state] :
      ( ( greater @ A @ B )
     => ( ( ( some_state @ B )
          = ( 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_280_unit__charact_I2_J,axiom,
    ( ( get_m @ unit )
    = empty_3446695950879338768t_char ) ).

% unit_charact(2)
thf(fact_281_compatible__options_Opelims_I2_J,axiom,
    ! [X3: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X3 @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa2 ) )
       => ( ! [A3: state] :
              ( ( X3
                = ( some_state @ A3 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) )
                   => ( A3 != B7 ) ) ) )
         => ( ( ( X3 = none_state )
             => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) )
           => ~ ( ( Xa2 = none_state )
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ none_state ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_282_compatible__options_Opelims_I2_J,axiom,
    ! [X3: option_val,Xa2: option_val] :
      ( ( compat7209469420100717861ns_val @ X3 @ Xa2 )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa2 ) )
       => ( ! [A3: val] :
              ( ( X3
                = ( some_val @ A3 ) )
             => ! [B7: val] :
                  ( ( Xa2
                    = ( some_val @ B7 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) )
                   => ( A3 != B7 ) ) ) )
         => ( ( ( X3 = none_val )
             => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa2 ) ) )
           => ~ ( ( Xa2 = none_val )
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_283_compatible__options_Opelims_I1_J,axiom,
    ! [X3: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa2 ) )
       => ( ! [A3: state] :
              ( ( X3
                = ( some_state @ A3 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( Y
                      = ( A3 = B7 ) )
                   => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) ) ) ) )
         => ( ( ( X3 = none_state )
             => ( Y
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
           => ~ ( ( Xa2 = none_state )
               => ( Y
                 => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ none_state ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_284_compatible__options_Opelims_I1_J,axiom,
    ! [X3: option_val,Xa2: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa2 ) )
       => ( ! [A3: val] :
              ( ( X3
                = ( some_val @ A3 ) )
             => ! [B7: val] :
                  ( ( Xa2
                    = ( some_val @ B7 ) )
                 => ( ( Y
                      = ( A3 = B7 ) )
                   => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) ) ) ) )
         => ( ( ( X3 = none_val )
             => ( Y
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa2 ) ) ) )
           => ~ ( ( Xa2 = none_val )
               => ( Y
                 => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ none_val ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_285_compatible__options_Opelims_I3_J,axiom,
    ! [X3: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X3 @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X3 @ Xa2 ) )
       => ~ ! [A3: state] :
              ( ( X3
                = ( some_state @ A3 ) )
             => ! [B7: state] :
                  ( ( Xa2
                    = ( some_state @ B7 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) )
                   => ( A3 = B7 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_286_compatible__options_Opelims_I3_J,axiom,
    ! [X3: option_val,Xa2: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X3 @ Xa2 )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X3 @ Xa2 ) )
       => ~ ! [A3: val] :
              ( ( X3
                = ( some_val @ A3 ) )
             => ! [B7: val] :
                  ( ( Xa2
                    = ( some_val @ B7 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) )
                   => ( A3 = B7 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_287_subrelI,axiom,
    ! [R4: set_Pr4231240917283394591_state,S: set_Pr4231240917283394591_state] :
      ( ! [X: state,Y3: produc3882902684156697774_state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ R4 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X @ Y3 ) @ S ) )
     => ( ord_le4032880777312351359_state @ R4 @ S ) ) ).

% subrelI
thf(fact_288_subrelI,axiom,
    ! [R4: set_Pr2391701600042252046_state,S: set_Pr2391701600042252046_state] :
      ( ! [X: state,Y3: state > state] :
          ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ R4 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X @ Y3 ) @ S ) )
     => ( ord_le6483787752022044334_state @ R4 @ S ) ) ).

% subrelI
thf(fact_289_subrelI,axiom,
    ! [R4: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ! [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ R4 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y3 ) @ S ) )
     => ( ord_le7224449410216889241on_val @ R4 @ S ) ) ).

% subrelI
thf(fact_290_subrelI,axiom,
    ! [R4: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
      ( ! [X: option_state,Y3: option_state] :
          ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ R4 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X @ Y3 ) @ S ) )
     => ( ord_le6423325748750870005_state @ R4 @ S ) ) ).

% subrelI
thf(fact_291_subrelI,axiom,
    ! [R4: set_Pr6039890582625207499_state,S: set_Pr6039890582625207499_state] :
      ( ! [X: prat,Y3: state] :
          ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ R4 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X @ Y3 ) @ S ) )
     => ( ord_le262197048489888875_state @ R4 @ S ) ) ).

% subrelI
thf(fact_292_subrelI,axiom,
    ! [R4: set_Pr6847263425858009217_state,S: set_Pr6847263425858009217_state] :
      ( ! [X: prat,Y3: set_state] :
          ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ R4 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X @ Y3 ) @ S ) )
     => ( ord_le5761494056910127137_state @ R4 @ S ) ) ).

% subrelI
thf(fact_293_subrelI,axiom,
    ! [R4: set_Pr7649655576376376530on_val,S: set_Pr7649655576376376530on_val] :
      ( ! [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ R4 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y3 ) @ S ) )
     => ( ord_le6900622908513426738on_val @ R4 @ S ) ) ).

% subrelI
thf(fact_294_sep__algebra_Osplus_Opelims,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: option_val,Xa2: option_val,Y: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( sep_splus_val @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ X3 @ Xa2 ) )
         => ( ( ( X3 = none_val )
             => ( ( Y = none_val )
               => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ none_val @ Xa2 ) ) ) )
           => ( ! [V2: val] :
                  ( ( X3
                    = ( some_val @ V2 ) )
                 => ( ( Xa2 = none_val )
                   => ( ( Y = none_val )
                     => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ ( some_val @ V2 ) @ none_val ) ) ) ) )
             => ~ ! [A3: val] :
                    ( ( X3
                      = ( some_val @ A3 ) )
                   => ! [B7: val] :
                        ( ( Xa2
                          = ( some_val @ B7 ) )
                       => ( ( Y
                            = ( Plus @ A3 @ B7 ) )
                         => ~ ( accp_P4269981079083053040on_val @ sep_splus_rel_val @ ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.pelims
thf(fact_295_sep__algebra_Osplus_Opelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: option_state,Xa2: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ X3 @ Xa2 ) )
         => ( ( ( X3 = none_state )
             => ( ( Y = none_state )
               => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
           => ( ! [V2: state] :
                  ( ( X3
                    = ( some_state @ V2 ) )
                 => ( ( Xa2 = none_state )
                   => ( ( Y = none_state )
                     => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) ) ) ) )
             => ~ ! [A3: state] :
                    ( ( X3
                      = ( some_state @ A3 ) )
                   => ! [B7: state] :
                        ( ( Xa2
                          = ( some_state @ B7 ) )
                       => ( ( Y
                            = ( Plus @ A3 @ B7 ) )
                         => ~ ( accp_P7183347322223220350_state @ sep_splus_rel_state @ ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.pelims
thf(fact_296_PartialSA_Omono__pure__condI,axiom,
    ! [B: state > $o] :
      ( ! [Phi4: state] :
          ( ( B @ Phi4 )
          = ( B @ ( core @ Phi4 ) ) )
     => ( ! [Phi4: state,Phi6: state,R3: state] :
            ( ( ( sep_pure_state @ plus @ R3 )
              & ( ( some_state @ Phi6 )
                = ( plus @ Phi4 @ R3 ) )
              & ~ ( B @ Phi4 ) )
           => ~ ( B @ Phi6 ) )
       => ( packag6595153354952283300_state @ plus @ core @ B ) ) ) ).

% PartialSA.mono_pure_condI
thf(fact_297_PartialSA_Omono__pure__cond__def,axiom,
    ! [B: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ B )
      = ( ! [Phi3: state] :
            ( ( B @ Phi3 )
            = ( B @ ( core @ Phi3 ) ) )
        & ! [Phi5: state,Phi3: state,R2: state] :
            ( ( ( sep_pure_state @ plus @ R2 )
              & ( ( some_state @ Phi5 )
                = ( plus @ Phi3 @ R2 ) )
              & ~ ( B @ Phi3 ) )
           => ~ ( B @ Phi5 ) ) ) ) ).

% PartialSA.mono_pure_cond_def
thf(fact_298_PartialSA_Omono__transformer__def,axiom,
    ! [T2: state > state] :
      ( ( packag4817583915586052229_state @ plus @ unit @ T2 )
      = ( ! [Phi3: state,Phi5: state] :
            ( ( greater @ Phi5 @ Phi3 )
           => ( greater @ ( T2 @ Phi5 ) @ ( T2 @ Phi3 ) ) )
        & ( ( T2 @ unit )
          = unit ) ) ) ).

% PartialSA.mono_transformer_def
thf(fact_299_sep__algebra_Ocommutative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Plus @ A @ B )
        = ( Plus @ B @ A ) ) ) ).

% sep_algebra.commutative
thf(fact_300_sep__algebra_Oasso1,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,Ab: val,C: val,Bc: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_val @ Ab ) )
          & ( ( Plus @ B @ C )
            = ( some_val @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A @ Bc ) ) ) ) ).

% sep_algebra.asso1
thf(fact_301_sep__algebra_Oasso1,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,Ab: state,C: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B @ C )
            = ( some_state @ Bc ) ) )
       => ( ( Plus @ Ab @ C )
          = ( Plus @ A @ Bc ) ) ) ) ).

% sep_algebra.asso1
thf(fact_302_sep__algebra_Ocore__max,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X3 )
          = ( Plus @ X3 @ C ) )
       => ? [R3: val] :
            ( ( some_val @ ( Core @ X3 ) )
            = ( Plus @ C @ R3 ) ) ) ) ).

% sep_algebra.core_max
thf(fact_303_sep__algebra_Ocore__max,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X3 )
          = ( Plus @ X3 @ C ) )
       => ? [R3: state] :
            ( ( some_state @ ( Core @ X3 ) )
            = ( Plus @ C @ R3 ) ) ) ) ).

% sep_algebra.core_max
thf(fact_304_sep__algebra_Ocore__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,C: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ C )
          = ( Plus @ A @ B ) )
       => ( ( some_val @ ( Core @ C ) )
          = ( Plus @ ( Core @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.core_sum
thf(fact_305_sep__algebra_Ocore__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C: state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C )
          = ( Plus @ A @ B ) )
       => ( ( some_state @ ( Core @ C ) )
          = ( Plus @ ( Core @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.core_sum
thf(fact_306_sep__algebra_Opositivity,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = ( some_val @ C ) )
       => ( ( ( some_val @ C )
            = ( Plus @ C @ C ) )
         => ( ( some_val @ A )
            = ( Plus @ A @ A ) ) ) ) ) ).

% sep_algebra.positivity
thf(fact_307_sep__algebra_Opositivity,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = ( some_state @ C ) )
       => ( ( ( some_state @ C )
            = ( Plus @ C @ C ) )
         => ( ( some_state @ A )
            = ( Plus @ A @ A ) ) ) ) ) ).

% sep_algebra.positivity
thf(fact_308_sep__algebra_Ocancellative,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,X3: val,Y: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ X3 ) )
       => ( ( ( some_val @ A )
            = ( Plus @ B @ Y ) )
         => ( ( ( Core @ X3 )
              = ( Core @ Y ) )
           => ( X3 = Y ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_309_sep__algebra_Ocancellative,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,X3: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B @ X3 ) )
       => ( ( ( some_state @ A )
            = ( Plus @ B @ Y ) )
         => ( ( ( Core @ X3 )
              = ( Core @ Y ) )
           => ( X3 = Y ) ) ) ) ) ).

% sep_algebra.cancellative
thf(fact_310_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( some_val @ ( Core @ X3 ) )
        = ( Plus @ ( Core @ X3 ) @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_311_sep__algebra_Ocore__is__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ ( Core @ X3 ) )
        = ( Plus @ ( Core @ X3 ) @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_pure
thf(fact_312_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( some_val @ X3 )
        = ( Plus @ X3 @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_313_sep__algebra_Ocore__is__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( some_state @ X3 )
        = ( Plus @ X3 @ ( Core @ X3 ) ) ) ) ).

% sep_algebra.core_is_smaller
thf(fact_314_sep__algebra_Osplus__asso,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: option_state,B: option_state,C: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ A @ B ) @ C )
        = ( sep_splus_state @ Plus @ A @ ( sep_splus_state @ Plus @ B @ C ) ) ) ) ).

% sep_algebra.splus_asso
thf(fact_315_sep__algebra_Osplus__comm,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: option_state,B: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ A @ B )
        = ( sep_splus_state @ Plus @ B @ A ) ) ) ).

% sep_algebra.splus_comm
thf(fact_316_sep__algebra__def,axiom,
    ( sep_algebra_val
    = ( ^ [Plus2: val > val > option_val,Core2: val > val] :
          ( ! [A4: val,B3: val] :
              ( ( Plus2 @ A4 @ B3 )
              = ( Plus2 @ B3 @ A4 ) )
          & ! [A4: val,B3: val,Ab2: val,C3: val,Bc2: val] :
              ( ( ( ( Plus2 @ A4 @ B3 )
                  = ( some_val @ Ab2 ) )
                & ( ( Plus2 @ B3 @ C3 )
                  = ( some_val @ Bc2 ) ) )
             => ( ( Plus2 @ Ab2 @ C3 )
                = ( Plus2 @ A4 @ Bc2 ) ) )
          & ! [A4: val,B3: val,Ab2: val,C3: val] :
              ( ( ( ( Plus2 @ A4 @ B3 )
                  = ( some_val @ Ab2 ) )
                & ( ( Plus2 @ B3 @ C3 )
                  = none_val ) )
             => ( ( Plus2 @ Ab2 @ C3 )
                = none_val ) )
          & ! [X5: val] :
              ( ( some_val @ X5 )
              = ( Plus2 @ X5 @ ( Core2 @ X5 ) ) )
          & ! [X5: val] :
              ( ( some_val @ ( Core2 @ X5 ) )
              = ( Plus2 @ ( Core2 @ X5 ) @ ( Core2 @ X5 ) ) )
          & ! [X5: val,C3: val] :
              ( ( ( some_val @ X5 )
                = ( Plus2 @ X5 @ C3 ) )
             => ? [R2: val] :
                  ( ( some_val @ ( Core2 @ X5 ) )
                  = ( Plus2 @ C3 @ R2 ) ) )
          & ! [C3: val,A4: val,B3: val] :
              ( ( ( some_val @ C3 )
                = ( Plus2 @ A4 @ B3 ) )
             => ( ( some_val @ ( Core2 @ C3 ) )
                = ( Plus2 @ ( Core2 @ A4 ) @ ( Core2 @ B3 ) ) ) )
          & ! [A4: val,B3: val,C3: val] :
              ( ( ( Plus2 @ A4 @ B3 )
                = ( some_val @ C3 ) )
             => ( ( ( some_val @ C3 )
                  = ( Plus2 @ C3 @ C3 ) )
               => ( ( some_val @ A4 )
                  = ( Plus2 @ A4 @ A4 ) ) ) )
          & ! [A4: val,B3: val,X5: val,Y4: val] :
              ( ( ( some_val @ A4 )
                = ( Plus2 @ B3 @ X5 ) )
             => ( ( ( some_val @ A4 )
                  = ( Plus2 @ B3 @ Y4 ) )
               => ( ( ( Core2 @ X5 )
                    = ( Core2 @ Y4 ) )
                 => ( X5 = Y4 ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_317_sep__algebra__def,axiom,
    ( sep_algebra_state
    = ( ^ [Plus2: state > state > option_state,Core2: state > state] :
          ( ! [A4: state,B3: state] :
              ( ( Plus2 @ A4 @ B3 )
              = ( Plus2 @ B3 @ A4 ) )
          & ! [A4: state,B3: state,Ab2: state,C3: state,Bc2: state] :
              ( ( ( ( Plus2 @ A4 @ B3 )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus2 @ B3 @ C3 )
                  = ( some_state @ Bc2 ) ) )
             => ( ( Plus2 @ Ab2 @ C3 )
                = ( Plus2 @ A4 @ Bc2 ) ) )
          & ! [A4: state,B3: state,Ab2: state,C3: state] :
              ( ( ( ( Plus2 @ A4 @ B3 )
                  = ( some_state @ Ab2 ) )
                & ( ( Plus2 @ B3 @ C3 )
                  = none_state ) )
             => ( ( Plus2 @ Ab2 @ C3 )
                = none_state ) )
          & ! [X5: state] :
              ( ( some_state @ X5 )
              = ( Plus2 @ X5 @ ( Core2 @ X5 ) ) )
          & ! [X5: state] :
              ( ( some_state @ ( Core2 @ X5 ) )
              = ( Plus2 @ ( Core2 @ X5 ) @ ( Core2 @ X5 ) ) )
          & ! [X5: state,C3: state] :
              ( ( ( some_state @ X5 )
                = ( Plus2 @ X5 @ C3 ) )
             => ? [R2: state] :
                  ( ( some_state @ ( Core2 @ X5 ) )
                  = ( Plus2 @ C3 @ R2 ) ) )
          & ! [C3: state,A4: state,B3: state] :
              ( ( ( some_state @ C3 )
                = ( Plus2 @ A4 @ B3 ) )
             => ( ( some_state @ ( Core2 @ C3 ) )
                = ( Plus2 @ ( Core2 @ A4 ) @ ( Core2 @ B3 ) ) ) )
          & ! [A4: state,B3: state,C3: state] :
              ( ( ( Plus2 @ A4 @ B3 )
                = ( some_state @ C3 ) )
             => ( ( ( some_state @ C3 )
                  = ( Plus2 @ C3 @ C3 ) )
               => ( ( some_state @ A4 )
                  = ( Plus2 @ A4 @ A4 ) ) ) )
          & ! [A4: state,B3: state,X5: state,Y4: state] :
              ( ( ( some_state @ A4 )
                = ( Plus2 @ B3 @ X5 ) )
             => ( ( ( some_state @ A4 )
                  = ( Plus2 @ B3 @ Y4 ) )
               => ( ( ( Core2 @ X5 )
                    = ( Core2 @ Y4 ) )
                 => ( X5 = Y4 ) ) ) ) ) ) ) ).

% sep_algebra_def
thf(fact_318_sep__algebra_Ointro,axiom,
    ! [Plus: val > val > option_val,Core: val > val] :
      ( ! [A3: val,B7: val] :
          ( ( Plus @ A3 @ B7 )
          = ( Plus @ B7 @ A3 ) )
     => ( ! [A3: val,B7: val,Ab3: val,C4: val,Bc3: val] :
            ( ( ( ( Plus @ A3 @ B7 )
                = ( some_val @ Ab3 ) )
              & ( ( Plus @ B7 @ C4 )
                = ( some_val @ Bc3 ) ) )
           => ( ( Plus @ Ab3 @ C4 )
              = ( Plus @ A3 @ Bc3 ) ) )
       => ( ! [A3: val,B7: val,Ab3: val,C4: val] :
              ( ( ( ( Plus @ A3 @ B7 )
                  = ( some_val @ Ab3 ) )
                & ( ( Plus @ B7 @ C4 )
                  = none_val ) )
             => ( ( Plus @ Ab3 @ C4 )
                = none_val ) )
         => ( ! [X: val] :
                ( ( some_val @ X )
                = ( Plus @ X @ ( Core @ X ) ) )
           => ( ! [X: val] :
                  ( ( some_val @ ( Core @ X ) )
                  = ( Plus @ ( Core @ X ) @ ( Core @ X ) ) )
             => ( ! [X: val,C4: val] :
                    ( ( ( some_val @ X )
                      = ( Plus @ X @ C4 ) )
                   => ? [R5: val] :
                        ( ( some_val @ ( Core @ X ) )
                        = ( Plus @ C4 @ R5 ) ) )
               => ( ! [C4: val,A3: val,B7: val] :
                      ( ( ( some_val @ C4 )
                        = ( Plus @ A3 @ B7 ) )
                     => ( ( some_val @ ( Core @ C4 ) )
                        = ( Plus @ ( Core @ A3 ) @ ( Core @ B7 ) ) ) )
                 => ( ! [A3: val,B7: val,C4: val] :
                        ( ( ( Plus @ A3 @ B7 )
                          = ( some_val @ C4 ) )
                       => ( ( ( some_val @ C4 )
                            = ( Plus @ C4 @ C4 ) )
                         => ( ( some_val @ A3 )
                            = ( Plus @ A3 @ A3 ) ) ) )
                   => ( ! [A3: val,B7: val,X: val,Y3: val] :
                          ( ( ( some_val @ A3 )
                            = ( Plus @ B7 @ X ) )
                         => ( ( ( some_val @ A3 )
                              = ( Plus @ B7 @ Y3 ) )
                           => ( ( ( Core @ X )
                                = ( Core @ Y3 ) )
                             => ( X = Y3 ) ) ) )
                     => ( sep_algebra_val @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_319_sep__algebra_Ointro,axiom,
    ! [Plus: state > state > option_state,Core: state > state] :
      ( ! [A3: state,B7: state] :
          ( ( Plus @ A3 @ B7 )
          = ( Plus @ B7 @ A3 ) )
     => ( ! [A3: state,B7: state,Ab3: state,C4: state,Bc3: state] :
            ( ( ( ( Plus @ A3 @ B7 )
                = ( some_state @ Ab3 ) )
              & ( ( Plus @ B7 @ C4 )
                = ( some_state @ Bc3 ) ) )
           => ( ( Plus @ Ab3 @ C4 )
              = ( Plus @ A3 @ Bc3 ) ) )
       => ( ! [A3: state,B7: state,Ab3: state,C4: state] :
              ( ( ( ( Plus @ A3 @ B7 )
                  = ( some_state @ Ab3 ) )
                & ( ( Plus @ B7 @ C4 )
                  = none_state ) )
             => ( ( Plus @ Ab3 @ C4 )
                = none_state ) )
         => ( ! [X: state] :
                ( ( some_state @ X )
                = ( Plus @ X @ ( Core @ X ) ) )
           => ( ! [X: state] :
                  ( ( some_state @ ( Core @ X ) )
                  = ( Plus @ ( Core @ X ) @ ( Core @ X ) ) )
             => ( ! [X: state,C4: state] :
                    ( ( ( some_state @ X )
                      = ( Plus @ X @ C4 ) )
                   => ? [R5: state] :
                        ( ( some_state @ ( Core @ X ) )
                        = ( Plus @ C4 @ R5 ) ) )
               => ( ! [C4: state,A3: state,B7: state] :
                      ( ( ( some_state @ C4 )
                        = ( Plus @ A3 @ B7 ) )
                     => ( ( some_state @ ( Core @ C4 ) )
                        = ( Plus @ ( Core @ A3 ) @ ( Core @ B7 ) ) ) )
                 => ( ! [A3: state,B7: state,C4: state] :
                        ( ( ( Plus @ A3 @ B7 )
                          = ( some_state @ C4 ) )
                       => ( ( ( some_state @ C4 )
                            = ( Plus @ C4 @ C4 ) )
                         => ( ( some_state @ A3 )
                            = ( Plus @ A3 @ A3 ) ) ) )
                   => ( ! [A3: state,B7: state,X: state,Y3: state] :
                          ( ( ( some_state @ A3 )
                            = ( Plus @ B7 @ X ) )
                         => ( ( ( some_state @ A3 )
                              = ( Plus @ B7 @ Y3 ) )
                           => ( ( ( Core @ X )
                                = ( Core @ Y3 ) )
                             => ( X = Y3 ) ) ) )
                     => ( sep_algebra_state @ Plus @ Core ) ) ) ) ) ) ) ) ) ) ).

% sep_algebra.intro
thf(fact_320_sep__algebra_Oasso3,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,Bc: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = none_val )
       => ( ( ( Plus @ B @ C )
            = ( some_val @ Bc ) )
         => ( ( Plus @ A @ Bc )
            = none_val ) ) ) ) ).

% sep_algebra.asso3
thf(fact_321_sep__algebra_Oasso3,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,Bc: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( Plus @ A @ B )
          = none_state )
       => ( ( ( Plus @ B @ C )
            = ( some_state @ Bc ) )
         => ( ( Plus @ A @ Bc )
            = none_state ) ) ) ) ).

% sep_algebra.asso3
thf(fact_322_sep__algebra_Oasso2,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,Ab: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_val @ Ab ) )
          & ( ( Plus @ B @ C )
            = none_val ) )
       => ( ( Plus @ Ab @ C )
          = none_val ) ) ) ).

% sep_algebra.asso2
thf(fact_323_sep__algebra_Oasso2,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,Ab: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( Plus @ A @ B )
            = ( some_state @ Ab ) )
          & ( ( Plus @ B @ C )
            = none_state ) )
       => ( ( Plus @ Ab @ C )
          = none_state ) ) ) ).

% sep_algebra.asso2
thf(fact_324_PartialSA_Osep__algebra__axioms,axiom,
    sep_algebra_state @ plus @ core ).

% PartialSA.sep_algebra_axioms
thf(fact_325_sep__algebra_Osplus__develop,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,D: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ C ) )
       => ( ( Plus @ A @ D )
          = ( sep_splus_val @ Plus @ ( sep_splus_val @ Plus @ ( some_val @ B ) @ ( some_val @ C ) ) @ ( some_val @ D ) ) ) ) ) ).

% sep_algebra.splus_develop
thf(fact_326_sep__algebra_Osplus__develop,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,D: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B @ C ) )
       => ( ( Plus @ A @ D )
          = ( sep_splus_state @ Plus @ ( sep_splus_state @ Plus @ ( some_state @ B ) @ ( some_state @ C ) ) @ ( some_state @ D ) ) ) ) ) ).

% sep_algebra.splus_develop
thf(fact_327_sep__algebra_Osplus_Osimps_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ ( some_val @ A ) @ ( some_val @ B ) )
        = ( Plus @ A @ B ) ) ) ).

% sep_algebra.splus.simps(3)
thf(fact_328_sep__algebra_Osplus_Osimps_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ A ) @ ( some_state @ B ) )
        = ( Plus @ A @ B ) ) ) ).

% sep_algebra.splus.simps(3)
thf(fact_329_sep__algebra_Opure__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_pure_val @ Plus @ A )
        = ( ( some_val @ A )
          = ( Plus @ A @ A ) ) ) ) ).

% sep_algebra.pure_def
thf(fact_330_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_331_sep__algebra_Opure__stable,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_pure_val @ Plus @ A )
       => ( ( sep_pure_val @ Plus @ B )
         => ( ( ( some_val @ C )
              = ( Plus @ A @ B ) )
           => ( sep_pure_val @ Plus @ C ) ) ) ) ) ).

% sep_algebra.pure_stable
thf(fact_332_sep__algebra_Opure__stable,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A )
       => ( ( sep_pure_state @ Plus @ B )
         => ( ( ( some_state @ C )
              = ( Plus @ A @ B ) )
           => ( sep_pure_state @ Plus @ C ) ) ) ) ) ).

% sep_algebra.pure_stable
thf(fact_333_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Uu: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ none_val @ Uu )
        = none_val ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_334_sep__algebra_Osplus_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Uu: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ none_state @ Uu )
        = none_state ) ) ).

% sep_algebra.splus.simps(1)
thf(fact_335_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: option_val,Xa2: option_val,Y: option_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( sep_splus_val @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( ( X3 = none_val )
           => ( Y != none_val ) )
         => ( ( ? [V2: val] :
                  ( X3
                  = ( some_val @ V2 ) )
             => ( ( Xa2 = none_val )
               => ( Y != none_val ) ) )
           => ~ ! [A3: val] :
                  ( ( X3
                    = ( some_val @ A3 ) )
                 => ! [B7: val] :
                      ( ( Xa2
                        = ( some_val @ B7 ) )
                     => ( Y
                       != ( Plus @ A3 @ B7 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_336_sep__algebra_Osplus_Oelims,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: option_state,Xa2: option_state,Y: option_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( sep_splus_state @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( ( X3 = none_state )
           => ( Y != none_state ) )
         => ( ( ? [V2: state] :
                  ( X3
                  = ( some_state @ V2 ) )
             => ( ( Xa2 = none_state )
               => ( Y != none_state ) ) )
           => ~ ! [A3: state] :
                  ( ( X3
                    = ( some_state @ A3 ) )
                 => ! [B7: state] :
                      ( ( Xa2
                        = ( some_state @ B7 ) )
                     => ( Y
                       != ( Plus @ A3 @ B7 ) ) ) ) ) ) ) ) ).

% sep_algebra.splus.elims
thf(fact_337_sep__algebra_Osplus_Osimps_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,V: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_splus_val @ Plus @ ( some_val @ V ) @ none_val )
        = none_val ) ) ).

% sep_algebra.splus.simps(2)
thf(fact_338_sep__algebra_Osplus_Osimps_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,V: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_splus_state @ Plus @ ( some_state @ V ) @ none_state )
        = none_state ) ) ).

% sep_algebra.splus.simps(2)
thf(fact_339_sep__algebra_Osplus_Ocases,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: produc5122021904355189735on_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ! [Uu2: option_val] :
            ( X3
           != ( produc904852209897547607on_val @ none_val @ Uu2 ) )
       => ( ! [V2: val] :
              ( X3
             != ( produc904852209897547607on_val @ ( some_val @ V2 ) @ none_val ) )
         => ~ ! [A3: val,B7: val] :
                ( X3
               != ( produc904852209897547607on_val @ ( some_val @ A3 ) @ ( some_val @ B7 ) ) ) ) ) ) ).

% sep_algebra.splus.cases
thf(fact_340_sep__algebra_Osplus_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: produc3142500478612311029_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [Uu2: option_state] :
            ( X3
           != ( produc9160152616361873709_state @ none_state @ Uu2 ) )
       => ( ! [V2: state] :
              ( X3
             != ( produc9160152616361873709_state @ ( some_state @ V2 ) @ none_state ) )
         => ~ ! [A3: state,B7: state] :
                ( X3
               != ( produc9160152616361873709_state @ ( some_state @ A3 ) @ ( some_state @ B7 ) ) ) ) ) ) ).

% sep_algebra.splus.cases
thf(fact_341_PartialSA_Ovalid__package__setI,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state] :
      ( ! [A3: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( defined @ A3 @ U )
            & ( greater @ ( core @ A3 ) @ ( core @ U ) )
            & ( packag4817583915586052229_state @ plus @ unit @ T3 )
            & ( greater @ A3 @ ( core @ ( T3 @ F ) ) ) ) )
     => ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F ) ) ).

% PartialSA.valid_package_setI
thf(fact_342_PartialSA_Omono__pure__cond__conj,axiom,
    ! [Pc: state > $o,B: state > $o] :
      ( ( packag6595153354952283300_state @ plus @ core @ Pc )
     => ( ( packag6595153354952283300_state @ plus @ core @ B )
       => ( packag6595153354952283300_state @ plus @ core @ ( packag1330488657391168505_state @ Pc @ B ) ) ) ) ).

% PartialSA.mono_pure_cond_conj
thf(fact_343_PartialSA_Ominus__equiv__def,axiom,
    ! [B: state,A: state] :
      ( ( greater @ B @ A )
     => ( ( ( some_state @ B )
          = ( plus @ A @ ( minus @ B @ A ) ) )
        & ( greater @ ( minus @ B @ A ) @ ( core @ B ) ) ) ) ).

% PartialSA.minus_equiv_def
thf(fact_344_PartialSA_OminusI,axiom,
    ! [B: state,A: state,X3: state] :
      ( ( ( some_state @ B )
        = ( plus @ A @ X3 ) )
     => ( ( greater @ X3 @ ( core @ B ) )
       => ( X3
          = ( minus @ B @ A ) ) ) ) ).

% PartialSA.minusI
thf(fact_345_PartialSA_Omono__pruner__def,axiom,
    ! [P2: state > $o] :
      ( ( packag2456304381420842418_state @ plus @ P2 )
      = ( ! [Phi5: state,Phi3: state,R2: state] :
            ( ( ( sep_pure_state @ plus @ R2 )
              & ( P2 @ Phi3 )
              & ( ( some_state @ Phi5 )
                = ( plus @ Phi3 @ R2 ) ) )
           => ( P2 @ Phi5 ) ) ) ) ).

% PartialSA.mono_pruner_def
thf(fact_346_PartialSA_Omax__projection__prop__pure__core,axiom,
    sep_ma8214210560313151521_state @ plus @ ( sep_pure_state @ plus ) @ core ).

% PartialSA.max_projection_prop_pure_core
thf(fact_347_PartialSA_Obigger__the,axiom,
    ! [A: state,X6: state,Y: state,X3: state] :
      ( ( ( some_state @ A )
        = ( plus @ X6 @ Y ) )
     => ( ( greater @ X6 @ X3 )
       => ( greater @ ( the_state @ ( plus @ ( core @ A ) @ X6 ) ) @ ( the_state @ ( plus @ ( core @ A ) @ X3 ) ) ) ) ) ).

% PartialSA.bigger_the
thf(fact_348_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_349_option_Ocollapse,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( ( some_val @ ( the_val @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_350_sep__algebra_Omax__projection__prop_Ocong,axiom,
    sep_ma8214210560313151521_state = sep_ma8214210560313151521_state ).

% sep_algebra.max_projection_prop.cong
thf(fact_351_PartialSA_Obool__conj__def,axiom,
    ( packag1330488657391168505_state
    = ( ^ [A4: state > $o,B3: state > $o,X5: state] :
          ( ( A4 @ X5 )
          & ( B3 @ X5 ) ) ) ) ).

% PartialSA.bool_conj_def
thf(fact_352_option_Osel,axiom,
    ! [X2: state] :
      ( ( the_state @ ( some_state @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_353_option_Osel,axiom,
    ! [X2: val] :
      ( ( the_val @ ( some_val @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_354_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_355_option_Oexpand,axiom,
    ! [Option: option_val,Option2: option_val] :
      ( ( ( Option = none_val )
        = ( Option2 = none_val ) )
     => ( ( ( Option != none_val )
         => ( ( Option2 != none_val )
           => ( ( the_val @ Option )
              = ( the_val @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_356_PartialSA_Ominus__core,axiom,
    ! [A: state,B: state] :
      ( ( core @ ( minus @ A @ B ) )
      = ( core @ A ) ) ).

% PartialSA.minus_core
thf(fact_357_PartialSA_Ominus__core__weaker,axiom,
    ! [A: state,B: state] :
      ( ( core @ ( minus @ A @ B ) )
      = ( minus @ ( core @ A ) @ ( core @ B ) ) ) ).

% PartialSA.minus_core_weaker
thf(fact_358_PartialSA_Ominus__default,axiom,
    ! [B: state,A: state] :
      ( ~ ( greater @ B @ A )
     => ( ( minus @ B @ A )
        = B ) ) ).

% PartialSA.minus_default
thf(fact_359_PartialSA_Ominus__smaller,axiom,
    ! [X3: state,A: state] :
      ( ( greater @ X3 @ A )
     => ( greater @ X3 @ ( minus @ X3 @ A ) ) ) ).

% PartialSA.minus_smaller
thf(fact_360_PartialSA_Ogreater__minus__trans,axiom,
    ! [Y: state,X3: state,A: state] :
      ( ( greater @ Y @ X3 )
     => ( ( greater @ X3 @ A )
       => ( greater @ ( minus @ Y @ A ) @ ( minus @ X3 @ A ) ) ) ) ).

% PartialSA.greater_minus_trans
thf(fact_361_PartialSA_Ompp__invo,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( F @ ( F @ X3 ) )
        = ( F @ X3 ) ) ) ).

% PartialSA.mpp_invo
thf(fact_362_PartialSA_Ompp__prop,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( P @ ( F @ X3 ) ) ) ).

% PartialSA.mpp_prop
thf(fact_363_sep__algebra_Ompp__invo,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( F @ ( F @ X3 ) )
          = ( F @ X3 ) ) ) ) ).

% sep_algebra.mpp_invo
thf(fact_364_sep__algebra_Ompp__prop,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( P @ ( F @ X3 ) ) ) ) ).

% sep_algebra.mpp_prop
thf(fact_365_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_366_option_Oexhaust__sel,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( Option
        = ( some_val @ ( the_val @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_367_PartialSA_Omax__projection__prop__def,axiom,
    ! [P: state > $o,F: state > state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
      = ( ! [X5: state] :
            ( ( greater @ X5 @ ( F @ X5 ) )
            & ( P @ ( F @ X5 ) )
            & ! [P3: state] :
                ( ( ( P @ P3 )
                  & ( greater @ X5 @ P3 ) )
               => ( greater @ ( F @ X5 ) @ P3 ) ) ) ) ) ).

% PartialSA.max_projection_prop_def
thf(fact_368_PartialSA_Omax__projection__propI,axiom,
    ! [F: state > state,P: state > $o] :
      ( ! [X: state] : ( greater @ X @ ( F @ X ) )
     => ( ! [X: state] : ( P @ ( F @ X ) )
       => ( ! [X: state,P4: state] :
              ( ( ( P @ P4 )
                & ( greater @ X @ P4 ) )
             => ( greater @ ( F @ X ) @ P4 ) )
         => ( sep_ma8214210560313151521_state @ plus @ P @ F ) ) ) ) ).

% PartialSA.max_projection_propI
thf(fact_369_PartialSA_Ompp__smaller,axiom,
    ! [P: state > $o,F: state > state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( greater @ X3 @ ( F @ X3 ) ) ) ).

% PartialSA.mpp_smaller
thf(fact_370_PartialSA_Ompp__mono,axiom,
    ! [P: state > $o,F: state > state,A: state,B: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A @ B )
       => ( greater @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% PartialSA.mpp_mono
thf(fact_371_PartialSA_OmppI,axiom,
    ! [P: state > $o,F: state > state,A: state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( greater @ A @ X3 )
       => ( ( P @ X3 )
         => ( ( greater @ X3 @ ( F @ A ) )
           => ( X3
              = ( F @ A ) ) ) ) ) ) ).

% PartialSA.mppI
thf(fact_372_PartialSA_Omax__projection__propE_I3_J,axiom,
    ! [P: state > $o,F: state > state,P2: state,X3: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( ( P @ P2 )
          & ( greater @ X3 @ P2 ) )
       => ( greater @ ( F @ X3 ) @ P2 ) ) ) ).

% PartialSA.max_projection_propE(3)
thf(fact_373_PartialSA_Ompp__compatible,axiom,
    ! [P: state > $o,F: state > state,A: state,B: state] :
      ( ( sep_ma8214210560313151521_state @ plus @ P @ F )
     => ( ( defined @ A @ B )
       => ( defined @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% PartialSA.mpp_compatible
thf(fact_374_PartialSA_Ominus__equiv__def__any__elem,axiom,
    ! [X3: state,A: state,B: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ A @ B ) )
     => ( ( some_state @ ( minus @ X3 @ A ) )
        = ( plus @ B @ ( core @ X3 ) ) ) ) ).

% PartialSA.minus_equiv_def_any_elem
thf(fact_375_PartialSA_Ominus__sum,axiom,
    ! [A: state,B: state,C: state,X3: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ C ) )
     => ( ( greater @ X3 @ A )
       => ( ( minus @ X3 @ A )
          = ( minus @ ( minus @ X3 @ B ) @ C ) ) ) ) ).

% PartialSA.minus_sum
thf(fact_376_PartialSA_Ominus__some,axiom,
    ! [A: state,B: state] :
      ( ( greater @ A @ B )
     => ( ( some_state @ A )
        = ( plus @ B @ ( minus @ A @ B ) ) ) ) ).

% PartialSA.minus_some
thf(fact_377_PartialSA_Ominus__bigger,axiom,
    ! [X3: state,A: state,B: state] :
      ( ( ( some_state @ X3 )
        = ( plus @ A @ B ) )
     => ( greater @ ( minus @ X3 @ A ) @ B ) ) ).

% PartialSA.minus_bigger
thf(fact_378_PartialSA_Ominus__and__plus,axiom,
    ! [Omega: state,Omega2: state,R4: state,A: state] :
      ( ( ( some_state @ Omega )
        = ( plus @ Omega2 @ R4 ) )
     => ( ( greater @ Omega2 @ A )
       => ( ( some_state @ ( minus @ Omega @ A ) )
          = ( plus @ ( minus @ Omega2 @ A ) @ R4 ) ) ) ) ).

% PartialSA.minus_and_plus
thf(fact_379_PartialSA_Oprove__last__completeness,axiom,
    ! [A6: state,A: state,Nf1: state,F2: state] :
      ( ( greater @ A6 @ A )
     => ( ( ( some_state @ A )
          = ( plus @ Nf1 @ F2 ) )
       => ( greater @ ( minus @ A6 @ Nf1 ) @ F2 ) ) ) ).

% PartialSA.prove_last_completeness
thf(fact_380_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_381_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ X3 @ ( sep_ad3908482799505143387_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc7272280185086996969_state,B3: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ A4 @ A2 )
              & ( member7461423308196221312_state @ B3 @ B2 )
              & ( ( some_P6297086604321900740_state @ X3 )
                = ( sep_sp9219505543755390023_state @ Plus @ ( some_P6297086604321900740_state @ A4 ) @ ( some_P6297086604321900740_state @ B3 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_382_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,X3: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ X3 @ ( sep_ad8320153009520438926on_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ A4 @ A2 )
              & ( member2879030624002546611on_val @ B3 @ B2 )
              & ( ( some_P2581650809551911671on_val @ X3 )
                = ( sep_sp4146280900173761658on_val @ Plus @ ( some_P2581650809551911671on_val @ A4 ) @ ( some_P2581650809551911671on_val @ B3 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_383_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ X3 @ ( sep_ad8203089552910907580_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc3882902684156697774_state,B3: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ A4 @ A2 )
              & ( member6715010751053357271_state @ B3 @ B2 )
              & ( ( some_P5383513176685068947_state @ X3 )
                = ( sep_sp7584349535955199440_state @ Plus @ ( some_P5383513176685068947_state @ A4 ) @ ( some_P5383513176685068947_state @ B3 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_384_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val,A2: set_val,B2: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ X3 @ ( sep_add_set_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: val,B3: val] :
              ( ( member_val @ A4 @ A2 )
              & ( member_val @ B3 @ B2 )
              & ( ( some_val @ X3 )
                = ( sep_splus_val @ Plus @ ( some_val @ A4 ) @ ( some_val @ B3 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_385_sep__algebra_Ox__elem__set__product__splus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X3 @ ( sep_add_set_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: state,B3: state] :
              ( ( member_state @ A4 @ A2 )
              & ( member_state @ B3 @ B2 )
              & ( ( some_state @ X3 )
                = ( sep_splus_state @ Plus @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product_splus
thf(fact_386_PartialSA_Opackage__rhs__connectionI,axiom,
    ! [Phi: state,F: state,Phi2: state,F3: state,S2: set_Pr4231240917283394591_state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state] :
      ( ( ( plus @ Phi @ F )
        = ( plus @ Phi2 @ F3 ) )
     => ( ( stable @ F3 )
       => ( ( defined @ Phi @ F )
         => ( ( greater @ F3 @ F )
           => ( ! [A3: state,U: state,T3: state > state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
                 => ? [Au: state] :
                      ( ( ( some_state @ Au )
                        = ( plus @ A3 @ U ) )
                      & ( ( defined @ Au @ ( minus @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                       => ? [A8: state,U2: state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A8 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                            & ( greater @ ( core @ A8 ) @ ( core @ A3 ) )
                            & ( ( plus @ Au @ ( minus @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                              = ( plus @ A8 @ U2 ) )
                            & ( greater @ U2 @ U )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A2 @ A8 @ U2 @ U ) ) ) ) )
             => ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 ) ) ) ) ) ) ).

% PartialSA.package_rhs_connectionI
thf(fact_387_PartialSA_Opackage__rhs__connection__instantiate,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A2: 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 @ A2 @ 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 ) ) )
                 => ? [A9: state,U4: state] :
                      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A9 @ ( produc8632743443909064678_state @ U4 @ T2 ) ) @ S3 )
                      & ( greater @ ( core @ A9 ) @ ( core @ A ) )
                      & ( ( plus @ Au2 @ ( minus @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                        = ( plus @ A9 @ U4 ) )
                      & ( greater @ U4 @ U3 )
                      & ( packag8070727096299822599_state @ plus @ core @ Pc @ A2 @ A9 @ U4 @ U3 ) ) ) ) ) ) ).

% PartialSA.package_rhs_connection_instantiate
thf(fact_388_minus__def,axiom,
    ( minus
    = ( sep_minus_state @ plus @ core ) ) ).

% minus_def
thf(fact_389_plus__ab__defined,axiom,
    ! [Phi: state,Phi2: state] :
      ( ( ( plus @ Phi @ Phi2 )
       != none_state )
      = ( ( compatible_heaps @ ( get_h @ Phi ) @ ( get_h @ Phi2 ) )
        & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ Phi ) @ ( get_m @ Phi2 ) ) ) ) ) ).

% plus_ab_defined
thf(fact_390_compatible__heapsI,axiom,
    ! [H3: produc1457211279475724562t_char > option_val,H4: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char,A3: val,B7: val] :
          ( ( ( H3 @ Hl2 )
            = ( some_val @ A3 ) )
         => ( ( ( H4 @ Hl2 )
              = ( some_val @ B7 ) )
           => ( A3 = B7 ) ) )
     => ( compatible_heaps @ H3 @ H4 ) ) ).

% compatible_heapsI
thf(fact_391_compatible__heaps__def,axiom,
    ( compatible_heaps
    = ( ^ [H5: produc1457211279475724562t_char > option_val,H6: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H5 @ Hl ) @ ( H6 @ Hl ) ) ) ) ).

% compatible_heaps_def
thf(fact_392_sep__algebra_Oadd__set_Ocong,axiom,
    sep_add_set_state = sep_add_set_state ).

% sep_algebra.add_set.cong
thf(fact_393_sep__algebra_Ominus_Ocong,axiom,
    sep_minus_state = sep_minus_state ).

% sep_algebra.minus.cong
thf(fact_394_sep__algebra_Ominus__core__weaker,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B ) )
        = ( sep_minus_state @ Plus @ Core @ ( Core @ A ) @ ( Core @ B ) ) ) ) ).

% sep_algebra.minus_core_weaker
thf(fact_395_sep__algebra_Ominus__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( Core @ ( sep_minus_state @ Plus @ Core @ A @ B ) )
        = ( Core @ A ) ) ) ).

% sep_algebra.minus_core
thf(fact_396_stable__unit,axiom,
    stable @ unit ).

% stable_unit
thf(fact_397_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C ) ) ) ) ).

% upper_valid
thf(fact_398_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_399_sep__algebra_Oadd__set__commm,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ A2 @ B2 )
        = ( sep_add_set_state @ Plus @ B2 @ A2 ) ) ) ).

% sep_algebra.add_set_commm
thf(fact_400_sep__algebra_Oadd__set__asso,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ ( sep_add_set_state @ Plus @ A2 @ B2 ) @ C2 )
        = ( sep_add_set_state @ Plus @ A2 @ ( sep_add_set_state @ Plus @ B2 @ C2 ) ) ) ) ).

% sep_algebra.add_set_asso
thf(fact_401_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_402_stable__sum,axiom,
    ! [A: state,B: state,X3: state] :
      ( ( stable @ A )
     => ( ( stable @ B )
       => ( ( ( some_state @ X3 )
            = ( plus @ A @ B ) )
         => ( stable @ X3 ) ) ) ) ).

% stable_sum
thf(fact_403_compatible__heaps__comm,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( map_ad2908576761495157028ar_val @ A @ B )
        = ( map_ad2908576761495157028ar_val @ B @ A ) ) ) ).

% compatible_heaps_comm
thf(fact_404_compatible__heaps__sum,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( compatible_heaps @ A @ C )
       => ( compatible_heaps @ A @ ( map_ad2908576761495157028ar_val @ B @ C ) ) ) ) ).

% compatible_heaps_sum
thf(fact_405_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X3 )
          = ( Plus @ A @ B ) )
       => ( ( some_val @ ( sep_minus_val @ Plus @ Core @ X3 @ A ) )
          = ( Plus @ B @ ( Core @ X3 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_406_sep__algebra_Ominus__equiv__def__any__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X3 )
          = ( Plus @ A @ B ) )
       => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ X3 @ A ) )
          = ( Plus @ B @ ( Core @ X3 ) ) ) ) ) ).

% sep_algebra.minus_equiv_def_any_elem
thf(fact_407_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,Phi: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ Phi @ ( sep_ad3908482799505143387_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc7272280185086996969_state,B3: produc7272280185086996969_state] :
              ( ( ( some_P6297086604321900740_state @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( member7461423308196221312_state @ A4 @ A2 )
              & ( member7461423308196221312_state @ B3 @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_408_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,Phi: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ Phi @ ( sep_ad8320153009520438926on_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
              ( ( ( some_P2581650809551911671on_val @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( member2879030624002546611on_val @ A4 @ A2 )
              & ( member2879030624002546611on_val @ B3 @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_409_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,Phi: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ Phi @ ( sep_ad8203089552910907580_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc3882902684156697774_state,B3: produc3882902684156697774_state] :
              ( ( ( some_P5383513176685068947_state @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( member6715010751053357271_state @ A4 @ A2 )
              & ( member6715010751053357271_state @ B3 @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_410_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Phi: val,A2: set_val,B2: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ Phi @ ( sep_add_set_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: val,B3: val] :
              ( ( ( some_val @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( member_val @ A4 @ A2 )
              & ( member_val @ B3 @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_411_sep__algebra_Oadd__set__elem,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Phi: state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ Phi @ ( sep_add_set_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: state,B3: state] :
              ( ( ( some_state @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( member_state @ A4 @ A2 )
              & ( member_state @ B3 @ B2 ) ) ) ) ) ).

% sep_algebra.add_set_elem
thf(fact_412_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( member7461423308196221312_state @ X3 @ ( sep_ad3908482799505143387_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc7272280185086996969_state,B3: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ A4 @ A2 )
              & ( member7461423308196221312_state @ B3 @ B2 )
              & ( ( some_P6297086604321900740_state @ X3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_413_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,X3: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( member2879030624002546611on_val @ X3 @ ( sep_ad8320153009520438926on_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ A4 @ A2 )
              & ( member2879030624002546611on_val @ B3 @ B2 )
              & ( ( some_P2581650809551911671on_val @ X3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_414_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( member6715010751053357271_state @ X3 @ ( sep_ad8203089552910907580_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: produc3882902684156697774_state,B3: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ A4 @ A2 )
              & ( member6715010751053357271_state @ B3 @ B2 )
              & ( ( some_P5383513176685068947_state @ X3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_415_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val,A2: set_val,B2: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( member_val @ X3 @ ( sep_add_set_val @ Plus @ A2 @ B2 ) )
        = ( ? [A4: val,B3: val] :
              ( ( member_val @ A4 @ A2 )
              & ( member_val @ B3 @ B2 )
              & ( ( some_val @ X3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_416_sep__algebra_Ox__elem__set__product,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( member_state @ X3 @ ( sep_add_set_state @ Plus @ A2 @ B2 ) )
        = ( ? [A4: state,B3: state] :
              ( ( member_state @ A4 @ A2 )
              & ( member_state @ B3 @ B2 )
              & ( ( some_state @ X3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.x_elem_set_product
thf(fact_417_add__set__def,axiom,
    ( add_set
    = ( sep_add_set_state @ plus ) ) ).

% add_set_def
thf(fact_418_compatibleI,axiom,
    ! [A: state,B: state] :
      ( ( compatible_heaps @ ( get_h @ A ) @ ( get_h @ B ) )
     => ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) )
       => ( defined @ A @ B ) ) ) ).

% compatibleI
thf(fact_419_compatible__same__mask,axiom,
    ! [A: produc1457211279475724562t_char > prat,W: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ ( add_ma6586698345353345811t_char @ A @ W ) )
     => ( W
        = ( comp_min_mask @ A @ W ) ) ) ).

% compatible_same_mask
thf(fact_420_valid__mask__add__comp__min,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( valid_mask @ ( add_ma6586698345353345811t_char @ ( comp_min_mask @ B @ A ) @ B ) ) ) ) ).

% valid_mask_add_comp_min
thf(fact_421_valid__state_Oelims_I3_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X3 )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ( ( valid_mask @ Pi )
              & ( valid_heap @ Pi @ H ) ) ) ) ).

% valid_state.elims(3)
thf(fact_422_valid__state_Oelims_I2_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ( valid_state @ X3 )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ~ ( ( valid_mask @ Pi )
                & ( valid_heap @ Pi @ H ) ) ) ) ).

% valid_state.elims(2)
thf(fact_423_valid__state_Oelims_I1_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X3 )
        = Y )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X3
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ( Y
              = ( ~ ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.elims(1)
thf(fact_424_valid__state_Osimps,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ Pi3 @ H3 ) )
      = ( ( valid_mask @ Pi3 )
        & ( valid_heap @ Pi3 @ H3 ) ) ) ).

% valid_state.simps
thf(fact_425_PartialSA_Opackage__sat__def,axiom,
    ! [Pc: state > $o,A2: aassertion_state,A6: state,U5: state,U3: state] :
      ( ( packag8070727096299822599_state @ plus @ core @ Pc @ A2 @ A6 @ U5 @ U3 )
      = ( ( Pc @ ( core @ A6 ) )
       => ? [X5: state] :
            ( ( ( some_state @ X5 )
              = ( plus @ ( core @ A6 ) @ ( minus @ U5 @ U3 ) ) )
            & ( package_sat_state @ plus @ A2 @ X5 ) ) ) ) ).

% PartialSA.package_sat_def
thf(fact_426_PartialSA_Opackage__satI,axiom,
    ! [Pc: state > $o,A6: state,U5: state,U3: state,A2: aassertion_state] :
      ( ( ( Pc @ ( core @ A6 ) )
       => ? [X4: state] :
            ( ( ( some_state @ X4 )
              = ( plus @ ( core @ A6 ) @ ( minus @ U5 @ U3 ) ) )
            & ( package_sat_state @ plus @ A2 @ X4 ) ) )
     => ( packag8070727096299822599_state @ plus @ core @ Pc @ A2 @ A6 @ U5 @ U3 ) ) ).

% PartialSA.package_satI
thf(fact_427_PartialSA_Ois__footprint__generalI,axiom,
    ! [A2: aassertion_state,R: state > state > state,W: state,B2: aassertion_state] :
      ( ! [A3: state,B7: state] :
          ( ( ( package_sat_state @ plus @ A2 @ A3 )
            & ( ( some_state @ B7 )
              = ( plus @ A3 @ ( R @ A3 @ W ) ) ) )
         => ( package_sat_state @ plus @ B2 @ B7 ) )
     => ( packag6192661653625646554_state @ plus @ W @ R @ A2 @ B2 ) ) ).

% PartialSA.is_footprint_generalI
thf(fact_428_PartialSA_Ois__footprint__standardI,axiom,
    ! [A2: aassertion_state,W: state,B2: aassertion_state] :
      ( ! [A3: state,B7: state] :
          ( ( ( package_sat_state @ plus @ A2 @ A3 )
            & ( ( some_state @ B7 )
              = ( plus @ A3 @ W ) ) )
         => ( package_sat_state @ plus @ B2 @ B7 ) )
     => ( packag7096211993527611489_state @ plus @ W @ A2 @ B2 ) ) ).

% PartialSA.is_footprint_standardI
thf(fact_429_PartialSA_Ois__footprint__general__def,axiom,
    ! [W: state,R: state > state > state,A2: aassertion_state,B2: aassertion_state] :
      ( ( packag6192661653625646554_state @ plus @ W @ R @ A2 @ B2 )
      = ( ! [A4: state,B3: state] :
            ( ( ( package_sat_state @ plus @ A2 @ A4 )
              & ( ( some_state @ B3 )
                = ( plus @ A4 @ ( R @ A4 @ W ) ) ) )
           => ( package_sat_state @ plus @ B2 @ B3 ) ) ) ) ).

% PartialSA.is_footprint_general_def
thf(fact_430_PartialSA_Ois__footprint__standard__def,axiom,
    ! [W: state,A2: aassertion_state,B2: aassertion_state] :
      ( ( packag7096211993527611489_state @ plus @ W @ A2 @ B2 )
      = ( ! [A4: state,B3: state] :
            ( ( ( package_sat_state @ plus @ A2 @ A4 )
              & ( ( some_state @ B3 )
                = ( plus @ A4 @ W ) ) )
           => ( package_sat_state @ plus @ B2 @ B3 ) ) ) ) ).

% PartialSA.is_footprint_standard_def
thf(fact_431_PartialSA_Owf__assertion__and__the,axiom,
    ! [A: state,B: state,A2: aassertion_state] :
      ( ( defined @ ( core @ A ) @ B )
     => ( ( package_sat_state @ plus @ A2 @ B )
       => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
         => ( package_sat_state @ plus @ A2 @ ( the_state @ ( plus @ ( core @ A ) @ B ) ) ) ) ) ) ).

% PartialSA.wf_assertion_and_the
thf(fact_432_PartialSA_Owf__assertion__sat__larger__pure,axiom,
    ! [A2: aassertion_state,Phi: state,Phi2: state,R4: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ A2 )
     => ( ( package_sat_state @ plus @ A2 @ Phi )
       => ( ( ( some_state @ Phi2 )
            = ( plus @ Phi @ R4 ) )
         => ( ( sep_pure_state @ plus @ R4 )
           => ( package_sat_state @ plus @ A2 @ Phi2 ) ) ) ) ) ).

% PartialSA.wf_assertion_sat_larger_pure
thf(fact_433_PartialSA_Opackage__rhs__proof,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
       => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( stable @ F )
             => ( ( defined @ Phi @ F )
               => ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
                  & ( packag3877571396724326311_state @ plus @ core @ unit @ S3 @ F3 ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs_proof
thf(fact_434_PartialSA_Osat__star__exists__bigger,axiom,
    ! [A2: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ A2 @ B2 ) @ Phi )
     => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
       => ( ( packag7298478212905792770_state @ plus @ core @ B2 )
         => ? [A3: state,B7: state] :
              ( ( greater @ ( core @ A3 ) @ ( core @ Phi ) )
              & ( greater @ ( core @ B7 ) @ ( core @ Phi ) )
              & ( ( some_state @ Phi )
                = ( plus @ A3 @ B7 ) )
              & ( package_sat_state @ plus @ A2 @ A3 )
              & ( package_sat_state @ plus @ B2 @ B7 ) ) ) ) ) ).

% PartialSA.sat_star_exists_bigger
thf(fact_435_PartialSA_Owf__assertion_Osimps_I3_J,axiom,
    ! [A2: state > $o] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aSem_state @ A2 ) )
      = ( packag2456304381420842418_state @ plus @ A2 ) ) ).

% PartialSA.wf_assertion.simps(3)
thf(fact_436_PartialSA_Owf__assertion_Osimps_I2_J,axiom,
    ! [B: state > $o,Aa: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ ( aImp_state @ B @ Aa ) )
      = ( ( packag6595153354952283300_state @ plus @ core @ B )
        & ( packag7298478212905792770_state @ plus @ core @ Aa ) ) ) ).

% PartialSA.wf_assertion.simps(2)
thf(fact_437_valid__mask__full__mask,axiom,
    valid_mask @ full_mask ).

% valid_mask_full_mask
thf(fact_438_sep__algebra_Oequiv__stable__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_equiv_state @ Plus @ A2 @ B2 )
       => ( sep_equiv_state @ Plus @ ( sep_add_set_state @ Plus @ A2 @ C2 ) @ ( sep_add_set_state @ Plus @ B2 @ C2 ) ) ) ) ).

% sep_algebra.equiv_stable_sum
thf(fact_439_sep__algebra_Oequiv_Ocong,axiom,
    sep_equiv_state = sep_equiv_state ).

% sep_algebra.equiv.cong
thf(fact_440_PartialSA_Owf__assertion_Ocases,axiom,
    ! [X3: aassertion_state] :
      ( ! [A10: aassertion_state,B8: aassertion_state] :
          ( X3
         != ( aStar_state @ A10 @ B8 ) )
     => ( ! [B7: state > $o,A10: aassertion_state] :
            ( X3
           != ( aImp_state @ B7 @ A10 ) )
       => ~ ! [A10: state > $o] :
              ( X3
             != ( aSem_state @ A10 ) ) ) ) ).

% PartialSA.wf_assertion.cases
thf(fact_441_PartialSA_Osat_Ocases,axiom,
    ! [X3: produc2756261550280135502_state] :
      ( ! [A10: aassertion_state,B8: aassertion_state,Phi4: state] :
          ( X3
         != ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Phi4 ) )
     => ( ! [B7: state > $o,A10: aassertion_state,Phi4: state] :
            ( X3
           != ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Phi4 ) )
       => ~ ! [A10: state > $o,Phi4: state] :
              ( X3
             != ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Phi4 ) ) ) ) ).

% PartialSA.sat.cases
thf(fact_442_PartialSA_Osat_Oelims_I1_J,axiom,
    ! [X3: aassertion_state,Xa2: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X3 @ Xa2 )
        = Y )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ( Y
              = ( ~ ? [A4: state,B3: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( plus @ A4 @ B3 ) )
                      & ( package_sat_state @ plus @ A10 @ A4 )
                      & ( package_sat_state @ plus @ B8 @ B3 ) ) ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ( Y
                = ( ~ ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A10 @ Xa2 ) ) ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ( Y
                  = ( ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.elims(1)
thf(fact_443_PartialSA_Osat_Oelims_I2_J,axiom,
    ! [X3: aassertion_state,Xa2: state] :
      ( ( package_sat_state @ plus @ X3 @ Xa2 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ~ ? [A3: state,B7: state] :
                  ( ( ( some_state @ Xa2 )
                    = ( plus @ A3 @ B7 ) )
                  & ( package_sat_state @ plus @ A10 @ A3 )
                  & ( package_sat_state @ plus @ B8 @ B7 ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ~ ( ( B7 @ Xa2 )
                 => ( package_sat_state @ plus @ A10 @ Xa2 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ~ ( A10 @ Xa2 ) ) ) ) ) ).

% PartialSA.sat.elims(2)
thf(fact_444_PartialSA_Osat_Oelims_I3_J,axiom,
    ! [X3: aassertion_state,Xa2: state] :
      ( ~ ( package_sat_state @ plus @ X3 @ Xa2 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ? [A5: state,B9: state] :
                ( ( ( some_state @ Xa2 )
                  = ( plus @ A5 @ B9 ) )
                & ( package_sat_state @ plus @ A10 @ A5 )
                & ( package_sat_state @ plus @ B8 @ B9 ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ( ( B7 @ Xa2 )
               => ( package_sat_state @ plus @ A10 @ Xa2 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ( A10 @ Xa2 ) ) ) ) ) ).

% PartialSA.sat.elims(3)
thf(fact_445_PartialSA_OAStar,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,B2: aassertion_state,Phi7: state,F4: state,S4: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ B2 @ Phi7 @ F4 @ S4 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aStar_state @ A2 @ B2 ) @ Phi7 @ F4 @ S4 ) ) ) ).

% PartialSA.AStar
thf(fact_446_PartialSA_Opackage__rhs_Ointros_I1_J,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state,Ba2: aassertion_state,Phi7: state,F4: state,S4: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ Ba2 @ Phi7 @ F4 @ S4 )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aStar_state @ A2 @ Ba2 ) @ Phi7 @ F4 @ S4 ) ) ) ).

% PartialSA.package_rhs.intros(1)
thf(fact_447_PartialSA_Owf__assertion_Oelims_I3_J,axiom,
    ! [X3: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X3 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ( ( packag7298478212905792770_state @ plus @ core @ A10 )
              & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(3)
thf(fact_448_PartialSA_Owf__assertion_Oelims_I2_J,axiom,
    ! [X3: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X3 )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                  & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(2)
thf(fact_449_PartialSA_Owf__assertion_Oelims_I1_J,axiom,
    ! [X3: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X3 )
        = Y )
     => ( ! [A10: aassertion_state,B8: aassertion_state] :
            ( ( X3
              = ( aStar_state @ A10 @ B8 ) )
           => ( Y
              = ( ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( X3
                = ( aImp_state @ B7 @ A10 ) )
             => ( Y
                = ( ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) ) )
         => ~ ! [A10: state > $o] :
                ( ( X3
                  = ( aSem_state @ A10 ) )
               => ( Y
                  = ( ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.elims(1)
thf(fact_450_PartialSA_Osat_Osimps_I3_J,axiom,
    ! [A2: state > $o,Phi: state] :
      ( ( package_sat_state @ plus @ ( aSem_state @ A2 ) @ Phi )
      = ( A2 @ Phi ) ) ).

% PartialSA.sat.simps(3)
thf(fact_451_PartialSA_Osat_Osimps_I2_J,axiom,
    ! [B: state > $o,Aa: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aImp_state @ B @ Aa ) @ Phi )
      = ( ( B @ Phi )
       => ( package_sat_state @ plus @ Aa @ Phi ) ) ) ).

% PartialSA.sat.simps(2)
thf(fact_452_PartialSA_OAImp,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,B: state > $o,A2: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ ( packag1330488657391168505_state @ Pc @ B ) @ A2 @ Phi2 @ F3 @ S3 )
     => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aImp_state @ B @ A2 ) @ Phi2 @ F3 @ S3 ) ) ).

% PartialSA.AImp
thf(fact_453_PartialSA_Osat_Osimps_I1_J,axiom,
    ! [Aa: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_sat_state @ plus @ ( aStar_state @ Aa @ B2 ) @ Phi )
      = ( ? [A4: state,B3: state] :
            ( ( ( some_state @ Phi )
              = ( plus @ A4 @ B3 ) )
            & ( package_sat_state @ plus @ Aa @ A4 )
            & ( package_sat_state @ plus @ B2 @ B3 ) ) ) ) ).

% PartialSA.sat.simps(1)
thf(fact_454_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_455_PartialSA_Oequiv__stable__sum,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_equiv_state @ plus @ A2 @ B2 )
     => ( sep_equiv_state @ plus @ ( add_set @ A2 @ C2 ) @ ( add_set @ B2 @ C2 ) ) ) ).

% PartialSA.equiv_stable_sum
thf(fact_456_sep__algebra_Omono__prop__set__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_mono_prop_state @ Plus @ P )
       => ( ( sep_equiv_state @ Plus @ A2 @ B2 )
         => ( ( sep_setify_state @ P @ A2 )
            = ( sep_setify_state @ P @ B2 ) ) ) ) ) ).

% sep_algebra.mono_prop_set_equiv
thf(fact_457_PartialSA_Ogeneral__completeness,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state,A2: aassertion_state,Phi: state,Phi2: state] :
      ( ! [A3: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ! [X: state] :
              ( ( ( some_state @ X )
                = ( plus @ A3 @ ( T3 @ F ) ) )
             => ( package_sat_state @ plus @ A2 @ X ) ) )
     => ( ( ( some_state @ Phi )
          = ( plus @ Phi2 @ F ) )
       => ( ( stable @ F )
         => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ unit )
           => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ unit @ S2
                  @ ^ [Uu3: state] : $true
                  @ A2
                  @ Phi2
                  @ F
                  @ X_1 ) ) ) ) ) ) ).

% PartialSA.general_completeness
thf(fact_458_sep__algebra_Oequiv__up__closed__subset,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A2 )
       => ( ( sep_equiv_state @ Plus @ B2 @ C2 )
         => ( ( ord_le2494988322063910608_state @ B2 @ A2 )
            = ( ord_le2494988322063910608_state @ C2 @ A2 ) ) ) ) ) ).

% sep_algebra.equiv_up_closed_subset
thf(fact_459_PartialSA_Owf__assertion_Opelims_I3_J,axiom,
    ! [X3: aassertion_state] :
      ( ~ ( packag7298478212905792770_state @ plus @ core @ X3 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X3 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) )
               => ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                  & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A10 ) )
                 => ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                    & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
                   => ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(3)
thf(fact_460_PartialSA_Owf__assertion_Opelims_I2_J,axiom,
    ! [X3: aassertion_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ X3 )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X3 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) )
               => ~ ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A10 ) )
                 => ~ ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) )
                   => ~ ( packag2456304381420842418_state @ plus @ A10 ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(2)
thf(fact_461_PartialSA_Owf__assertion_Opelims_I1_J,axiom,
    ! [X3: aassertion_state,Y: $o] :
      ( ( ( packag7298478212905792770_state @ plus @ core @ X3 )
        = Y )
     => ( ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ X3 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( Y
                  = ( ( packag7298478212905792770_state @ plus @ core @ A10 )
                    & ( packag7298478212905792770_state @ plus @ core @ B8 ) ) )
               => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aStar_state @ A10 @ B8 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( Y
                    = ( ( packag6595153354952283300_state @ plus @ core @ B7 )
                      & ( packag7298478212905792770_state @ plus @ core @ A10 ) ) )
                 => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aImp_state @ B7 @ A10 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( Y
                      = ( packag2456304381420842418_state @ plus @ A10 ) )
                   => ~ ( accp_a4361485425079813866_state @ packag4834522500328176237_state @ ( aSem_state @ A10 ) ) ) ) ) ) ) ) ).

% PartialSA.wf_assertion.pelims(1)
thf(fact_462_empty__map__add,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val
        @ ^ [X5: produc1457211279475724562t_char] : none_val
        @ M )
      = M ) ).

% empty_map_add
thf(fact_463_map__add__empty,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ M
        @ ^ [X5: produc1457211279475724562t_char] : none_val )
      = M ) ).

% map_add_empty
thf(fact_464_empty__eq__map__add__iff,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( ^ [X5: produc1457211279475724562t_char] : none_val )
        = ( map_ad2908576761495157028ar_val @ F @ G ) )
      = ( ( F
          = ( ^ [X5: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X5: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% empty_eq_map_add_iff
thf(fact_465_case__optionE,axiom,
    ! [P: $o,Q: state > $o,X3: option_state] :
      ( ( case_option_o_state @ P @ Q @ X3 )
     => ( ( ( X3 = none_state )
         => ~ P )
       => ~ ! [Y3: state] :
              ( ( X3
                = ( some_state @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_466_case__optionE,axiom,
    ! [P: $o,Q: val > $o,X3: option_val] :
      ( ( case_option_o_val @ P @ Q @ X3 )
     => ( ( ( X3 = none_val )
         => ~ P )
       => ~ ! [Y3: val] :
              ( ( X3
                = ( some_val @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_467_map__add__def,axiom,
    ( map_ad2908576761495157028ar_val
    = ( ^ [M12: produc1457211279475724562t_char > option_val,M22: produc1457211279475724562t_char > option_val,X5: produc1457211279475724562t_char] : ( case_o7794706883426356101al_val @ ( M12 @ X5 ) @ some_val @ ( M22 @ X5 ) ) ) ) ).

% map_add_def
thf(fact_468_pred__equals__eq2,axiom,
    ! [R: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ( ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_469_pred__equals__eq2,axiom,
    ! [R: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ( ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_470_pred__equals__eq2,axiom,
    ! [R: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( ( ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_471_pred__equals__eq2,axiom,
    ! [R: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( ( ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_472_pred__equals__eq2,axiom,
    ! [R: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( ( ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_473_pred__equals__eq2,axiom,
    ! [R: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( ( ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_474_pred__equals__eq2,axiom,
    ! [R: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ( ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ R ) )
        = ( ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_475_less__eq__set__def,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
          ( ord_le7281427340513060550tate_o
          @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ A7 )
          @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_476_less__eq__set__def,axiom,
    ( ord_le6900622908513426738on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,B4: set_Pr7649655576376376530on_val] :
          ( ord_le6140313152057869779_val_o
          @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ A7 )
          @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_477_less__eq__set__def,axiom,
    ( ord_le6483787752022044334_state
    = ( ^ [A7: set_Pr2391701600042252046_state,B4: set_Pr2391701600042252046_state] :
          ( ord_le8077393285996101039tate_o
          @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ A7 )
          @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_478_less__eq__set__def,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A7: set_state,B4: set_state] :
          ( ord_less_eq_state_o
          @ ^ [X5: state] : ( member_state @ X5 @ A7 )
          @ ^ [X5: state] : ( member_state @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_479_Collect__subset,axiom,
    ! [A2: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ord_le6483787752022044334_state
      @ ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X5 @ A2 )
            & ( P @ X5 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_480_Collect__subset,axiom,
    ! [A2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ord_le4032880777312351359_state
      @ ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ A2 )
            & ( P @ X5 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_481_Collect__subset,axiom,
    ! [A2: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X5: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X5 @ A2 )
            & ( P @ X5 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_482_Collect__subset,axiom,
    ! [A2: set_state,P: state > $o] :
      ( ord_le2494988322063910608_state
      @ ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ A2 )
            & ( P @ X5 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_483_sep__algebra_Oup__closed_Ocong,axiom,
    sep_up_closed_state = sep_up_closed_state ).

% sep_algebra.up_closed.cong
thf(fact_484_pred__subset__eq,axiom,
    ! [R: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ord_le7281427340513060550tate_o
        @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ R )
        @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ S2 ) )
      = ( ord_le4032880777312351359_state @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_485_pred__subset__eq,axiom,
    ! [R: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ord_le6140313152057869779_val_o
        @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ R )
        @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ S2 ) )
      = ( ord_le6900622908513426738on_val @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_486_pred__subset__eq,axiom,
    ! [R: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ord_le8077393285996101039tate_o
        @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ R )
        @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ S2 ) )
      = ( ord_le6483787752022044334_state @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_487_pred__subset__eq,axiom,
    ! [R: set_state,S2: set_state] :
      ( ( ord_less_eq_state_o
        @ ^ [X5: state] : ( member_state @ X5 @ R )
        @ ^ [X5: state] : ( member_state @ X5 @ S2 ) )
      = ( ord_le2494988322063910608_state @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_488_pred__subset__eq2,axiom,
    ! [R: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( ord_le6704609105599929480tate_o
        @ ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le4032880777312351359_state @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_489_pred__subset__eq2,axiom,
    ! [R: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( ord_le1106678366802930969tate_o
        @ ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le6483787752022044334_state @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_490_pred__subset__eq2,axiom,
    ! [R: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( ord_le616650860478780142_val_o
        @ ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ R )
        @ ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le7224449410216889241on_val @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_491_pred__subset__eq2,axiom,
    ! [R: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( ord_le2052508915978315794tate_o
        @ ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le6423325748750870005_state @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_492_pred__subset__eq2,axiom,
    ! [R: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( ord_le9040380983013460636tate_o
        @ ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le262197048489888875_state @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_493_pred__subset__eq2,axiom,
    ! [R: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( ord_le8830457724327557734tate_o
        @ ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le5761494056910127137_state @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_494_pred__subset__eq2,axiom,
    ! [R: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( ord_le3852884270828607079_val_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ R )
        @ ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ S2 ) )
      = ( ord_le6900622908513426738on_val @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_495_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_496_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
      = ( case_option_o_val @ $false
        @ ^ [Uu3: val] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_497_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_498_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_val] :
      ( ( Option = none_val )
      = ( case_option_o_val @ $true
        @ ^ [Uu3: val] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_499_PartialSA_Oup__close__equiv,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( sep_up_closed_state @ plus @ A2 )
     => ( ( sep_up_closed_state @ plus @ B2 )
       => ( ( sep_equiv_state @ plus @ A2 @ B2 )
          = ( A2 = B2 ) ) ) ) ).

% PartialSA.up_close_equiv
thf(fact_500_sep__algebra_Osetify__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_setify_state @ P @ A2 )
        = ( ! [X5: state] :
              ( ( member_state @ X5 @ A2 )
             => ( P @ X5 ) ) ) ) ) ).

% sep_algebra.setify_def
thf(fact_501_PartialSA_Oup__closedI,axiom,
    ! [A2: set_state] :
      ( ! [Phi6: state,Phi4: state] :
          ( ( ( greater @ Phi6 @ Phi4 )
            & ( member_state @ Phi4 @ A2 ) )
         => ( member_state @ Phi6 @ A2 ) )
     => ( sep_up_closed_state @ plus @ A2 ) ) ).

% PartialSA.up_closedI
thf(fact_502_PartialSA_Oup__closed__def,axiom,
    ! [A2: set_state] :
      ( ( sep_up_closed_state @ plus @ A2 )
      = ( ! [Phi5: state] :
            ( ? [X5: state] :
                ( ( member_state @ X5 @ A2 )
                & ( greater @ Phi5 @ X5 ) )
           => ( member_state @ Phi5 @ A2 ) ) ) ) ).

% PartialSA.up_closed_def
thf(fact_503_PartialSA_Oup__closed__sum,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( sep_up_closed_state @ plus @ A2 )
     => ( sep_up_closed_state @ plus @ ( add_set @ A2 @ B2 ) ) ) ).

% PartialSA.up_closed_sum
thf(fact_504_PartialSA_Oequiv__up__closed__subset,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_up_closed_state @ plus @ A2 )
     => ( ( sep_equiv_state @ plus @ B2 @ C2 )
       => ( ( ord_le2494988322063910608_state @ B2 @ A2 )
          = ( ord_le2494988322063910608_state @ C2 @ A2 ) ) ) ) ).

% PartialSA.equiv_up_closed_subset
thf(fact_505_sep__algebra_Oup__closed__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A2 )
       => ( sep_up_closed_state @ Plus @ ( sep_add_set_state @ Plus @ A2 @ B2 ) ) ) ) ).

% sep_algebra.up_closed_sum
thf(fact_506_PartialSA_Omono__prop__set__equiv,axiom,
    ! [P: state > $o,A2: set_state,B2: set_state] :
      ( ( sep_mono_prop_state @ plus @ P )
     => ( ( sep_equiv_state @ plus @ A2 @ B2 )
       => ( ( sep_setify_state @ P @ A2 )
          = ( sep_setify_state @ P @ B2 ) ) ) ) ).

% PartialSA.mono_prop_set_equiv
thf(fact_507_sep__algebra_Oup__close__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A2 )
       => ( ( sep_up_closed_state @ Plus @ B2 )
         => ( ( sep_equiv_state @ Plus @ A2 @ B2 )
            = ( A2 = B2 ) ) ) ) ) ).

% sep_algebra.up_close_equiv
thf(fact_508_PartialSA_Ocompleteness,axiom,
    ! [B2: aassertion_state,W: state,R: state > state > state,A2: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ( ( stable @ W )
          & ( packag6192661653625646554_state @ plus @ W @ R @ A2 @ B2 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ( ! [A3: state] :
                ( ( package_sat_state @ plus @ A2 @ A3 )
               => ( packag4817583915586052229_state @ plus @ unit @ ( R @ A3 ) ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ unit @ ( R @ A4 ) ) ) )
                      & ( package_sat_state @ plus @ A2 @ A4 ) ) )
                @ ^ [Uu3: state] : $true
                @ B2
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% PartialSA.completeness
thf(fact_509_PartialSA_Osoundness,axiom,
    ! [B2: aassertion_state,A2: aassertion_state,S2: set_state,R: state > state > state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ! [A3: state] :
            ( ( package_sat_state @ plus @ A2 @ A3 )
           => ( member_state @ A3 @ S2 ) )
       => ( ! [A3: state] :
              ( ( member_state @ A3 @ S2 )
             => ( packag4817583915586052229_state @ plus @ unit @ ( R @ A3 ) ) )
         => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ unit @ ( R @ A4 ) ) ) )
                    & ( member_state @ A4 @ 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 @ R @ A2 @ B2 ) ) ) ) ) ) ) ).

% PartialSA.soundness
thf(fact_510_PartialSA_Ominus__def,axiom,
    ( minus
    = ( ^ [B3: state,A4: state] :
          ( fun_TH8221542053827138557_state @ B3
          @ ^ [X5: state] :
              ( ( ( some_state @ B3 )
                = ( plus @ A4 @ X5 ) )
              & ( greater @ X5 @ ( core @ B3 ) ) ) ) ) ) ).

% PartialSA.minus_def
thf(fact_511_sep__algebra_Oupper__closure__up__closed,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_up_closed_state @ Plus @ ( sep_up1246176804924251236_state @ Plus @ A2 ) ) ) ).

% sep_algebra.upper_closure_up_closed
thf(fact_512_PartialSA_Ocompleteness__paper,axiom,
    ! [B2: aassertion_state,W: state,A2: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ( ( stable @ W )
          & ( packag7096211993527611489_state @ plus @ W @ A2 @ B2 ) )
       => ( ( ( some_state @ Sigma )
            = ( plus @ Sigma2 @ W ) )
         => ? [X_1: set_Pr4231240917283394591_state] :
              ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                    & ( package_sat_state @ plus @ A2 @ A4 ) ) )
              @ ^ [Uu3: state] : $true
              @ B2
              @ Sigma2
              @ W
              @ X_1 ) ) ) ) ).

% PartialSA.completeness_paper
thf(fact_513_PartialSA_Osetify__def,axiom,
    ( sep_setify_state
    = ( ^ [P6: state > $o,A7: set_state] :
        ! [X5: state] :
          ( ( member_state @ X5 @ A7 )
         => ( P6 @ X5 ) ) ) ) ).

% PartialSA.setify_def
thf(fact_514_sep__algebra_Oupper__closure_Ocong,axiom,
    sep_up1246176804924251236_state = sep_up1246176804924251236_state ).

% sep_algebra.upper_closure.cong
thf(fact_515_PartialSA_Oupper__closure__def,axiom,
    ! [A2: set_state] :
      ( ( sep_up1246176804924251236_state @ plus @ A2 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Phi5: state,Phi3: state] :
            ( ( Uu3 = Phi5 )
            & ( greater @ Phi5 @ Phi3 )
            & ( member_state @ Phi3 @ A2 ) ) ) ) ).

% PartialSA.upper_closure_def
thf(fact_516_PartialSA_Oupper__closure__up__closed,axiom,
    ! [A2: set_state] : ( sep_up_closed_state @ plus @ ( sep_up1246176804924251236_state @ plus @ A2 ) ) ).

% PartialSA.upper_closure_up_closed
thf(fact_517_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_ad8203089552910907580_state @ Plus @ A2 @ B2 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Phi3: produc3882902684156697774_state,A4: produc3882902684156697774_state,B3: produc3882902684156697774_state] :
              ( ( Uu3 = Phi3 )
              & ( member6715010751053357271_state @ A4 @ A2 )
              & ( member6715010751053357271_state @ B3 @ B2 )
              & ( ( some_P5383513176685068947_state @ Phi3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_518_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_ad3908482799505143387_state @ Plus @ A2 @ B2 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Phi3: produc7272280185086996969_state,A4: produc7272280185086996969_state,B3: produc7272280185086996969_state] :
              ( ( Uu3 = Phi3 )
              & ( member7461423308196221312_state @ A4 @ A2 )
              & ( member7461423308196221312_state @ B3 @ B2 )
              & ( ( some_P6297086604321900740_state @ Phi3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_519_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_ad8320153009520438926on_val @ Plus @ A2 @ B2 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi3: produc9003781588162623004on_val,A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi3 )
              & ( member2879030624002546611on_val @ A4 @ A2 )
              & ( member2879030624002546611on_val @ B3 @ B2 )
              & ( ( some_P2581650809551911671on_val @ Phi3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_520_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A2: set_val,B2: set_val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_add_set_val @ Plus @ A2 @ B2 )
        = ( collect_val
          @ ^ [Uu3: val] :
            ? [Phi3: val,A4: val,B3: val] :
              ( ( Uu3 = Phi3 )
              & ( member_val @ A4 @ A2 )
              & ( member_val @ B3 @ B2 )
              & ( ( some_val @ Phi3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_521_sep__algebra_Oadd__set__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_add_set_state @ Plus @ A2 @ B2 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi3: state,A4: state,B3: state] :
              ( ( Uu3 = Phi3 )
              & ( member_state @ A4 @ A2 )
              & ( member_state @ B3 @ B2 )
              & ( ( some_state @ Phi3 )
                = ( Plus @ A4 @ B3 ) ) ) ) ) ) ).

% sep_algebra.add_set_def
thf(fact_522_PartialSA_Oadd__set__def,axiom,
    ( add_set
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi3: state,A4: state,B3: state] :
              ( ( Uu3 = Phi3 )
              & ( member_state @ A4 @ A7 )
              & ( member_state @ B3 @ B4 )
              & ( ( some_state @ Phi3 )
                = ( plus @ A4 @ B3 ) ) ) ) ) ) ).

% PartialSA.add_set_def
thf(fact_523_PartialSA_Osoundness__paper,axiom,
    ! [B2: aassertion_state,A2: aassertion_state,S2: set_state,Sigma: state,Sigma2: state,W: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag7298478212905792770_state @ plus @ core @ B2 )
     => ( ! [A3: state] :
            ( ( package_sat_state @ plus @ A2 @ A3 )
           => ( member_state @ A3 @ S2 ) )
       => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Sigma @ unit
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A4: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ unit @ id_state ) ) )
                  & ( member_state @ A4 @ 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 @ A2 @ B2 ) ) ) ) ) ) ).

% PartialSA.soundness_paper
thf(fact_524_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_525_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_526_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_527_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_528_accp__subset,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( ord_le616650860478780142_val_o @ R1 @ R22 )
     => ( ord_le6140313152057869779_val_o @ ( accp_P4405556307444930259on_val @ R22 ) @ ( accp_P4405556307444930259on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_529_PartialSA_OAddFromOutside,axiom,
    ! [Phi: state,Phi2: state,M: state,F3: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi7: state,F4: state,S4: set_Pr4231240917283394591_state,F: state,S2: set_Pr4231240917283394591_state] :
      ( ( ( some_state @ Phi )
        = ( plus @ Phi2 @ M ) )
     => ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi2 @ F3 @ S3 @ Pc @ A2 @ Phi7 @ F4 @ S4 )
       => ( ( stable @ M )
         => ( ( ( some_state @ F3 )
              = ( plus @ F @ M ) )
           => ( ( S3
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state,U6: state,T4: state > state,R2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                      & ( ( some_state @ R2 )
                        = ( plus @ A4 @ ( minus @ ( T4 @ F3 ) @ ( T4 @ F ) ) ) )
                      & ( defined @ R2 @ U6 ) ) ) )
             => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi7 @ F4 @ S4 ) ) ) ) ) ) ).

% PartialSA.AddFromOutside
thf(fact_530_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,A2: aasser7019605857572776418_state,S2: set_Pr4231240917283394591_state,R: 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 )
       => ( ! [A3: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A2 @ A3 )
             => ( member7461423308196221312_state @ A3 @ S2 ) )
         => ( ! [A3: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ A3 @ S2 )
               => ( packag3884151059541177780_state @ Plus @ Unit @ ( R @ A3 ) ) )
           => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec7677102015826254172_state
                  @ ^ [Uu3: produc3057782296003698823_state] :
                    ? [A4: produc7272280185086996969_state] :
                      ( ( Uu3
                        = ( produc1904032489304825401_state @ A4 @ ( produc7648782313880683123_state @ Unit @ ( R @ A4 ) ) ) )
                      & ( member7461423308196221312_state @ A4 @ 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 @ R @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_531_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,A2: aasser1163550960519286933on_val,S2: set_Pr7649655576376376530on_val,R: 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 )
       => ( ! [A3: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A2 @ A3 )
             => ( member2879030624002546611on_val @ A3 @ S2 ) )
         => ( ! [A3: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ A3 @ S2 )
               => ( packag5955012701910386663on_val @ Plus @ Unit @ ( R @ A3 ) ) )
           => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec3787697688312515330on_val
                  @ ^ [Uu3: produc217507841372899885on_val] :
                    ? [A4: produc9003781588162623004on_val] :
                      ( ( Uu3
                        = ( produc6985948203474072671on_val @ A4 @ ( produc6949276712527400972on_val @ Unit @ ( R @ A4 ) ) ) )
                      & ( member2879030624002546611on_val @ A4 @ 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 @ R @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_532_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,A2: aasser3179708585289803787_state,S2: set_Pr2391701600042252046_state,R: 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 )
       => ( ! [A3: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A2 @ A3 )
             => ( member6715010751053357271_state @ A3 @ S2 ) )
         => ( ! [A3: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ A3 @ S2 )
               => ( packag5827594511758291363_state @ Plus @ Unit @ ( R @ A3 ) ) )
           => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec6806307004801468532_state
                  @ ^ [Uu3: produc6455357576889117855_state] :
                    ? [A4: produc3882902684156697774_state] :
                      ( ( Uu3
                        = ( produc5427679348311974809_state @ A4 @ ( produc3987245858496418492_state @ Unit @ ( R @ A4 ) ) ) )
                      & ( member6715010751053357271_state @ A4 @ 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 @ R @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_533_package__logic_Osoundness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B2: aassertion_val,A2: aassertion_val,S2: set_val,R: val > val > val,Sigma: val,Sigma2: val,W: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B2 )
       => ( ! [A3: val] :
              ( ( package_sat_val @ Plus @ A2 @ A3 )
             => ( member_val @ A3 @ S2 ) )
         => ( ! [A3: val] :
                ( ( member_val @ A3 @ S2 )
               => ( packag4175576287423129206er_val @ Plus @ Unit @ ( R @ A3 ) ) )
           => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A4: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ Unit @ ( R @ A4 ) ) ) )
                      & ( member_val @ A4 @ S2 ) ) )
                @ ^ [Uu3: val] : $true
                @ B2
                @ Sigma2
                @ W
                @ S3 )
             => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ B2 ) )
                  | ( packag3202507263771882441al_val @ Plus @ S3 ) )
               => ( ( Stable @ W )
                  & ( ( some_val @ Sigma )
                    = ( Plus @ Sigma2 @ W ) )
                  & ( packag4897621287243763553al_val @ Plus @ W @ R @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_534_package__logic_Osoundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,A2: aassertion_state,S2: set_state,R: 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 )
       => ( ! [A3: state] :
              ( ( package_sat_state @ Plus @ A2 @ A3 )
             => ( member_state @ A3 @ S2 ) )
         => ( ! [A3: state] :
                ( ( member_state @ A3 @ S2 )
               => ( packag4817583915586052229_state @ Plus @ Unit @ ( R @ A3 ) ) )
           => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ Unit @ ( R @ A4 ) ) ) )
                      & ( member_state @ A4 @ 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 @ R @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% package_logic.soundness
thf(fact_535_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,A2: 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 )
       => ( ! [A3: produc7272280185086996969_state] :
              ( ( packag4865960244449834244_state @ Plus @ A2 @ A3 )
             => ( member7461423308196221312_state @ A3 @ S2 ) )
         => ( ( packag3630742828623380589_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec7677102015826254172_state
                @ ^ [Uu3: produc3057782296003698823_state] :
                  ? [A4: produc7272280185086996969_state] :
                    ( ( Uu3
                      = ( produc1904032489304825401_state @ A4 @ ( produc7648782313880683123_state @ Unit @ id_Pro2933125508726022968_state ) ) )
                    & ( member7461423308196221312_state @ A4 @ 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 @ A2 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_536_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,A2: 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 )
       => ( ! [A3: produc9003781588162623004on_val] :
              ( ( packag3677137384183703351on_val @ Plus @ A2 @ A3 )
             => ( member2879030624002546611on_val @ A3 @ S2 ) )
         => ( ( packag6843733491101478048on_val @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec3787697688312515330on_val
                @ ^ [Uu3: produc217507841372899885on_val] :
                  ? [A4: produc9003781588162623004on_val] :
                    ( ( Uu3
                      = ( produc6985948203474072671on_val @ A4 @ ( produc6949276712527400972on_val @ Unit @ id_Pro1402184692240358763on_val ) ) )
                    & ( member2879030624002546611on_val @ A4 @ 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 @ A2 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_537_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,A2: 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 )
       => ( ! [A3: produc3882902684156697774_state] :
              ( ( packag2518615827549365843_state @ Plus @ A2 @ A3 )
             => ( member6715010751053357271_state @ A3 @ S2 ) )
         => ( ( packag4061699384143886122_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec6806307004801468532_state
                @ ^ [Uu3: produc6455357576889117855_state] :
                  ? [A4: produc3882902684156697774_state] :
                    ( ( Uu3
                      = ( produc5427679348311974809_state @ A4 @ ( produc3987245858496418492_state @ Unit @ id_Pro689249233197853983_state ) ) )
                    & ( member6715010751053357271_state @ A4 @ 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 @ A2 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_538_package__logic_Osoundness__paper,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B2: aassertion_val,A2: aassertion_val,S2: set_val,Sigma: val,Sigma2: val,W: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B2 )
       => ( ! [A3: val] :
              ( ( package_sat_val @ Plus @ A2 @ A3 )
             => ( member_val @ A3 @ S2 ) )
         => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec8300311316619386842al_val
                @ ^ [Uu3: produc2072440392815185925al_val] :
                  ? [A4: val] :
                    ( ( Uu3
                      = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ Unit @ id_val ) ) )
                    & ( member_val @ A4 @ S2 ) ) )
              @ ^ [Uu3: val] : $true
              @ B2
              @ Sigma2
              @ W
              @ S3 )
           => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ B2 ) )
                | ( packag3202507263771882441al_val @ Plus @ S3 ) )
             => ( ( Stable @ W )
                & ( ( some_val @ Sigma )
                  = ( Plus @ Sigma2 @ W ) )
                & ( packag685230357167683994rd_val @ Plus @ W @ A2 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_539_package__logic_Osoundness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,A2: 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 )
       => ( ! [A3: state] :
              ( ( package_sat_state @ Plus @ A2 @ A3 )
             => ( member_state @ A3 @ S2 ) )
         => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                    & ( member_state @ A4 @ 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 @ A2 @ B2 ) ) ) ) ) ) ) ).

% package_logic.soundness_paper
thf(fact_540_PartialSA_Ounder__def,axiom,
    ! [A2: set_state,Omega2: state] :
      ( ( sep_under_state @ plus @ A2 @ Omega2 )
      = ( collect_state
        @ ^ [Uu3: state] :
          ? [Omega3: state] :
            ( ( Uu3 = Omega3 )
            & ( member_state @ Omega3 @ A2 )
            & ( greater @ Omega2 @ Omega3 ) ) ) ) ).

% PartialSA.under_def
thf(fact_541_package__logic_Ounit__neutral,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( some_val @ A )
        = ( Plus @ A @ Unit ) ) ) ).

% package_logic.unit_neutral
thf(fact_542_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_543_package__logic_Ostable__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,X3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A )
       => ( ( Stable @ B )
         => ( ( ( some_val @ X3 )
              = ( Plus @ A @ B ) )
           => ( Stable @ X3 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_544_package__logic_Ostable__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,X3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( Stable @ A )
       => ( ( Stable @ B )
         => ( ( ( some_state @ X3 )
              = ( Plus @ A @ B ) )
           => ( Stable @ X3 ) ) ) ) ) ).

% package_logic.stable_sum
thf(fact_545_sep__algebra_Ounder_Ocong,axiom,
    sep_under_state = sep_under_state ).

% sep_algebra.under.cong
thf(fact_546_package__logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A2: aassertion_val,B2: aassertion_val,Phi: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ ( aStar_val @ A2 @ B2 ) @ Phi )
        = ( ? [A4: val,B3: val] :
              ( ( ( some_val @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( package_sat_val @ Plus @ A2 @ A4 )
              & ( package_sat_val @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.sat.simps(1)
thf(fact_547_package__logic_Osat_Osimps_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A2: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A2 @ B2 ) @ Phi )
        = ( ? [A4: state,B3: state] :
              ( ( ( some_state @ Phi )
                = ( Plus @ A4 @ B3 ) )
              & ( package_sat_state @ Plus @ A2 @ A4 )
              & ( package_sat_state @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.sat.simps(1)
thf(fact_548_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6714868506247462167nd_val @ Plus @ Core @ B )
        = ( ! [Phi3: val] :
              ( ( B @ Phi3 )
              = ( B @ ( Core @ Phi3 ) ) )
          & ! [Phi5: val,Phi3: val,R2: val] :
              ( ( ( sep_pure_val @ Plus @ R2 )
                & ( ( some_val @ Phi5 )
                  = ( Plus @ Phi3 @ R2 ) )
                & ~ ( B @ Phi3 ) )
             => ~ ( B @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_549_package__logic_Omono__pure__cond__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6595153354952283300_state @ Plus @ Core @ B )
        = ( ! [Phi3: state] :
              ( ( B @ Phi3 )
              = ( B @ ( Core @ Phi3 ) ) )
          & ! [Phi5: state,Phi3: state,R2: state] :
              ( ( ( sep_pure_state @ Plus @ R2 )
                & ( ( some_state @ Phi5 )
                  = ( Plus @ Phi3 @ R2 ) )
                & ~ ( B @ Phi3 ) )
             => ~ ( B @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pure_cond_def
thf(fact_550_package__logic_Omono__pure__condI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi4: val] :
            ( ( B @ Phi4 )
            = ( B @ ( Core @ Phi4 ) ) )
       => ( ! [Phi4: val,Phi6: val,R3: val] :
              ( ( ( sep_pure_val @ Plus @ R3 )
                & ( ( some_val @ Phi6 )
                  = ( Plus @ Phi4 @ R3 ) )
                & ~ ( B @ Phi4 ) )
             => ~ ( B @ Phi6 ) )
         => ( packag6714868506247462167nd_val @ Plus @ Core @ B ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_551_package__logic_Omono__pure__condI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B: state > $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [Phi4: state] :
            ( ( B @ Phi4 )
            = ( B @ ( Core @ Phi4 ) ) )
       => ( ! [Phi4: state,Phi6: state,R3: state] :
              ( ( ( sep_pure_state @ Plus @ R3 )
                & ( ( some_state @ Phi6 )
                  = ( Plus @ Phi4 @ R3 ) )
                & ~ ( B @ Phi4 ) )
             => ~ ( B @ Phi6 ) )
         => ( packag6595153354952283300_state @ Plus @ Core @ B ) ) ) ) ).

% package_logic.mono_pure_condI
thf(fact_552_package__logic_Ois__footprint__generalI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A2: aassertion_val,R: val > val > val,W: val,B2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: val,B7: val] :
            ( ( ( package_sat_val @ Plus @ A2 @ A3 )
              & ( ( some_val @ B7 )
                = ( Plus @ A3 @ ( R @ A3 @ W ) ) ) )
           => ( package_sat_val @ Plus @ B2 @ B7 ) )
       => ( packag4897621287243763553al_val @ Plus @ W @ R @ A2 @ B2 ) ) ) ).

% package_logic.is_footprint_generalI
thf(fact_553_package__logic_Ois__footprint__generalI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A2: aassertion_state,R: state > state > state,W: state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: state,B7: state] :
            ( ( ( package_sat_state @ Plus @ A2 @ A3 )
              & ( ( some_state @ B7 )
                = ( Plus @ A3 @ ( R @ A3 @ W ) ) ) )
           => ( package_sat_state @ Plus @ B2 @ B7 ) )
       => ( packag6192661653625646554_state @ Plus @ W @ R @ A2 @ B2 ) ) ) ).

% package_logic.is_footprint_generalI
thf(fact_554_package__logic_Ois__footprint__standardI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A2: aassertion_val,W: val,B2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: val,B7: val] :
            ( ( ( package_sat_val @ Plus @ A2 @ A3 )
              & ( ( some_val @ B7 )
                = ( Plus @ A3 @ W ) ) )
           => ( package_sat_val @ Plus @ B2 @ B7 ) )
       => ( packag685230357167683994rd_val @ Plus @ W @ A2 @ B2 ) ) ) ).

% package_logic.is_footprint_standardI
thf(fact_555_package__logic_Ois__footprint__standardI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A2: aassertion_state,W: state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: state,B7: state] :
            ( ( ( package_sat_state @ Plus @ A2 @ A3 )
              & ( ( some_state @ B7 )
                = ( Plus @ A3 @ W ) ) )
           => ( package_sat_state @ Plus @ B2 @ B7 ) )
       => ( packag7096211993527611489_state @ Plus @ W @ A2 @ B2 ) ) ) ).

% package_logic.is_footprint_standardI
thf(fact_556_package__logic_Ois__footprint__general__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,W: val,R: val > val > val,A2: aassertion_val,B2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag4897621287243763553al_val @ Plus @ W @ R @ A2 @ B2 )
        = ( ! [A4: val,B3: val] :
              ( ( ( package_sat_val @ Plus @ A2 @ A4 )
                & ( ( some_val @ B3 )
                  = ( Plus @ A4 @ ( R @ A4 @ W ) ) ) )
             => ( package_sat_val @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.is_footprint_general_def
thf(fact_557_package__logic_Ois__footprint__general__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,R: state > state > state,A2: aassertion_state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag6192661653625646554_state @ Plus @ W @ R @ A2 @ B2 )
        = ( ! [A4: state,B3: state] :
              ( ( ( package_sat_state @ Plus @ A2 @ A4 )
                & ( ( some_state @ B3 )
                  = ( Plus @ A4 @ ( R @ A4 @ W ) ) ) )
             => ( package_sat_state @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.is_footprint_general_def
thf(fact_558_package__logic_Ois__footprint__standard__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,W: val,A2: aassertion_val,B2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag685230357167683994rd_val @ Plus @ W @ A2 @ B2 )
        = ( ! [A4: val,B3: val] :
              ( ( ( package_sat_val @ Plus @ A2 @ A4 )
                & ( ( some_val @ B3 )
                  = ( Plus @ A4 @ W ) ) )
             => ( package_sat_val @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.is_footprint_standard_def
thf(fact_559_package__logic_Ois__footprint__standard__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,W: state,A2: aassertion_state,B2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7096211993527611489_state @ Plus @ W @ A2 @ B2 )
        = ( ! [A4: state,B3: state] :
              ( ( ( package_sat_state @ Plus @ A2 @ A4 )
                & ( ( some_state @ B3 )
                  = ( Plus @ A4 @ W ) ) )
             => ( package_sat_state @ Plus @ B2 @ B3 ) ) ) ) ) ).

% package_logic.is_footprint_standard_def
thf(fact_560_package__logic_Omono__pruner__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,P2: val > $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8699728003424501129er_val @ Plus @ P2 )
        = ( ! [Phi5: val,Phi3: val,R2: val] :
              ( ( ( sep_pure_val @ Plus @ R2 )
                & ( P2 @ Phi3 )
                & ( ( some_val @ Phi5 )
                  = ( Plus @ Phi3 @ R2 ) ) )
             => ( P2 @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_561_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 )
        = ( ! [Phi5: state,Phi3: state,R2: state] :
              ( ( ( sep_pure_state @ Plus @ R2 )
                & ( P2 @ Phi3 )
                & ( ( some_state @ Phi5 )
                  = ( Plus @ Phi3 @ R2 ) ) )
             => ( P2 @ Phi5 ) ) ) ) ) ).

% package_logic.mono_pruner_def
thf(fact_562_Rep__state__induct,axiom,
    ! [Y: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi3: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi3 )
              & ( valid_state @ Phi3 ) ) ) )
     => ( ! [X: state] : ( P @ ( rep_state @ X ) )
       => ( P @ Y ) ) ) ).

% Rep_state_induct
thf(fact_563_Rep__state__cases,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y @ ( collec7475294502966882545on_val @ valid_state ) )
     => ~ ! [X: state] :
            ( Y
           != ( rep_state @ X ) ) ) ).

% Rep_state_cases
thf(fact_564_Rep__state,axiom,
    ! [X3: state] :
      ( member2879030624002546611on_val @ ( rep_state @ X3 )
      @ ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [Phi3: produc9003781588162623004on_val] :
            ( ( Uu3 = Phi3 )
            & ( valid_state @ Phi3 ) ) ) ) ).

% Rep_state
thf(fact_565_Abs__state__inject,axiom,
    ! [X3: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ X3
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi3: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi3 )
              & ( valid_state @ Phi3 ) ) ) )
     => ( ( member2879030624002546611on_val @ Y
          @ ( collec7475294502966882545on_val
            @ ^ [Uu3: produc9003781588162623004on_val] :
              ? [Phi3: produc9003781588162623004on_val] :
                ( ( Uu3 = Phi3 )
                & ( valid_state @ Phi3 ) ) ) )
       => ( ( ( abs_state @ X3 )
            = ( abs_state @ Y ) )
          = ( X3 = Y ) ) ) ) ).

% Abs_state_inject
thf(fact_566_Abs__state__induct,axiom,
    ! [P: state > $o,X3: state] :
      ( ! [Y3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ Y3
            @ ( collec7475294502966882545on_val
              @ ^ [Uu3: produc9003781588162623004on_val] :
                ? [Phi3: produc9003781588162623004on_val] :
                  ( ( Uu3 = Phi3 )
                  & ( valid_state @ Phi3 ) ) ) )
         => ( P @ ( abs_state @ Y3 ) ) )
     => ( P @ X3 ) ) ).

% Abs_state_induct
thf(fact_567_Abs__state__cases,axiom,
    ! [X3: state] :
      ~ ! [Y3: produc9003781588162623004on_val] :
          ( ( X3
            = ( abs_state @ Y3 ) )
         => ~ ( member2879030624002546611on_val @ Y3 @ ( collec7475294502966882545on_val @ valid_state ) ) ) ).

% Abs_state_cases
thf(fact_568_package__logic_Owf__assertion__sat__larger__pure,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A2: aassertion_val,Phi: val,Phi2: val,R4: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ A2 )
       => ( ( package_sat_val @ Plus @ A2 @ Phi )
         => ( ( ( some_val @ Phi2 )
              = ( Plus @ Phi @ R4 ) )
           => ( ( sep_pure_val @ Plus @ R4 )
             => ( package_sat_val @ Plus @ A2 @ Phi2 ) ) ) ) ) ) ).

% package_logic.wf_assertion_sat_larger_pure
thf(fact_569_package__logic_Owf__assertion__sat__larger__pure,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A2: aassertion_state,Phi: state,Phi2: state,R4: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ A2 )
       => ( ( package_sat_state @ Plus @ A2 @ Phi )
         => ( ( ( some_state @ Phi2 )
              = ( Plus @ Phi @ R4 ) )
           => ( ( sep_pure_state @ Plus @ R4 )
             => ( package_sat_state @ Plus @ A2 @ Phi2 ) ) ) ) ) ) ).

% package_logic.wf_assertion_sat_larger_pure
thf(fact_570_package__logic_Osat_Ocases,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: produc2756261550280135502_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A10: aassertion_state,B8: aassertion_state,Phi4: state] :
            ( X3
           != ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Phi4 ) )
       => ( ! [B7: state > $o,A10: aassertion_state,Phi4: state] :
              ( X3
             != ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Phi4 ) )
         => ~ ! [A10: state > $o,Phi4: state] :
                ( X3
               != ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Phi4 ) ) ) ) ) ).

% package_logic.sat.cases
thf(fact_571_package__logic_Opackage__sat__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Pc: val > $o,A2: aassertion_val,A6: val,U5: val,U3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A2 @ A6 @ U5 @ U3 )
        = ( ( Pc @ ( Core @ A6 ) )
         => ? [X5: val] :
              ( ( ( some_val @ X5 )
                = ( Plus @ ( Core @ A6 ) @ ( sep_minus_val @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_val @ Plus @ A2 @ X5 ) ) ) ) ) ).

% package_logic.package_sat_def
thf(fact_572_package__logic_Opackage__sat__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A2: aassertion_state,A6: state,U5: state,U3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A2 @ A6 @ U5 @ U3 )
        = ( ( Pc @ ( Core @ A6 ) )
         => ? [X5: state] :
              ( ( ( some_state @ X5 )
                = ( Plus @ ( Core @ A6 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_state @ Plus @ A2 @ X5 ) ) ) ) ) ).

% package_logic.package_sat_def
thf(fact_573_package__logic_Opackage__satI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Pc: val > $o,A6: val,U5: val,U3: val,A2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Pc @ ( Core @ A6 ) )
         => ? [X4: val] :
              ( ( ( some_val @ X4 )
                = ( Plus @ ( Core @ A6 ) @ ( sep_minus_val @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_val @ Plus @ A2 @ X4 ) ) )
       => ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A2 @ A6 @ U5 @ U3 ) ) ) ).

% package_logic.package_satI
thf(fact_574_package__logic_Opackage__satI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Pc: state > $o,A6: state,U5: state,U3: state,A2: aassertion_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Pc @ ( Core @ A6 ) )
         => ? [X4: state] :
              ( ( ( some_state @ X4 )
                = ( Plus @ ( Core @ A6 ) @ ( sep_minus_state @ Plus @ Core @ U5 @ U3 ) ) )
              & ( package_sat_state @ Plus @ A2 @ X4 ) ) )
       => ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A2 @ A6 @ U5 @ U3 ) ) ) ).

% package_logic.package_satI
thf(fact_575_accp_Ocases,axiom,
    ! [R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R4 @ A )
     => ! [Y7: produc1317171644805857433on_val] :
          ( ( R4 @ Y7 @ A )
         => ( accp_P6797916352740768290on_val @ R4 @ Y7 ) ) ) ).

% accp.cases
thf(fact_576_accp_Ocases,axiom,
    ! [R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,A: produc3142500478612311029_state] :
      ( ( accp_P7183347322223220350_state @ R4 @ A )
     => ! [Y7: produc3142500478612311029_state] :
          ( ( R4 @ Y7 @ A )
         => ( accp_P7183347322223220350_state @ R4 @ Y7 ) ) ) ).

% accp.cases
thf(fact_577_accp_Ocases,axiom,
    ! [R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,A: produc4844402304972151915_state] :
      ( ( accp_P6233459008282737652_state @ R4 @ A )
     => ! [Y7: produc4844402304972151915_state] :
          ( ( R4 @ Y7 @ A )
         => ( accp_P6233459008282737652_state @ R4 @ Y7 ) ) ) ).

% accp.cases
thf(fact_578_accp_Ocases,axiom,
    ! [R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,A: produc6251377953716854561_state] :
      ( ( accp_P6554740477754186410_state @ R4 @ A )
     => ! [Y7: produc6251377953716854561_state] :
          ( ( R4 @ Y7 @ A )
         => ( accp_P6554740477754186410_state @ R4 @ Y7 ) ) ) ).

% accp.cases
thf(fact_579_accp_Ocases,axiom,
    ! [R4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R4 @ A )
     => ! [Y7: produc9003781588162623004on_val] :
          ( ( R4 @ Y7 @ A )
         => ( accp_P4405556307444930259on_val @ R4 @ Y7 ) ) ) ).

% accp.cases
thf(fact_580_accp_Osimps,axiom,
    ( accp_P6797916352740768290on_val
    = ( ^ [R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A4: produc1317171644805857433on_val] :
        ? [X5: produc1317171644805857433on_val] :
          ( ( A4 = X5 )
          & ! [Y4: produc1317171644805857433on_val] :
              ( ( R2 @ Y4 @ X5 )
             => ( accp_P6797916352740768290on_val @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_581_accp_Osimps,axiom,
    ( accp_P7183347322223220350_state
    = ( ^ [R2: produc3142500478612311029_state > produc3142500478612311029_state > $o,A4: produc3142500478612311029_state] :
        ? [X5: produc3142500478612311029_state] :
          ( ( A4 = X5 )
          & ! [Y4: produc3142500478612311029_state] :
              ( ( R2 @ Y4 @ X5 )
             => ( accp_P7183347322223220350_state @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_582_accp_Osimps,axiom,
    ( accp_P6233459008282737652_state
    = ( ^ [R2: produc4844402304972151915_state > produc4844402304972151915_state > $o,A4: produc4844402304972151915_state] :
        ? [X5: produc4844402304972151915_state] :
          ( ( A4 = X5 )
          & ! [Y4: produc4844402304972151915_state] :
              ( ( R2 @ Y4 @ X5 )
             => ( accp_P6233459008282737652_state @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_583_accp_Osimps,axiom,
    ( accp_P6554740477754186410_state
    = ( ^ [R2: produc6251377953716854561_state > produc6251377953716854561_state > $o,A4: produc6251377953716854561_state] :
        ? [X5: produc6251377953716854561_state] :
          ( ( A4 = X5 )
          & ! [Y4: produc6251377953716854561_state] :
              ( ( R2 @ Y4 @ X5 )
             => ( accp_P6554740477754186410_state @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_584_accp_Osimps,axiom,
    ( accp_P4405556307444930259on_val
    = ( ^ [R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A4: produc9003781588162623004on_val] :
        ? [X5: produc9003781588162623004on_val] :
          ( ( A4 = X5 )
          & ! [Y4: produc9003781588162623004on_val] :
              ( ( R2 @ Y4 @ X5 )
             => ( accp_P4405556307444930259on_val @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_585_accpI,axiom,
    ! [R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ! [Y3: produc1317171644805857433on_val] :
          ( ( R4 @ Y3 @ X3 )
         => ( accp_P6797916352740768290on_val @ R4 @ Y3 ) )
     => ( accp_P6797916352740768290on_val @ R4 @ X3 ) ) ).

% accpI
thf(fact_586_accpI,axiom,
    ! [R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,X3: produc3142500478612311029_state] :
      ( ! [Y3: produc3142500478612311029_state] :
          ( ( R4 @ Y3 @ X3 )
         => ( accp_P7183347322223220350_state @ R4 @ Y3 ) )
     => ( accp_P7183347322223220350_state @ R4 @ X3 ) ) ).

% accpI
thf(fact_587_accpI,axiom,
    ! [R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,X3: produc4844402304972151915_state] :
      ( ! [Y3: produc4844402304972151915_state] :
          ( ( R4 @ Y3 @ X3 )
         => ( accp_P6233459008282737652_state @ R4 @ Y3 ) )
     => ( accp_P6233459008282737652_state @ R4 @ X3 ) ) ).

% accpI
thf(fact_588_accpI,axiom,
    ! [R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,X3: produc6251377953716854561_state] :
      ( ! [Y3: produc6251377953716854561_state] :
          ( ( R4 @ Y3 @ X3 )
         => ( accp_P6554740477754186410_state @ R4 @ Y3 ) )
     => ( accp_P6554740477754186410_state @ R4 @ X3 ) ) ).

% accpI
thf(fact_589_accpI,axiom,
    ! [R4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val] :
      ( ! [Y3: produc9003781588162623004on_val] :
          ( ( R4 @ Y3 @ X3 )
         => ( accp_P4405556307444930259on_val @ R4 @ Y3 ) )
     => ( accp_P4405556307444930259on_val @ R4 @ X3 ) ) ).

% accpI
thf(fact_590_accp__induct,axiom,
    ! [R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R4 @ A )
     => ( ! [X: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R4 @ X )
           => ( ! [Y7: produc1317171644805857433on_val] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_591_accp__induct,axiom,
    ! [R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,A: produc3142500478612311029_state,P: produc3142500478612311029_state > $o] :
      ( ( accp_P7183347322223220350_state @ R4 @ A )
     => ( ! [X: produc3142500478612311029_state] :
            ( ( accp_P7183347322223220350_state @ R4 @ X )
           => ( ! [Y7: produc3142500478612311029_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_592_accp__induct,axiom,
    ! [R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,A: produc4844402304972151915_state,P: produc4844402304972151915_state > $o] :
      ( ( accp_P6233459008282737652_state @ R4 @ A )
     => ( ! [X: produc4844402304972151915_state] :
            ( ( accp_P6233459008282737652_state @ R4 @ X )
           => ( ! [Y7: produc4844402304972151915_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_593_accp__induct,axiom,
    ! [R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,A: produc6251377953716854561_state,P: produc6251377953716854561_state > $o] :
      ( ( accp_P6554740477754186410_state @ R4 @ A )
     => ( ! [X: produc6251377953716854561_state] :
            ( ( accp_P6554740477754186410_state @ R4 @ X )
           => ( ! [Y7: produc6251377953716854561_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_594_accp__induct,axiom,
    ! [R4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R4 @ A )
     => ( ! [X: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R4 @ X )
           => ( ! [Y7: produc9003781588162623004on_val] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_595_accp__downward,axiom,
    ! [R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,B: produc1317171644805857433on_val,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R4 @ B )
     => ( ( R4 @ A @ B )
       => ( accp_P6797916352740768290on_val @ R4 @ A ) ) ) ).

% accp_downward
thf(fact_596_accp__downward,axiom,
    ! [R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,B: produc3142500478612311029_state,A: produc3142500478612311029_state] :
      ( ( accp_P7183347322223220350_state @ R4 @ B )
     => ( ( R4 @ A @ B )
       => ( accp_P7183347322223220350_state @ R4 @ A ) ) ) ).

% accp_downward
thf(fact_597_accp__downward,axiom,
    ! [R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,B: produc4844402304972151915_state,A: produc4844402304972151915_state] :
      ( ( accp_P6233459008282737652_state @ R4 @ B )
     => ( ( R4 @ A @ B )
       => ( accp_P6233459008282737652_state @ R4 @ A ) ) ) ).

% accp_downward
thf(fact_598_accp__downward,axiom,
    ! [R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,B: produc6251377953716854561_state,A: produc6251377953716854561_state] :
      ( ( accp_P6554740477754186410_state @ R4 @ B )
     => ( ( R4 @ A @ B )
       => ( accp_P6554740477754186410_state @ R4 @ A ) ) ) ).

% accp_downward
thf(fact_599_accp__downward,axiom,
    ! [R4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,B: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R4 @ B )
     => ( ( R4 @ A @ B )
       => ( accp_P4405556307444930259on_val @ R4 @ A ) ) ) ).

% accp_downward
thf(fact_600_not__accp__down,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val] :
      ( ~ ( accp_P6797916352740768290on_val @ R @ X3 )
     => ~ ! [Z3: produc1317171644805857433on_val] :
            ( ( R @ Z3 @ X3 )
           => ( accp_P6797916352740768290on_val @ R @ Z3 ) ) ) ).

% not_accp_down
thf(fact_601_not__accp__down,axiom,
    ! [R: produc3142500478612311029_state > produc3142500478612311029_state > $o,X3: produc3142500478612311029_state] :
      ( ~ ( accp_P7183347322223220350_state @ R @ X3 )
     => ~ ! [Z3: produc3142500478612311029_state] :
            ( ( R @ Z3 @ X3 )
           => ( accp_P7183347322223220350_state @ R @ Z3 ) ) ) ).

% not_accp_down
thf(fact_602_not__accp__down,axiom,
    ! [R: produc4844402304972151915_state > produc4844402304972151915_state > $o,X3: produc4844402304972151915_state] :
      ( ~ ( accp_P6233459008282737652_state @ R @ X3 )
     => ~ ! [Z3: produc4844402304972151915_state] :
            ( ( R @ Z3 @ X3 )
           => ( accp_P6233459008282737652_state @ R @ Z3 ) ) ) ).

% not_accp_down
thf(fact_603_not__accp__down,axiom,
    ! [R: produc6251377953716854561_state > produc6251377953716854561_state > $o,X3: produc6251377953716854561_state] :
      ( ~ ( accp_P6554740477754186410_state @ R @ X3 )
     => ~ ! [Z3: produc6251377953716854561_state] :
            ( ( R @ Z3 @ X3 )
           => ( accp_P6554740477754186410_state @ R @ Z3 ) ) ) ).

% not_accp_down
thf(fact_604_not__accp__down,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val] :
      ( ~ ( accp_P4405556307444930259on_val @ R @ X3 )
     => ~ ! [Z3: produc9003781588162623004on_val] :
            ( ( R @ Z3 @ X3 )
           => ( accp_P4405556307444930259on_val @ R @ Z3 ) ) ) ).

% not_accp_down
thf(fact_605_accp__induct__rule,axiom,
    ! [R4: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R4 @ A )
     => ( ! [X: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R4 @ X )
           => ( ! [Y7: produc1317171644805857433on_val] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_606_accp__induct__rule,axiom,
    ! [R4: produc3142500478612311029_state > produc3142500478612311029_state > $o,A: produc3142500478612311029_state,P: produc3142500478612311029_state > $o] :
      ( ( accp_P7183347322223220350_state @ R4 @ A )
     => ( ! [X: produc3142500478612311029_state] :
            ( ( accp_P7183347322223220350_state @ R4 @ X )
           => ( ! [Y7: produc3142500478612311029_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_607_accp__induct__rule,axiom,
    ! [R4: produc4844402304972151915_state > produc4844402304972151915_state > $o,A: produc4844402304972151915_state,P: produc4844402304972151915_state > $o] :
      ( ( accp_P6233459008282737652_state @ R4 @ A )
     => ( ! [X: produc4844402304972151915_state] :
            ( ( accp_P6233459008282737652_state @ R4 @ X )
           => ( ! [Y7: produc4844402304972151915_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_608_accp__induct__rule,axiom,
    ! [R4: produc6251377953716854561_state > produc6251377953716854561_state > $o,A: produc6251377953716854561_state,P: produc6251377953716854561_state > $o] :
      ( ( accp_P6554740477754186410_state @ R4 @ A )
     => ( ! [X: produc6251377953716854561_state] :
            ( ( accp_P6554740477754186410_state @ R4 @ X )
           => ( ! [Y7: produc6251377953716854561_state] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_609_accp__induct__rule,axiom,
    ! [R4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R4 @ A )
     => ( ! [X: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R4 @ X )
           => ( ! [Y7: produc9003781588162623004on_val] :
                  ( ( R4 @ Y7 @ X )
                 => ( P @ Y7 ) )
             => ( P @ X ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_610_package__logic_Osat_Oelims_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_val @ Plus @ X3 @ Xa2 )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X3
                = ( aStar_val @ A10 @ B8 ) )
             => ? [A5: val,B9: val] :
                  ( ( ( some_val @ Xa2 )
                    = ( Plus @ A5 @ B9 ) )
                  & ( package_sat_val @ Plus @ A10 @ A5 )
                  & ( package_sat_val @ Plus @ B8 @ B9 ) ) )
         => ( ! [B7: val > $o,A10: aassertion_val] :
                ( ( X3
                  = ( aImp_val @ B7 @ A10 ) )
               => ( ( B7 @ Xa2 )
                 => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X3
                    = ( aSem_val @ A10 ) )
                 => ( A10 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(3)
thf(fact_611_package__logic_Osat_Oelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X3 @ Xa2 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ? [A5: state,B9: state] :
                  ( ( ( some_state @ Xa2 )
                    = ( Plus @ A5 @ B9 ) )
                  & ( package_sat_state @ Plus @ A10 @ A5 )
                  & ( package_sat_state @ Plus @ B8 @ B9 ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( B7 @ Xa2 )
                 => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( A10 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(3)
thf(fact_612_package__logic_Osat_Oelims_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ X3 @ Xa2 )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X3
                = ( aStar_val @ A10 @ B8 ) )
             => ~ ? [A3: val,B7: val] :
                    ( ( ( some_val @ Xa2 )
                      = ( Plus @ A3 @ B7 ) )
                    & ( package_sat_val @ Plus @ A10 @ A3 )
                    & ( package_sat_val @ Plus @ B8 @ B7 ) ) )
         => ( ! [B7: val > $o,A10: aassertion_val] :
                ( ( X3
                  = ( aImp_val @ B7 @ A10 ) )
               => ~ ( ( B7 @ Xa2 )
                   => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X3
                    = ( aSem_val @ A10 ) )
                 => ~ ( A10 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(2)
thf(fact_613_package__logic_Osat_Oelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X3 @ Xa2 )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ~ ? [A3: state,B7: state] :
                    ( ( ( some_state @ Xa2 )
                      = ( Plus @ A3 @ B7 ) )
                    & ( package_sat_state @ Plus @ A10 @ A3 )
                    & ( package_sat_state @ Plus @ B8 @ B7 ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ~ ( ( B7 @ Xa2 )
                   => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ~ ( A10 @ Xa2 ) ) ) ) ) ) ).

% package_logic.sat.elims(2)
thf(fact_614_package__logic_Osat_Oelims_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val,Y: $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_val @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ! [A10: aassertion_val,B8: aassertion_val] :
              ( ( X3
                = ( aStar_val @ A10 @ B8 ) )
             => ( Y
                = ( ~ ? [A4: val,B3: val] :
                        ( ( ( some_val @ Xa2 )
                          = ( Plus @ A4 @ B3 ) )
                        & ( package_sat_val @ Plus @ A10 @ A4 )
                        & ( package_sat_val @ Plus @ B8 @ B3 ) ) ) ) )
         => ( ! [B7: val > $o,A10: aassertion_val] :
                ( ( X3
                  = ( aImp_val @ B7 @ A10 ) )
               => ( Y
                  = ( ~ ( ( B7 @ Xa2 )
                       => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) ) ) )
           => ~ ! [A10: val > $o] :
                  ( ( X3
                    = ( aSem_val @ A10 ) )
                 => ( Y
                    = ( ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.elims(1)
thf(fact_615_package__logic_Osat_Oelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( Y
                = ( ~ ? [A4: state,B3: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( Plus @ A4 @ B3 ) )
                        & ( package_sat_state @ Plus @ A10 @ A4 )
                        & ( package_sat_state @ Plus @ B8 @ B3 ) ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( Y
                  = ( ~ ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( Y
                    = ( ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.elims(1)
thf(fact_616_Abs__state__inverse,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi3: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi3 )
              & ( valid_state @ Phi3 ) ) ) )
     => ( ( rep_state @ ( abs_state @ Y ) )
        = Y ) ) ).

% Abs_state_inverse
thf(fact_617_PartialSA_Opackage__logic__axioms,axiom,
    package_logic_state @ plus @ core @ unit @ stable ).

% PartialSA.package_logic_axioms
thf(fact_618_package__logic_Ogeneral__completeness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,S2: set_Pr4723104092121316667al_val,F: val,A2: aassertion_val,Phi: val,Phi2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: val,U: val,T3: val > val] :
            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A3 @ ( produc303288909735261391al_val @ U @ T3 ) ) @ S2 )
           => ! [X: val] :
                ( ( ( some_val @ X )
                  = ( Plus @ A3 @ ( T3 @ F ) ) )
               => ( package_sat_val @ Plus @ A2 @ X ) ) )
       => ( ( ( some_val @ Phi )
            = ( Plus @ Phi2 @ F ) )
         => ( ( Stable @ F )
           => ( ( packag7517340229427718612et_val @ Plus @ Core @ Unit @ S2 @ Unit )
             => ( ( packag5732365195153136185on_val @ Plus @ Core @ A2 )
               => ? [X_1: set_Pr4723104092121316667al_val] :
                    ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi @ Unit @ S2
                    @ ^ [Uu3: val] : $true
                    @ A2
                    @ Phi2
                    @ F
                    @ X_1 ) ) ) ) ) ) ) ).

% package_logic.general_completeness
thf(fact_619_package__logic_Ogeneral__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,S2: set_Pr4231240917283394591_state,F: state,A2: aassertion_state,Phi: state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ! [X: state] :
                ( ( ( some_state @ X )
                  = ( Plus @ A3 @ ( T3 @ F ) ) )
               => ( package_sat_state @ Plus @ A2 @ X ) ) )
       => ( ( ( some_state @ Phi )
            = ( Plus @ Phi2 @ F ) )
         => ( ( Stable @ F )
           => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ Unit )
             => ( ( packag7298478212905792770_state @ Plus @ Core @ A2 )
               => ? [X_1: set_Pr4231240917283394591_state] :
                    ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ Unit @ S2
                    @ ^ [Uu3: state] : $true
                    @ A2
                    @ Phi2
                    @ F
                    @ X_1 ) ) ) ) ) ) ) ).

% package_logic.general_completeness
thf(fact_620_package__logic_Ocompleteness__paper,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B2: aassertion_val,W: val,A2: aassertion_val,Sigma: val,Sigma2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag685230357167683994rd_val @ Plus @ W @ A2 @ B2 ) )
         => ( ( ( some_val @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ? [X_1: set_Pr4723104092121316667al_val] :
                ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A4: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ Unit @ id_val ) ) )
                      & ( package_sat_val @ Plus @ A2 @ A4 ) ) )
                @ ^ [Uu3: val] : $true
                @ B2
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% package_logic.completeness_paper
thf(fact_621_package__logic_Ocompleteness__paper,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,W: state,A2: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag7096211993527611489_state @ Plus @ W @ A2 @ B2 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ? [X_1: set_Pr4231240917283394591_state] :
                ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ Unit @ id_state ) ) )
                      & ( package_sat_state @ Plus @ A2 @ A4 ) ) )
                @ ^ [Uu3: state] : $true
                @ B2
                @ Sigma2
                @ W
                @ X_1 ) ) ) ) ) ).

% package_logic.completeness_paper
thf(fact_622_package__logic_Ocompleteness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,B2: aassertion_val,W: val,R: val > val > val,A2: aassertion_val,Sigma: val,Sigma2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag5732365195153136185on_val @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag4897621287243763553al_val @ Plus @ W @ R @ A2 @ B2 ) )
         => ( ( ( some_val @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ( ! [A3: val] :
                  ( ( package_sat_val @ Plus @ A2 @ A3 )
                 => ( packag4175576287423129206er_val @ Plus @ Unit @ ( R @ A3 ) ) )
             => ? [X_1: set_Pr4723104092121316667al_val] :
                  ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Sigma @ Unit
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A4: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ Unit @ ( R @ A4 ) ) ) )
                        & ( package_sat_val @ Plus @ A2 @ A4 ) ) )
                  @ ^ [Uu3: val] : $true
                  @ B2
                  @ Sigma2
                  @ W
                  @ X_1 ) ) ) ) ) ) ).

% package_logic.completeness
thf(fact_623_package__logic_Ocompleteness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,B2: aassertion_state,W: state,R: state > state > state,A2: aassertion_state,Sigma: state,Sigma2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
       => ( ( ( Stable @ W )
            & ( packag6192661653625646554_state @ Plus @ W @ R @ A2 @ B2 ) )
         => ( ( ( some_state @ Sigma )
              = ( Plus @ Sigma2 @ W ) )
           => ( ! [A3: state] :
                  ( ( package_sat_state @ Plus @ A2 @ A3 )
                 => ( packag4817583915586052229_state @ Plus @ Unit @ ( R @ A3 ) ) )
             => ? [X_1: set_Pr4231240917283394591_state] :
                  ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Sigma @ Unit
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A4: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ Unit @ ( R @ A4 ) ) ) )
                        & ( package_sat_state @ Plus @ A2 @ A4 ) ) )
                  @ ^ [Uu3: state] : $true
                  @ B2
                  @ Sigma2
                  @ W
                  @ X_1 ) ) ) ) ) ) ).

% package_logic.completeness
thf(fact_624_accp__subset__induct,axiom,
    ! [D2: produc1317171644805857433on_val > $o,R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X3: produc1317171644805857433on_val,P: produc1317171644805857433on_val > $o] :
      ( ( ord_le8528861841627966596_val_o @ D2 @ ( accp_P6797916352740768290on_val @ R ) )
     => ( ! [X: produc1317171644805857433on_val,Z3: produc1317171644805857433on_val] :
            ( ( D2 @ X )
           => ( ( R @ Z3 @ X )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X: produc1317171644805857433on_val] :
                ( ( D2 @ X )
               => ( ! [Z4: produc1317171644805857433on_val] :
                      ( ( R @ Z4 @ X )
                     => ( P @ Z4 ) )
                 => ( P @ X ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_625_accp__subset__induct,axiom,
    ! [D2: produc3142500478612311029_state > $o,R: produc3142500478612311029_state > produc3142500478612311029_state > $o,X3: produc3142500478612311029_state,P: produc3142500478612311029_state > $o] :
      ( ( ord_le5366583112655343912tate_o @ D2 @ ( accp_P7183347322223220350_state @ R ) )
     => ( ! [X: produc3142500478612311029_state,Z3: produc3142500478612311029_state] :
            ( ( D2 @ X )
           => ( ( R @ Z3 @ X )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X: produc3142500478612311029_state] :
                ( ( D2 @ X )
               => ( ! [Z4: produc3142500478612311029_state] :
                      ( ( R @ Z4 @ X )
                     => ( P @ Z4 ) )
                 => ( P @ X ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_626_accp__subset__induct,axiom,
    ! [D2: produc4844402304972151915_state > $o,R: produc4844402304972151915_state > produc4844402304972151915_state > $o,X3: produc4844402304972151915_state,P: produc4844402304972151915_state > $o] :
      ( ( ord_le5681876185599785522tate_o @ D2 @ ( accp_P6233459008282737652_state @ R ) )
     => ( ! [X: produc4844402304972151915_state,Z3: produc4844402304972151915_state] :
            ( ( D2 @ X )
           => ( ( R @ Z3 @ X )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X: produc4844402304972151915_state] :
                ( ( D2 @ X )
               => ( ! [Z4: produc4844402304972151915_state] :
                      ( ( R @ Z4 @ X )
                     => ( P @ Z4 ) )
                 => ( P @ X ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_627_accp__subset__induct,axiom,
    ! [D2: produc6251377953716854561_state > $o,R: produc6251377953716854561_state > produc6251377953716854561_state > $o,X3: produc6251377953716854561_state,P: produc6251377953716854561_state > $o] :
      ( ( ord_le8699832056682887676tate_o @ D2 @ ( accp_P6554740477754186410_state @ R ) )
     => ( ! [X: produc6251377953716854561_state,Z3: produc6251377953716854561_state] :
            ( ( D2 @ X )
           => ( ( R @ Z3 @ X )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X: produc6251377953716854561_state] :
                ( ( D2 @ X )
               => ( ! [Z4: produc6251377953716854561_state] :
                      ( ( R @ Z4 @ X )
                     => ( P @ Z4 ) )
                 => ( P @ X ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_628_accp__subset__induct,axiom,
    ! [D2: produc9003781588162623004on_val > $o,R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( ord_le6140313152057869779_val_o @ D2 @ ( accp_P4405556307444930259on_val @ R ) )
     => ( ! [X: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] :
            ( ( D2 @ X )
           => ( ( R @ Z3 @ X )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X3 )
         => ( ! [X: produc9003781588162623004on_val] :
                ( ( D2 @ X )
               => ( ! [Z4: produc9003781588162623004on_val] :
                      ( ( R @ Z4 @ X )
                     => ( P @ Z4 ) )
                 => ( P @ X ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_629_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 )
      = ( ? [Phi3: state,F5: state,S5: set_Pr4231240917283394591_state,Pc2: state > $o,A7: aassertion_state,Phi5: state,F6: state,S6: set_Pr4231240917283394591_state,B4: aassertion_state,Phi8: state,F7: state,S7: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi3 )
            & ( A22 = F5 )
            & ( A32 = S5 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aStar_state @ A7 @ B4 ) )
            & ( A62 = Phi8 )
            & ( A72 = F7 )
            & ( A82 = S7 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F5 @ S5 @ Pc2 @ A7 @ Phi5 @ F6 @ S6 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F6 @ S6 @ Pc2 @ B4 @ Phi8 @ F7 @ S7 ) )
        | ? [Phi3: state,F5: state,S5: set_Pr4231240917283394591_state,Pc2: state > $o,B3: state > $o,A7: aassertion_state,Phi5: state,F6: state,S6: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi3 )
            & ( A22 = F5 )
            & ( A32 = S5 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aImp_state @ B3 @ A7 ) )
            & ( A62 = Phi5 )
            & ( A72 = F6 )
            & ( A82 = S6 )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi3 @ F5 @ S5 @ ( packag1330488657391168505_state @ Pc2 @ B3 ) @ A7 @ Phi5 @ F6 @ S6 ) )
        | ? [S5: set_Pr4231240917283394591_state,Pc2: state > $o,Witness: produc7272280185086996969_state > state,B4: state > $o,S6: set_Pr4231240917283394591_state,Phi3: state,F5: state] :
            ( ( A1 = Phi3 )
            & ( A22 = F5 )
            & ( A32 = S5 )
            & ( A42 = Pc2 )
            & ( A52
              = ( aSem_state @ B4 ) )
            & ( A62 = Phi3 )
            & ( A72 = F5 )
            & ( A82 = S6 )
            & ! [X5: state,Y4: state,Z5: state > state,Aa2: state] :
                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ ( produc8632743443909064678_state @ Y4 @ Z5 ) ) @ S5 )
               => ( ( Pc2 @ X5 )
                 => ( ( Aa2
                      = ( Witness @ ( produc5886875603596089627_state @ X5 @ ( produc8632743443909064678_state @ Y4 @ Z5 ) ) ) )
                   => ( ( greater @ X5 @ Aa2 )
                      & ( B4 @ Aa2 ) ) ) ) )
            & ( S6
              = ( sup_su3837536599209510219_state
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state,U6: state,T4: state > state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                      & ~ ( Pc2 @ A4 ) ) )
                @ ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state,U6: state,T4: state > state,B3: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ ( minus @ A4 @ B3 ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B3 ) ) @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                      & ( Pc2 @ A4 )
                      & ( B3
                        = ( Witness @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) ) )
        | ? [Phi3: state,Phi5: state,M4: state,F6: state,S6: set_Pr4231240917283394591_state,Pc2: state > $o,A7: aassertion_state,Phi8: state,F7: state,S7: set_Pr4231240917283394591_state,F5: state,S5: set_Pr4231240917283394591_state] :
            ( ( A1 = Phi3 )
            & ( A22 = F5 )
            & ( A32 = S5 )
            & ( A42 = Pc2 )
            & ( A52 = A7 )
            & ( A62 = Phi8 )
            & ( A72 = F7 )
            & ( A82 = S7 )
            & ( ( some_state @ Phi3 )
              = ( plus @ Phi5 @ M4 ) )
            & ( packag3171892606229212990_state @ plus @ core @ stable @ Phi5 @ F6 @ S6 @ Pc2 @ A7 @ Phi8 @ F7 @ S7 )
            & ( stable @ M4 )
            & ( ( some_state @ F6 )
              = ( plus @ F5 @ M4 ) )
            & ( S6
              = ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state,U6: state,T4: state > state,R2: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                    & ( ( some_state @ R2 )
                      = ( plus @ A4 @ ( minus @ ( T4 @ F6 ) @ ( T4 @ F5 ) ) ) )
                    & ( defined @ R2 @ U6 ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.simps
thf(fact_630_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 )
     => ( ! [A10: aassertion_state,Phi6: state,F8: state,S8: set_Pr4231240917283394591_state,B8: aassertion_state] :
            ( ( A52
              = ( aStar_state @ A10 @ B8 ) )
           => ( ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ A42 @ A10 @ Phi6 @ F8 @ S8 )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F8 @ S8 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
       => ( ! [B7: state > $o,A10: aassertion_state] :
              ( ( A52
                = ( aImp_state @ B7 @ A10 ) )
             => ~ ( packag3171892606229212990_state @ plus @ core @ stable @ A1 @ A22 @ A32 @ ( packag1330488657391168505_state @ A42 @ B7 ) @ A10 @ A62 @ A72 @ A82 ) )
         => ( ! [Witness2: produc7272280185086996969_state > state,B8: state > $o] :
                ( ( A52
                  = ( aSem_state @ B8 ) )
               => ( ( A62 = A1 )
                 => ( ( A72 = A22 )
                   => ( ! [A5: state,U7: state,T5: state > state] :
                          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) @ A32 )
                         => ( ( A42 @ A5 )
                           => ( ( greater @ A5 @ ( Witness2 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) )
                              & ( B8 @ ( Witness2 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) ) ) ) )
                     => ( A82
                       != ( sup_su3837536599209510219_state
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A4: state,U6: state,T4: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                & ~ ( A42 @ A4 ) ) )
                          @ ( collec906845016635587262_state
                            @ ^ [Uu3: produc7272280185086996969_state] :
                              ? [A4: state,U6: state,T4: state > state] :
                                ( ( Uu3
                                  = ( produc5886875603596089627_state @ ( minus @ A4 @ ( Witness2 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ ( Witness2 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) @ T4 ) ) )
                                & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                & ( A42 @ A4 ) ) ) ) ) ) ) ) )
           => ~ ! [Phi6: state,M5: state] :
                  ( ( ( some_state @ A1 )
                    = ( plus @ Phi6 @ M5 ) )
                 => ! [F8: state,S8: set_Pr4231240917283394591_state] :
                      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi6 @ F8 @ S8 @ A42 @ A52 @ A62 @ A72 @ A82 )
                     => ( ( stable @ M5 )
                       => ( ( ( some_state @ F8 )
                            = ( plus @ A22 @ M5 ) )
                         => ( S8
                           != ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A4: state,U6: state,T4: state > state,R2: state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ( ( some_state @ R2 )
                                    = ( plus @ A4 @ ( minus @ ( T4 @ F8 ) @ ( T4 @ A22 ) ) ) )
                                  & ( defined @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% PartialSA.package_rhs.cases
thf(fact_631_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] :
      ( ! [A3: state,U: state,T3: state > state,B7: state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( Pc @ A3 )
           => ( ( B7
                = ( Witness3 @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) ) )
             => ( ( greater @ A3 @ B7 )
                & ( B2 @ B7 ) ) ) ) )
     => ( ( S3
          = ( sup_su3837536599209510219_state
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A4: state,U6: state,T4: state > state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                  & ~ ( Pc @ A4 ) ) )
            @ ( collec906845016635587262_state
              @ ^ [Uu3: produc7272280185086996969_state] :
                ? [A4: state,U6: state,T4: state > state,B3: state] :
                  ( ( Uu3
                    = ( produc5886875603596089627_state @ ( minus @ A4 @ B3 ) @ ( produc8632743443909064678_state @ ( the_state @ ( plus @ U6 @ B3 ) ) @ T4 ) ) )
                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                  & ( Pc @ A4 )
                  & ( B3
                    = ( Witness3 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) )
       => ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ ( aSem_state @ B2 ) @ Phi @ F @ S3 ) ) ) ).

% PartialSA.ASem
thf(fact_632_package__logic_Osat_Opelims_I1_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val,Y: $o] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_val @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X3
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( Y
                    = ( ? [A4: val,B3: val] :
                          ( ( ( some_val @ Xa2 )
                            = ( Plus @ A4 @ B3 ) )
                          & ( package_sat_val @ Plus @ A10 @ A4 )
                          & ( package_sat_val @ Plus @ B8 @ B3 ) ) ) )
                 => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa2 ) ) ) )
           => ( ! [B7: val > $o,A10: aassertion_val] :
                  ( ( X3
                    = ( aImp_val @ B7 @ A10 ) )
                 => ( ( Y
                      = ( ( B7 @ Xa2 )
                       => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) )
                   => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B7 @ A10 ) @ Xa2 ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X3
                      = ( aSem_val @ A10 ) )
                   => ( ( Y
                        = ( A10 @ Xa2 ) )
                     => ~ ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa2 ) ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(1)
thf(fact_633_package__logic_Osat_Opelims_I1_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state,Y: $o] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( package_sat_state @ Plus @ X3 @ Xa2 )
          = Y )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X3
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( Y
                    = ( ? [A4: state,B3: state] :
                          ( ( ( some_state @ Xa2 )
                            = ( Plus @ A4 @ B3 ) )
                          & ( package_sat_state @ Plus @ A10 @ A4 )
                          & ( package_sat_state @ Plus @ B8 @ B3 ) ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) ) ) )
           => ( ! [B7: state > $o,A10: aassertion_state] :
                  ( ( X3
                    = ( aImp_state @ B7 @ A10 ) )
                 => ( ( Y
                      = ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X3
                      = ( aSem_state @ A10 ) )
                   => ( ( Y
                        = ( A10 @ Xa2 ) )
                     => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(1)
thf(fact_634_UnCI,axiom,
    ! [C: state,B2: set_state,A2: set_state] :
      ( ( ~ ( member_state @ C @ B2 )
       => ( member_state @ C @ A2 ) )
     => ( member_state @ C @ ( sup_sup_set_state @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_635_UnCI,axiom,
    ! [C: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,A2: set_Pr7649655576376376530on_val] :
      ( ( ~ ( member2879030624002546611on_val @ C @ B2 )
       => ( member2879030624002546611on_val @ C @ A2 ) )
     => ( member2879030624002546611on_val @ C @ ( sup_su7506060154676209150on_val @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_636_UnCI,axiom,
    ! [C: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,A2: set_Pr2391701600042252046_state] :
      ( ( ~ ( member6715010751053357271_state @ C @ B2 )
       => ( member6715010751053357271_state @ C @ A2 ) )
     => ( member6715010751053357271_state @ C @ ( sup_su1984960714304006370_state @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_637_UnCI,axiom,
    ! [C: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] :
      ( ( ~ ( member7461423308196221312_state @ C @ B2 )
       => ( member7461423308196221312_state @ C @ A2 ) )
     => ( member7461423308196221312_state @ C @ ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_638_Un__iff,axiom,
    ! [C: state,A2: set_state,B2: set_state] :
      ( ( member_state @ C @ ( sup_sup_set_state @ A2 @ B2 ) )
      = ( ( member_state @ C @ A2 )
        | ( member_state @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_639_Un__iff,axiom,
    ! [C: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C @ ( sup_su7506060154676209150on_val @ A2 @ B2 ) )
      = ( ( member2879030624002546611on_val @ C @ A2 )
        | ( member2879030624002546611on_val @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_640_Un__iff,axiom,
    ! [C: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C @ ( sup_su1984960714304006370_state @ A2 @ B2 ) )
      = ( ( member6715010751053357271_state @ C @ A2 )
        | ( member6715010751053357271_state @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_641_Un__iff,axiom,
    ! [C: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
      = ( ( member7461423308196221312_state @ C @ A2 )
        | ( member7461423308196221312_state @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_642_Un__subset__iff,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A2 @ B2 ) @ C2 )
      = ( ( ord_le4032880777312351359_state @ A2 @ C2 )
        & ( ord_le4032880777312351359_state @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_643_Un__subset__iff,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A2 @ B2 ) @ C2 )
      = ( ( ord_le2494988322063910608_state @ A2 @ C2 )
        & ( ord_le2494988322063910608_state @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_644_UnE,axiom,
    ! [C: state,A2: set_state,B2: set_state] :
      ( ( member_state @ C @ ( sup_sup_set_state @ A2 @ B2 ) )
     => ( ~ ( member_state @ C @ A2 )
       => ( member_state @ C @ B2 ) ) ) ).

% UnE
thf(fact_645_UnE,axiom,
    ! [C: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C @ ( sup_su7506060154676209150on_val @ A2 @ B2 ) )
     => ( ~ ( member2879030624002546611on_val @ C @ A2 )
       => ( member2879030624002546611on_val @ C @ B2 ) ) ) ).

% UnE
thf(fact_646_UnE,axiom,
    ! [C: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C @ ( sup_su1984960714304006370_state @ A2 @ B2 ) )
     => ( ~ ( member6715010751053357271_state @ C @ A2 )
       => ( member6715010751053357271_state @ C @ B2 ) ) ) ).

% UnE
thf(fact_647_UnE,axiom,
    ! [C: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
     => ( ~ ( member7461423308196221312_state @ C @ A2 )
       => ( member7461423308196221312_state @ C @ B2 ) ) ) ).

% UnE
thf(fact_648_UnI1,axiom,
    ! [C: state,A2: set_state,B2: set_state] :
      ( ( member_state @ C @ A2 )
     => ( member_state @ C @ ( sup_sup_set_state @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_649_UnI1,axiom,
    ! [C: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C @ A2 )
     => ( member2879030624002546611on_val @ C @ ( sup_su7506060154676209150on_val @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_650_UnI1,axiom,
    ! [C: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C @ A2 )
     => ( member6715010751053357271_state @ C @ ( sup_su1984960714304006370_state @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_651_UnI1,axiom,
    ! [C: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C @ A2 )
     => ( member7461423308196221312_state @ C @ ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_652_UnI2,axiom,
    ! [C: state,B2: set_state,A2: set_state] :
      ( ( member_state @ C @ B2 )
     => ( member_state @ C @ ( sup_sup_set_state @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_653_UnI2,axiom,
    ! [C: produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val,A2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C @ B2 )
     => ( member2879030624002546611on_val @ C @ ( sup_su7506060154676209150on_val @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_654_UnI2,axiom,
    ! [C: produc3882902684156697774_state,B2: set_Pr2391701600042252046_state,A2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ C @ B2 )
     => ( member6715010751053357271_state @ C @ ( sup_su1984960714304006370_state @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_655_UnI2,axiom,
    ! [C: produc7272280185086996969_state,B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ C @ B2 )
     => ( member7461423308196221312_state @ C @ ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_656_bex__Un,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( ? [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
            & ( P @ X5 ) ) )
      = ( ? [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ A2 )
            & ( P @ X5 ) )
        | ? [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ B2 )
            & ( P @ X5 ) ) ) ) ).

% bex_Un
thf(fact_657_ball__Un,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
           => ( P @ X5 ) ) )
      = ( ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ A2 )
           => ( P @ X5 ) )
        & ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ B2 )
           => ( P @ X5 ) ) ) ) ).

% ball_Un
thf(fact_658_Un__assoc,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ ( sup_su3837536599209510219_state @ A2 @ B2 ) @ C2 )
      = ( sup_su3837536599209510219_state @ A2 @ ( sup_su3837536599209510219_state @ B2 @ C2 ) ) ) ).

% Un_assoc
thf(fact_659_Un__absorb,axiom,
    ! [A2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_660_Un__commute,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] : ( sup_su3837536599209510219_state @ B4 @ A7 ) ) ) ).

% Un_commute
thf(fact_661_Un__left__absorb,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A2 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
      = ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ).

% Un_left_absorb
thf(fact_662_Un__left__commute,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state] :
      ( ( sup_su3837536599209510219_state @ A2 @ ( sup_su3837536599209510219_state @ B2 @ C2 ) )
      = ( sup_su3837536599209510219_state @ B2 @ ( sup_su3837536599209510219_state @ A2 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_663_Un__mono,axiom,
    ! [A2: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state,D2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A2 @ C2 )
     => ( ( ord_le4032880777312351359_state @ B2 @ D2 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A2 @ B2 ) @ ( sup_su3837536599209510219_state @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_664_Un__mono,axiom,
    ! [A2: set_state,C2: set_state,B2: set_state,D2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ C2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ D2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A2 @ B2 ) @ ( sup_sup_set_state @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_665_Un__least,axiom,
    ! [A2: set_Pr4231240917283394591_state,C2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A2 @ C2 )
     => ( ( ord_le4032880777312351359_state @ B2 @ C2 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A2 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_666_Un__least,axiom,
    ! [A2: set_state,C2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ C2 )
     => ( ( ord_le2494988322063910608_state @ B2 @ C2 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A2 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_667_Un__upper1,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ A2 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ).

% Un_upper1
thf(fact_668_Un__upper1,axiom,
    ! [A2: set_state,B2: set_state] : ( ord_le2494988322063910608_state @ A2 @ ( sup_sup_set_state @ A2 @ B2 ) ) ).

% Un_upper1
thf(fact_669_Un__upper2,axiom,
    ! [B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ B2 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) ) ).

% Un_upper2
thf(fact_670_Un__upper2,axiom,
    ! [B2: set_state,A2: set_state] : ( ord_le2494988322063910608_state @ B2 @ ( sup_sup_set_state @ A2 @ B2 ) ) ).

% Un_upper2
thf(fact_671_Un__absorb1,axiom,
    ! [A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A2 @ B2 )
     => ( ( sup_su3837536599209510219_state @ A2 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_672_Un__absorb1,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ( sup_sup_set_state @ A2 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_673_Un__absorb2,axiom,
    ! [B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B2 @ A2 )
     => ( ( sup_su3837536599209510219_state @ A2 @ B2 )
        = A2 ) ) ).

% Un_absorb2
thf(fact_674_Un__absorb2,axiom,
    ! [B2: set_state,A2: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ A2 )
     => ( ( sup_sup_set_state @ A2 @ B2 )
        = A2 ) ) ).

% Un_absorb2
thf(fact_675_subset__UnE,axiom,
    ! [C2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C2 @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
     => ~ ! [A11: set_Pr4231240917283394591_state] :
            ( ( ord_le4032880777312351359_state @ A11 @ A2 )
           => ! [B10: set_Pr4231240917283394591_state] :
                ( ( ord_le4032880777312351359_state @ B10 @ B2 )
               => ( C2
                 != ( sup_su3837536599209510219_state @ A11 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_676_subset__UnE,axiom,
    ! [C2: set_state,A2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ C2 @ ( sup_sup_set_state @ A2 @ B2 ) )
     => ~ ! [A11: set_state] :
            ( ( ord_le2494988322063910608_state @ A11 @ A2 )
           => ! [B10: set_state] :
                ( ( ord_le2494988322063910608_state @ B10 @ B2 )
               => ( C2
                 != ( sup_sup_set_state @ A11 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_677_subset__Un__eq,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A7 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_678_subset__Un__eq,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A7: set_state,B4: set_state] :
          ( ( sup_sup_set_state @ A7 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_679_Un__def,axiom,
    ( sup_su1984960714304006370_state
    = ( ^ [A7: set_Pr2391701600042252046_state,B4: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ^ [X5: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X5 @ A7 )
              | ( member6715010751053357271_state @ X5 @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_680_Un__def,axiom,
    ( sup_sup_set_state
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ^ [X5: state] :
              ( ( member_state @ X5 @ A7 )
              | ( member_state @ X5 @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_681_Un__def,axiom,
    ( sup_su7506060154676209150on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,B4: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ^ [X5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X5 @ A7 )
              | ( member2879030624002546611on_val @ X5 @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_682_Un__def,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ^ [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ A7 )
              | ( member7461423308196221312_state @ X5 @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_683_Collect__disj__eq,axiom,
    ! [P: state > $o,Q: state > $o] :
      ( ( collect_state
        @ ^ [X5: state] :
            ( ( P @ X5 )
            | ( Q @ X5 ) ) )
      = ( sup_sup_set_state @ ( collect_state @ P ) @ ( collect_state @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_684_Collect__disj__eq,axiom,
    ! [P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [X5: produc9003781588162623004on_val] :
            ( ( P @ X5 )
            | ( Q @ X5 ) ) )
      = ( sup_su7506060154676209150on_val @ ( collec7475294502966882545on_val @ P ) @ ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_685_Collect__disj__eq,axiom,
    ! [P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( P @ X5 )
            | ( Q @ X5 ) ) )
      = ( sup_su3837536599209510219_state @ ( collec906845016635587262_state @ P ) @ ( collec906845016635587262_state @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_686_PartialSA_Osat_Opelims_I3_J,axiom,
    ! [X3: aassertion_state,Xa2: state] :
      ( ~ ( package_sat_state @ plus @ X3 @ Xa2 )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) )
               => ? [A5: state,B9: state] :
                    ( ( ( some_state @ Xa2 )
                      = ( plus @ A5 @ B9 ) )
                    & ( package_sat_state @ plus @ A10 @ A5 )
                    & ( package_sat_state @ plus @ B8 @ B9 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) )
                 => ( ( B7 @ Xa2 )
                   => ( package_sat_state @ plus @ A10 @ Xa2 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) )
                   => ( A10 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(3)
thf(fact_687_PartialSA_Osat_Opelims_I2_J,axiom,
    ! [X3: aassertion_state,Xa2: state] :
      ( ( package_sat_state @ plus @ X3 @ Xa2 )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) )
               => ~ ? [A3: state,B7: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( plus @ A3 @ B7 ) )
                      & ( package_sat_state @ plus @ A10 @ A3 )
                      & ( package_sat_state @ plus @ B8 @ B7 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) )
                 => ~ ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A10 @ Xa2 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) )
                   => ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(2)
thf(fact_688_PartialSA_Osat_Opelims_I1_J,axiom,
    ! [X3: aassertion_state,Xa2: state,Y: $o] :
      ( ( ( package_sat_state @ plus @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
       => ( ! [A10: aassertion_state,B8: aassertion_state] :
              ( ( X3
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( Y
                  = ( ? [A4: state,B3: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( plus @ A4 @ B3 ) )
                        & ( package_sat_state @ plus @ A10 @ A4 )
                        & ( package_sat_state @ plus @ B8 @ B3 ) ) ) )
               => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( X3
                  = ( aImp_state @ B7 @ A10 ) )
               => ( ( Y
                    = ( ( B7 @ Xa2 )
                     => ( package_sat_state @ plus @ A10 @ Xa2 ) ) )
                 => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) ) ) )
           => ~ ! [A10: state > $o] :
                  ( ( X3
                    = ( aSem_state @ A10 ) )
                 => ( ( Y
                      = ( A10 @ Xa2 ) )
                   => ~ ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% PartialSA.sat.pelims(1)
thf(fact_689_package__logic_Osat_Opelims_I3_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_val @ Plus @ X3 @ Xa2 )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X3
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa2 ) )
                 => ? [A5: val,B9: val] :
                      ( ( ( some_val @ Xa2 )
                        = ( Plus @ A5 @ B9 ) )
                      & ( package_sat_val @ Plus @ A10 @ A5 )
                      & ( package_sat_val @ Plus @ B8 @ B9 ) ) ) )
           => ( ! [B7: val > $o,A10: aassertion_val] :
                  ( ( X3
                    = ( aImp_val @ B7 @ A10 ) )
                 => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B7 @ A10 ) @ Xa2 ) )
                   => ( ( B7 @ Xa2 )
                     => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X3
                      = ( aSem_val @ A10 ) )
                   => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa2 ) )
                     => ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(3)
thf(fact_690_package__logic_Osat_Opelims_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ~ ( package_sat_state @ Plus @ X3 @ Xa2 )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X3
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) )
                 => ? [A5: state,B9: state] :
                      ( ( ( some_state @ Xa2 )
                        = ( Plus @ A5 @ B9 ) )
                      & ( package_sat_state @ Plus @ A10 @ A5 )
                      & ( package_sat_state @ Plus @ B8 @ B9 ) ) ) )
           => ( ! [B7: state > $o,A10: aassertion_state] :
                  ( ( X3
                    = ( aImp_state @ B7 @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) )
                   => ( ( B7 @ Xa2 )
                     => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X3
                      = ( aSem_state @ A10 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) )
                     => ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(3)
thf(fact_691_package__logic_Osat_Opelims_I2_J,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,X3: aassertion_val,Xa2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ X3 @ Xa2 )
       => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_val,B8: aassertion_val] :
                ( ( X3
                  = ( aStar_val @ A10 @ B8 ) )
               => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aStar_val @ A10 @ B8 ) @ Xa2 ) )
                 => ~ ? [A3: val,B7: val] :
                        ( ( ( some_val @ Xa2 )
                          = ( Plus @ A3 @ B7 ) )
                        & ( package_sat_val @ Plus @ A10 @ A3 )
                        & ( package_sat_val @ Plus @ B8 @ B7 ) ) ) )
           => ( ! [B7: val > $o,A10: aassertion_val] :
                  ( ( X3
                    = ( aImp_val @ B7 @ A10 ) )
                 => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aImp_val @ B7 @ A10 ) @ Xa2 ) )
                   => ~ ( ( B7 @ Xa2 )
                       => ( package_sat_val @ Plus @ A10 @ Xa2 ) ) ) )
             => ~ ! [A10: val > $o] :
                    ( ( X3
                      = ( aSem_val @ A10 ) )
                   => ( ( accp_P7007234769827254483al_val @ package_sat_rel_val @ ( produc1969469836606013754al_val @ ( aSem_val @ A10 ) @ Xa2 ) )
                     => ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(2)
thf(fact_692_package__logic_Osat_Opelims_I2_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,X3: aassertion_state,Xa2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ X3 @ Xa2 )
       => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ X3 @ Xa2 ) )
         => ( ! [A10: aassertion_state,B8: aassertion_state] :
                ( ( X3
                  = ( aStar_state @ A10 @ B8 ) )
               => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aStar_state @ A10 @ B8 ) @ Xa2 ) )
                 => ~ ? [A3: state,B7: state] :
                        ( ( ( some_state @ Xa2 )
                          = ( Plus @ A3 @ B7 ) )
                        & ( package_sat_state @ Plus @ A10 @ A3 )
                        & ( package_sat_state @ Plus @ B8 @ B7 ) ) ) )
           => ( ! [B7: state > $o,A10: aassertion_state] :
                  ( ( X3
                    = ( aImp_state @ B7 @ A10 ) )
                 => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aImp_state @ B7 @ A10 ) @ Xa2 ) )
                   => ~ ( ( B7 @ Xa2 )
                       => ( package_sat_state @ Plus @ A10 @ Xa2 ) ) ) )
             => ~ ! [A10: state > $o] :
                    ( ( X3
                      = ( aSem_state @ A10 ) )
                   => ( ( accp_P1644577663358802135_state @ packag1918864394588125018_state @ ( produc6654088501289988998_state @ ( aSem_state @ A10 ) @ Xa2 ) )
                     => ~ ( A10 @ Xa2 ) ) ) ) ) ) ) ) ).

% package_logic.sat.pelims(2)
thf(fact_693_le__sup__iff,axiom,
    ! [X3: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state,Z2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ X3 @ Y ) @ Z2 )
      = ( ( ord_le4032880777312351359_state @ X3 @ Z2 )
        & ( ord_le4032880777312351359_state @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_694_le__sup__iff,axiom,
    ! [X3: set_state,Y: set_state,Z2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ X3 @ Y ) @ Z2 )
      = ( ( ord_le2494988322063910608_state @ X3 @ Z2 )
        & ( ord_le2494988322063910608_state @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_695_sup_Obounded__iff,axiom,
    ! [B: set_Pr4231240917283394591_state,C: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B @ C ) @ A )
      = ( ( ord_le4032880777312351359_state @ B @ A )
        & ( ord_le4032880777312351359_state @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_696_sup_Obounded__iff,axiom,
    ! [B: set_state,C: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B @ C ) @ A )
      = ( ( ord_le2494988322063910608_state @ B @ A )
        & ( ord_le2494988322063910608_state @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_697_package__logic_OAddFromOutside,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,Phi2: val,M: val,F3: val,S3: set_Pr4723104092121316667al_val,Pc: val > $o,A2: aassertion_val,Phi7: val,F4: val,S4: set_Pr4723104092121316667al_val,F: val,S2: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ Phi )
          = ( Plus @ Phi2 @ M ) )
       => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi2 @ F3 @ S3 @ Pc @ A2 @ Phi7 @ F4 @ S4 )
         => ( ( Stable @ M )
           => ( ( ( some_val @ F3 )
                = ( Plus @ F @ M ) )
             => ( ( S3
                  = ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A4: val,U6: val,T4: val > val,R2: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ R2 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ S2 )
                        & ( ( some_val @ R2 )
                          = ( Plus @ A4 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) ) )
                        & ( sep_defined_val @ Plus @ R2 @ U6 ) ) ) )
               => ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi7 @ F4 @ S4 ) ) ) ) ) ) ) ).

% package_logic.AddFromOutside
thf(fact_698_package__logic_OAddFromOutside,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,Phi2: state,M: state,F3: state,S3: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi7: state,F4: state,S4: set_Pr4231240917283394591_state,F: state,S2: set_Pr4231240917283394591_state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi )
          = ( Plus @ Phi2 @ M ) )
       => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi2 @ F3 @ S3 @ Pc @ A2 @ Phi7 @ F4 @ S4 )
         => ( ( Stable @ M )
           => ( ( ( some_state @ F3 )
                = ( Plus @ F @ M ) )
             => ( ( S3
                  = ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A4: state,U6: state,T4: state > state,R2: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                        & ( ( some_state @ R2 )
                          = ( Plus @ A4 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) ) )
                        & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) )
               => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi7 @ F4 @ S4 ) ) ) ) ) ) ) ).

% package_logic.AddFromOutside
thf(fact_699_valid__state_Opelims_I1_J,axiom,
    ! [X3: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X3 )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( Y
                  = ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) )
               => ~ ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(1)
thf(fact_700_sep__algebra_Odefined_Ocong,axiom,
    sep_defined_state = sep_defined_state ).

% sep_algebra.defined.cong
thf(fact_701_sup__set__def,axiom,
    ( sup_su1984960714304006370_state
    = ( ^ [A7: set_Pr2391701600042252046_state,B4: set_Pr2391701600042252046_state] :
          ( collec394671971739345689_state
          @ ( sup_su1130956725435522171tate_o
            @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ A7 )
            @ ^ [X5: produc3882902684156697774_state] : ( member6715010751053357271_state @ X5 @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_702_sup__set__def,axiom,
    ( sup_sup_set_state
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ( sup_sup_state_o
            @ ^ [X5: state] : ( member_state @ X5 @ A7 )
            @ ^ [X5: state] : ( member_state @ X5 @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_703_sup__set__def,axiom,
    ( sup_su7506060154676209150on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,B4: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ( sup_su3200866414790943751_val_o
            @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ A7 )
            @ ^ [X5: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X5 @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_704_sup__set__def,axiom,
    ( sup_su3837536599209510219_state
    = ( ^ [A7: set_Pr4231240917283394591_state,B4: set_Pr4231240917283394591_state] :
          ( collec906845016635587262_state
          @ ( sup_su4345871077856951546tate_o
            @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ A7 )
            @ ^ [X5: produc7272280185086996969_state] : ( member7461423308196221312_state @ X5 @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_705_sup__Un__eq2,axiom,
    ! [R: set_Pr2391701600042252046_state,S2: set_Pr2391701600042252046_state] :
      ( ( sup_su1108609720783396685tate_o
        @ ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: state,Y4: state > state] : ( member6715010751053357271_state @ ( produc8632743443909064678_state @ X5 @ Y4 ) @ ( sup_su1984960714304006370_state @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_706_sup__Un__eq2,axiom,
    ! [R: set_Pr4305388979797279737on_val,S2: set_Pr4305388979797279737on_val] :
      ( ( sup_su177996417187360546_val_o
        @ ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ R )
        @ ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: produc9003781588162623004on_val,Y4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X5 @ Y4 ) @ ( sup_su3890347010539237325on_val @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_707_sup__Un__eq2,axiom,
    ! [R: set_Pr1688445902015331925_state,S2: set_Pr1688445902015331925_state] :
      ( ( sup_su2989194110892726342tate_o
        @ ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: option_state,Y4: option_state] : ( member3029510603097127326_state @ ( produc9160152616361873709_state @ X5 @ Y4 ) @ ( sup_su5006661989123481129_state @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_708_sup__Un__eq2,axiom,
    ! [R: set_Pr6039890582625207499_state,S2: set_Pr6039890582625207499_state] :
      ( ( sup_sup_prat_state_o
        @ ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: prat,Y4: state] : ( member6226390569306242836_state @ ( produc6205719116230455971_state @ X5 @ Y4 ) @ ( sup_su1826719071296284831_state @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_709_sup__Un__eq2,axiom,
    ! [R: set_Pr6847263425858009217_state,S2: set_Pr6847263425858009217_state] :
      ( ( sup_su7614437069348798106tate_o
        @ ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: prat,Y4: set_state] : ( member6708919011922255306_state @ ( produc9080284787853255769_state @ X5 @ Y4 ) @ ( sup_su5611131606495579221_state @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_710_sup__Un__eq2,axiom,
    ! [R: set_Pr7649655576376376530on_val,S2: set_Pr7649655576376376530on_val] :
      ( ( sup_su4175861339346331443_val_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ R )
        @ ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X5 @ Y4 ) @ ( sup_su7506060154676209150on_val @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_711_sup__Un__eq2,axiom,
    ! [R: set_Pr4231240917283394591_state,S2: set_Pr4231240917283394591_state] :
      ( ( sup_su1162340662190093652tate_o
        @ ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ R )
        @ ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ S2 ) )
      = ( ^ [X5: state,Y4: produc3882902684156697774_state] : ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ Y4 ) @ ( sup_su3837536599209510219_state @ R @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_712_sep__algebra_Odefined__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_defined_val @ Plus @ A @ B )
        = ( ( Plus @ A @ B )
         != none_val ) ) ) ).

% sep_algebra.defined_def
thf(fact_713_sep__algebra_Odefined__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_defined_state @ Plus @ A @ B )
        = ( ( Plus @ A @ B )
         != none_state ) ) ) ).

% sep_algebra.defined_def
thf(fact_714_package__logic_Odefined__sum__move,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,X3: val,Y: val,A6: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_val @ Plus @ A @ B )
       => ( ( ( some_val @ B )
            = ( Plus @ X3 @ Y ) )
         => ( ( ( some_val @ A6 )
              = ( Plus @ A @ X3 ) )
           => ( sep_defined_val @ Plus @ A6 @ Y ) ) ) ) ) ).

% package_logic.defined_sum_move
thf(fact_715_package__logic_Odefined__sum__move,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,X3: state,Y: state,A6: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_defined_state @ Plus @ A @ B )
       => ( ( ( some_state @ B )
            = ( Plus @ X3 @ Y ) )
         => ( ( ( some_state @ A6 )
              = ( Plus @ A @ X3 ) )
           => ( sep_defined_state @ Plus @ A6 @ Y ) ) ) ) ) ).

% package_logic.defined_sum_move
thf(fact_716_defined__def,axiom,
    ( defined
    = ( sep_defined_state @ plus ) ) ).

% defined_def
thf(fact_717_sep__algebra_Ompp__compatible,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_defined_state @ Plus @ A @ B )
         => ( sep_defined_state @ Plus @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% sep_algebra.mpp_compatible
thf(fact_718_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y @ ( sup_su3837536599209510219_state @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_719_inf__sup__ord_I4_J,axiom,
    ! [Y: set_state,X3: set_state] : ( ord_le2494988322063910608_state @ Y @ ( sup_sup_set_state @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_720_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X3 @ ( sup_su3837536599209510219_state @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_721_inf__sup__ord_I3_J,axiom,
    ! [X3: set_state,Y: set_state] : ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_722_le__supE,axiom,
    ! [A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B ) @ X3 )
     => ~ ( ( ord_le4032880777312351359_state @ A @ X3 )
         => ~ ( ord_le4032880777312351359_state @ B @ X3 ) ) ) ).

% le_supE
thf(fact_723_le__supE,axiom,
    ! [A: set_state,B: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B ) @ X3 )
     => ~ ( ( ord_le2494988322063910608_state @ A @ X3 )
         => ~ ( ord_le2494988322063910608_state @ B @ X3 ) ) ) ).

% le_supE
thf(fact_724_le__supI,axiom,
    ! [A: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ X3 )
     => ( ( ord_le4032880777312351359_state @ B @ X3 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_725_le__supI,axiom,
    ! [A: set_state,X3: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ X3 )
     => ( ( ord_le2494988322063910608_state @ B @ X3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_726_sup__ge1,axiom,
    ! [X3: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X3 @ ( sup_su3837536599209510219_state @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_727_sup__ge1,axiom,
    ! [X3: set_state,Y: set_state] : ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_728_sup__ge2,axiom,
    ! [Y: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y @ ( sup_su3837536599209510219_state @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_729_sup__ge2,axiom,
    ! [Y: set_state,X3: set_state] : ( ord_le2494988322063910608_state @ Y @ ( sup_sup_set_state @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_730_le__supI1,axiom,
    ! [X3: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X3 @ A )
     => ( ord_le4032880777312351359_state @ X3 @ ( sup_su3837536599209510219_state @ A @ B ) ) ) ).

% le_supI1
thf(fact_731_le__supI1,axiom,
    ! [X3: set_state,A: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ A )
     => ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ A @ B ) ) ) ).

% le_supI1
thf(fact_732_le__supI2,axiom,
    ! [X3: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X3 @ B )
     => ( ord_le4032880777312351359_state @ X3 @ ( sup_su3837536599209510219_state @ A @ B ) ) ) ).

% le_supI2
thf(fact_733_le__supI2,axiom,
    ! [X3: set_state,B: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ B )
     => ( ord_le2494988322063910608_state @ X3 @ ( sup_sup_set_state @ A @ B ) ) ) ).

% le_supI2
thf(fact_734_sup_Omono,axiom,
    ! [C: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,D: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C @ A )
     => ( ( ord_le4032880777312351359_state @ D @ B )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ C @ D ) @ ( sup_su3837536599209510219_state @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_735_sup_Omono,axiom,
    ! [C: set_state,A: set_state,D: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ A )
     => ( ( ord_le2494988322063910608_state @ D @ B )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ C @ D ) @ ( sup_sup_set_state @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_736_sup__mono,axiom,
    ! [A: set_Pr4231240917283394591_state,C: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state,D: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ C )
     => ( ( ord_le4032880777312351359_state @ B @ D )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ A @ B ) @ ( sup_su3837536599209510219_state @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_737_sup__mono,axiom,
    ! [A: set_state,C: set_state,B: set_state,D: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ C )
     => ( ( ord_le2494988322063910608_state @ B @ D )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ A @ B ) @ ( sup_sup_set_state @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_738_sup__least,axiom,
    ! [Y: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state,Z2: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ Y @ X3 )
     => ( ( ord_le4032880777312351359_state @ Z2 @ X3 )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_739_sup__least,axiom,
    ! [Y: set_state,X3: set_state,Z2: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X3 )
     => ( ( ord_le2494988322063910608_state @ Z2 @ X3 )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_740_le__iff__sup,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [X5: set_Pr4231240917283394591_state,Y4: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ X5 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_741_le__iff__sup,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [X5: set_state,Y4: set_state] :
          ( ( sup_sup_set_state @ X5 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_742_sup_OorderE,axiom,
    ! [B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B @ A )
     => ( A
        = ( sup_su3837536599209510219_state @ A @ B ) ) ) ).

% sup.orderE
thf(fact_743_sup_OorderE,axiom,
    ! [B: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B @ A )
     => ( A
        = ( sup_sup_set_state @ A @ B ) ) ) ).

% sup.orderE
thf(fact_744_sup_OorderI,axiom,
    ! [A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( A
        = ( sup_su3837536599209510219_state @ A @ B ) )
     => ( ord_le4032880777312351359_state @ B @ A ) ) ).

% sup.orderI
thf(fact_745_sup_OorderI,axiom,
    ! [A: set_state,B: set_state] :
      ( ( A
        = ( sup_sup_set_state @ A @ B ) )
     => ( ord_le2494988322063910608_state @ B @ A ) ) ).

% sup.orderI
thf(fact_746_sup__unique,axiom,
    ! [F: set_Pr4231240917283394591_state > set_Pr4231240917283394591_state > set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] :
      ( ! [X: set_Pr4231240917283394591_state,Y3: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ X @ ( F @ X @ Y3 ) )
     => ( ! [X: set_Pr4231240917283394591_state,Y3: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ Y3 @ ( F @ X @ Y3 ) )
       => ( ! [X: set_Pr4231240917283394591_state,Y3: set_Pr4231240917283394591_state,Z3: set_Pr4231240917283394591_state] :
              ( ( ord_le4032880777312351359_state @ Y3 @ X )
             => ( ( ord_le4032880777312351359_state @ Z3 @ X )
               => ( ord_le4032880777312351359_state @ ( F @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_su3837536599209510219_state @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_747_sup__unique,axiom,
    ! [F: set_state > set_state > set_state,X3: set_state,Y: set_state] :
      ( ! [X: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ X @ ( F @ X @ Y3 ) )
     => ( ! [X: set_state,Y3: set_state] : ( ord_le2494988322063910608_state @ Y3 @ ( F @ X @ Y3 ) )
       => ( ! [X: set_state,Y3: set_state,Z3: set_state] :
              ( ( ord_le2494988322063910608_state @ Y3 @ X )
             => ( ( ord_le2494988322063910608_state @ Z3 @ X )
               => ( ord_le2494988322063910608_state @ ( F @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_sup_set_state @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_748_sup_Oabsorb1,axiom,
    ! [B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B @ A )
     => ( ( sup_su3837536599209510219_state @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_749_sup_Oabsorb1,axiom,
    ! [B: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ B @ A )
     => ( ( sup_sup_set_state @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_750_sup_Oabsorb2,axiom,
    ! [A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ A @ B )
     => ( ( sup_su3837536599209510219_state @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_751_sup_Oabsorb2,axiom,
    ! [A: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ A @ B )
     => ( ( sup_sup_set_state @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_752_sup__absorb1,axiom,
    ! [Y: set_Pr4231240917283394591_state,X3: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ Y @ X3 )
     => ( ( sup_su3837536599209510219_state @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_753_sup__absorb1,axiom,
    ! [Y: set_state,X3: set_state] :
      ( ( ord_le2494988322063910608_state @ Y @ X3 )
     => ( ( sup_sup_set_state @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_754_sup__absorb2,axiom,
    ! [X3: set_Pr4231240917283394591_state,Y: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ X3 @ Y )
     => ( ( sup_su3837536599209510219_state @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_755_sup__absorb2,axiom,
    ! [X3: set_state,Y: set_state] :
      ( ( ord_le2494988322063910608_state @ X3 @ Y )
     => ( ( sup_sup_set_state @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_756_sup_OboundedE,axiom,
    ! [B: set_Pr4231240917283394591_state,C: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B @ C ) @ A )
     => ~ ( ( ord_le4032880777312351359_state @ B @ A )
         => ~ ( ord_le4032880777312351359_state @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_757_sup_OboundedE,axiom,
    ! [B: set_state,C: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B @ C ) @ A )
     => ~ ( ( ord_le2494988322063910608_state @ B @ A )
         => ~ ( ord_le2494988322063910608_state @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_758_sup_OboundedI,axiom,
    ! [B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,C: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ B @ A )
     => ( ( ord_le4032880777312351359_state @ C @ A )
       => ( ord_le4032880777312351359_state @ ( sup_su3837536599209510219_state @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_759_sup_OboundedI,axiom,
    ! [B: set_state,A: set_state,C: set_state] :
      ( ( ord_le2494988322063910608_state @ B @ A )
     => ( ( ord_le2494988322063910608_state @ C @ A )
       => ( ord_le2494988322063910608_state @ ( sup_sup_set_state @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_760_sup_Oorder__iff,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [B3: set_Pr4231240917283394591_state,A4: set_Pr4231240917283394591_state] :
          ( A4
          = ( sup_su3837536599209510219_state @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_761_sup_Oorder__iff,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B3: set_state,A4: set_state] :
          ( A4
          = ( sup_sup_set_state @ A4 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_762_sup_Ocobounded1,axiom,
    ! [A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ A @ ( sup_su3837536599209510219_state @ A @ B ) ) ).

% sup.cobounded1
thf(fact_763_sup_Ocobounded1,axiom,
    ! [A: set_state,B: set_state] : ( ord_le2494988322063910608_state @ A @ ( sup_sup_set_state @ A @ B ) ) ).

% sup.cobounded1
thf(fact_764_sup_Ocobounded2,axiom,
    ! [B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] : ( ord_le4032880777312351359_state @ B @ ( sup_su3837536599209510219_state @ A @ B ) ) ).

% sup.cobounded2
thf(fact_765_sup_Ocobounded2,axiom,
    ! [B: set_state,A: set_state] : ( ord_le2494988322063910608_state @ B @ ( sup_sup_set_state @ A @ B ) ) ).

% sup.cobounded2
thf(fact_766_sup_Oabsorb__iff1,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [B3: set_Pr4231240917283394591_state,A4: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_767_sup_Oabsorb__iff1,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [B3: set_state,A4: set_state] :
          ( ( sup_sup_set_state @ A4 @ B3 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_768_sup_Oabsorb__iff2,axiom,
    ( ord_le4032880777312351359_state
    = ( ^ [A4: set_Pr4231240917283394591_state,B3: set_Pr4231240917283394591_state] :
          ( ( sup_su3837536599209510219_state @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_769_sup_Oabsorb__iff2,axiom,
    ( ord_le2494988322063910608_state
    = ( ^ [A4: set_state,B3: set_state] :
          ( ( sup_sup_set_state @ A4 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_770_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C @ A )
     => ( ord_le4032880777312351359_state @ C @ ( sup_su3837536599209510219_state @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_771_sup_OcoboundedI1,axiom,
    ! [C: set_state,A: set_state,B: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ A )
     => ( ord_le2494988322063910608_state @ C @ ( sup_sup_set_state @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_772_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4231240917283394591_state,B: set_Pr4231240917283394591_state,A: set_Pr4231240917283394591_state] :
      ( ( ord_le4032880777312351359_state @ C @ B )
     => ( ord_le4032880777312351359_state @ C @ ( sup_su3837536599209510219_state @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_773_sup_OcoboundedI2,axiom,
    ! [C: set_state,B: set_state,A: set_state] :
      ( ( ord_le2494988322063910608_state @ C @ B )
     => ( ord_le2494988322063910608_state @ C @ ( sup_sup_set_state @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_774_valid__state_Opelims_I3_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X3 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) )
               => ( ( valid_mask @ Pi )
                  & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(3)
thf(fact_775_valid__state_Opelims_I2_J,axiom,
    ! [X3: produc9003781588162623004on_val] :
      ( ( valid_state @ X3 )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X3 )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X3
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) )
               => ~ ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(2)
thf(fact_776_package__logic_Opackage__rhs_Osimps,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A1: val,A22: val,A32: set_Pr4723104092121316667al_val,A42: val > $o,A52: aassertion_val,A62: val,A72: val,A82: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
        = ( ? [Phi3: val,F5: val,S5: set_Pr4723104092121316667al_val,Pc2: val > $o,A7: aassertion_val,Phi5: val,F6: val,S6: set_Pr4723104092121316667al_val,B4: aassertion_val,Phi8: val,F7: val,S7: set_Pr4723104092121316667al_val] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aStar_val @ A7 @ B4 ) )
              & ( A62 = Phi8 )
              & ( A72 = F7 )
              & ( A82 = S7 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi3 @ F5 @ S5 @ Pc2 @ A7 @ Phi5 @ F6 @ S6 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi5 @ F6 @ S6 @ Pc2 @ B4 @ Phi8 @ F7 @ S7 ) )
          | ? [Phi3: val,F5: val,S5: set_Pr4723104092121316667al_val,Pc2: val > $o,B3: val > $o,A7: aassertion_val,Phi5: val,F6: val,S6: set_Pr4723104092121316667al_val] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aImp_val @ B3 @ A7 ) )
              & ( A62 = Phi5 )
              & ( A72 = F6 )
              & ( A82 = S6 )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi3 @ F5 @ S5 @ ( packag6750194596558674946nj_val @ Pc2 @ B3 ) @ A7 @ Phi5 @ F6 @ S6 ) )
          | ? [S5: set_Pr4723104092121316667al_val,Pc2: val > $o,Witness: produc2072440392815185925al_val > val,B4: val > $o,S6: set_Pr4723104092121316667al_val,Phi3: val,F5: val] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aSem_val @ B4 ) )
              & ( A62 = Phi3 )
              & ( A72 = F5 )
              & ( A82 = S6 )
              & ! [X5: val,Y4: val,Z5: val > val,Aa2: val] :
                  ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ X5 @ ( produc303288909735261391al_val @ Y4 @ Z5 ) ) @ S5 )
                 => ( ( Pc2 @ X5 )
                   => ( ( Aa2
                        = ( Witness @ ( produc7187740677493149823al_val @ X5 @ ( produc303288909735261391al_val @ Y4 @ Z5 ) ) ) )
                     => ( ( sep_greater_val @ Plus @ X5 @ Aa2 )
                        & ( B4 @ Aa2 ) ) ) ) )
              & ( S6
                = ( sup_su4800132526891881063al_val
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A4: val,U6: val,T4: val > val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ S5 )
                        & ~ ( Pc2 @ A4 ) ) )
                  @ ( collec8300311316619386842al_val
                    @ ^ [Uu3: produc2072440392815185925al_val] :
                      ? [A4: val,U6: val,T4: val > val,B3: val] :
                        ( ( Uu3
                          = ( produc7187740677493149823al_val @ ( sep_minus_val @ Plus @ Core @ A4 @ B3 ) @ ( produc303288909735261391al_val @ ( the_val @ ( Plus @ U6 @ B3 ) ) @ T4 ) ) )
                        & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ S5 )
                        & ( Pc2 @ A4 )
                        & ( B3
                          = ( Witness @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) ) ) ) ) ) ) )
          | ? [Phi3: val,Phi5: val,M4: val,F6: val,S6: set_Pr4723104092121316667al_val,Pc2: val > $o,A7: aassertion_val,Phi8: val,F7: val,S7: set_Pr4723104092121316667al_val,F5: val,S5: set_Pr4723104092121316667al_val] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52 = A7 )
              & ( A62 = Phi8 )
              & ( A72 = F7 )
              & ( A82 = S7 )
              & ( ( some_val @ Phi3 )
                = ( Plus @ Phi5 @ M4 ) )
              & ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi5 @ F6 @ S6 @ Pc2 @ A7 @ Phi8 @ F7 @ S7 )
              & ( Stable @ M4 )
              & ( ( some_val @ F6 )
                = ( Plus @ F5 @ M4 ) )
              & ( S6
                = ( collec8300311316619386842al_val
                  @ ^ [Uu3: produc2072440392815185925al_val] :
                    ? [A4: val,U6: val,T4: val > val,R2: val] :
                      ( ( Uu3
                        = ( produc7187740677493149823al_val @ R2 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) )
                      & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ S5 )
                      & ( ( some_val @ R2 )
                        = ( Plus @ A4 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F6 ) @ ( T4 @ F5 ) ) ) )
                      & ( sep_defined_val @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.simps
thf(fact_777_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 )
        = ( ? [Phi3: state,F5: state,S5: set_Pr4231240917283394591_state,Pc2: state > $o,A7: aassertion_state,Phi5: state,F6: state,S6: set_Pr4231240917283394591_state,B4: aassertion_state,Phi8: state,F7: state,S7: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aStar_state @ A7 @ B4 ) )
              & ( A62 = Phi8 )
              & ( A72 = F7 )
              & ( A82 = S7 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi3 @ F5 @ S5 @ Pc2 @ A7 @ Phi5 @ F6 @ S6 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi5 @ F6 @ S6 @ Pc2 @ B4 @ Phi8 @ F7 @ S7 ) )
          | ? [Phi3: state,F5: state,S5: set_Pr4231240917283394591_state,Pc2: state > $o,B3: state > $o,A7: aassertion_state,Phi5: state,F6: state,S6: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aImp_state @ B3 @ A7 ) )
              & ( A62 = Phi5 )
              & ( A72 = F6 )
              & ( A82 = S6 )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi3 @ F5 @ S5 @ ( packag1330488657391168505_state @ Pc2 @ B3 ) @ A7 @ Phi5 @ F6 @ S6 ) )
          | ? [S5: set_Pr4231240917283394591_state,Pc2: state > $o,Witness: produc7272280185086996969_state > state,B4: state > $o,S6: set_Pr4231240917283394591_state,Phi3: state,F5: state] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52
                = ( aSem_state @ B4 ) )
              & ( A62 = Phi3 )
              & ( A72 = F5 )
              & ( A82 = S6 )
              & ! [X5: state,Y4: state,Z5: state > state,Aa2: state] :
                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ X5 @ ( produc8632743443909064678_state @ Y4 @ Z5 ) ) @ S5 )
                 => ( ( Pc2 @ X5 )
                   => ( ( Aa2
                        = ( Witness @ ( produc5886875603596089627_state @ X5 @ ( produc8632743443909064678_state @ Y4 @ Z5 ) ) ) )
                     => ( ( sep_greater_state @ Plus @ X5 @ Aa2 )
                        & ( B4 @ Aa2 ) ) ) ) )
              & ( S6
                = ( sup_su3837536599209510219_state
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A4: state,U6: state,T4: state > state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                        & ~ ( Pc2 @ A4 ) ) )
                  @ ( collec906845016635587262_state
                    @ ^ [Uu3: produc7272280185086996969_state] :
                      ? [A4: state,U6: state,T4: state > state,B3: state] :
                        ( ( Uu3
                          = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A4 @ B3 ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B3 ) ) @ T4 ) ) )
                        & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                        & ( Pc2 @ A4 )
                        & ( B3
                          = ( Witness @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) ) )
          | ? [Phi3: state,Phi5: state,M4: state,F6: state,S6: set_Pr4231240917283394591_state,Pc2: state > $o,A7: aassertion_state,Phi8: state,F7: state,S7: set_Pr4231240917283394591_state,F5: state,S5: set_Pr4231240917283394591_state] :
              ( ( A1 = Phi3 )
              & ( A22 = F5 )
              & ( A32 = S5 )
              & ( A42 = Pc2 )
              & ( A52 = A7 )
              & ( A62 = Phi8 )
              & ( A72 = F7 )
              & ( A82 = S7 )
              & ( ( some_state @ Phi3 )
                = ( Plus @ Phi5 @ M4 ) )
              & ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi5 @ F6 @ S6 @ Pc2 @ A7 @ Phi8 @ F7 @ S7 )
              & ( Stable @ M4 )
              & ( ( some_state @ F6 )
                = ( Plus @ F5 @ M4 ) )
              & ( S6
                = ( collec906845016635587262_state
                  @ ^ [Uu3: produc7272280185086996969_state] :
                    ? [A4: state,U6: state,T4: state > state,R2: state] :
                      ( ( Uu3
                        = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                      & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S5 )
                      & ( ( some_state @ R2 )
                        = ( Plus @ A4 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F6 ) @ ( T4 @ F5 ) ) ) )
                      & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.simps
thf(fact_778_package__logic_Opackage__rhs_Ocases,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A1: val,A22: val,A32: set_Pr4723104092121316667al_val,A42: val > $o,A52: aassertion_val,A62: val,A72: val,A82: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A52 @ A62 @ A72 @ A82 )
       => ( ! [A10: aassertion_val,Phi6: val,F8: val,S8: set_Pr4723104092121316667al_val,B8: aassertion_val] :
              ( ( A52
                = ( aStar_val @ A10 @ B8 ) )
             => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A10 @ Phi6 @ F8 @ S8 )
               => ~ ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi6 @ F8 @ S8 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
         => ( ! [B7: val > $o,A10: aassertion_val] :
                ( ( A52
                  = ( aImp_val @ B7 @ A10 ) )
               => ~ ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ ( packag6750194596558674946nj_val @ A42 @ B7 ) @ A10 @ A62 @ A72 @ A82 ) )
           => ( ! [Witness2: produc2072440392815185925al_val > val,B8: val > $o] :
                  ( ( A52
                    = ( aSem_val @ B8 ) )
                 => ( ( A62 = A1 )
                   => ( ( A72 = A22 )
                     => ( ! [A5: val,U7: val,T5: val > val] :
                            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U7 @ T5 ) ) @ A32 )
                           => ( ( A42 @ A5 )
                             => ( ( sep_greater_val @ Plus @ A5 @ ( Witness2 @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U7 @ T5 ) ) ) )
                                & ( B8 @ ( Witness2 @ ( produc7187740677493149823al_val @ A5 @ ( produc303288909735261391al_val @ U7 @ T5 ) ) ) ) ) ) )
                       => ( A82
                         != ( sup_su4800132526891881063al_val
                            @ ( collec8300311316619386842al_val
                              @ ^ [Uu3: produc2072440392815185925al_val] :
                                ? [A4: val,U6: val,T4: val > val] :
                                  ( ( Uu3
                                    = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) )
                                  & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ A32 )
                                  & ~ ( A42 @ A4 ) ) )
                            @ ( collec8300311316619386842al_val
                              @ ^ [Uu3: produc2072440392815185925al_val] :
                                ? [A4: val,U6: val,T4: val > val] :
                                  ( ( Uu3
                                    = ( produc7187740677493149823al_val @ ( sep_minus_val @ Plus @ Core @ A4 @ ( Witness2 @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) ) ) @ ( produc303288909735261391al_val @ ( the_val @ ( Plus @ U6 @ ( Witness2 @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) ) ) ) @ T4 ) ) )
                                  & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ A32 )
                                  & ( A42 @ A4 ) ) ) ) ) ) ) ) )
             => ~ ! [Phi6: val,M5: val] :
                    ( ( ( some_val @ A1 )
                      = ( Plus @ Phi6 @ M5 ) )
                   => ! [F8: val,S8: set_Pr4723104092121316667al_val] :
                        ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi6 @ F8 @ S8 @ A42 @ A52 @ A62 @ A72 @ A82 )
                       => ( ( Stable @ M5 )
                         => ( ( ( some_val @ F8 )
                              = ( Plus @ A22 @ M5 ) )
                           => ( S8
                             != ( collec8300311316619386842al_val
                                @ ^ [Uu3: produc2072440392815185925al_val] :
                                  ? [A4: val,U6: val,T4: val > val,R2: val] :
                                    ( ( Uu3
                                      = ( produc7187740677493149823al_val @ R2 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) )
                                    & ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ A32 )
                                    & ( ( some_val @ R2 )
                                      = ( Plus @ A4 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F8 ) @ ( T4 @ A22 ) ) ) )
                                    & ( sep_defined_val @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.cases
thf(fact_779_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 )
       => ( ! [A10: aassertion_state,Phi6: state,F8: state,S8: set_Pr4231240917283394591_state,B8: aassertion_state] :
              ( ( A52
                = ( aStar_state @ A10 @ B8 ) )
             => ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ A42 @ A10 @ Phi6 @ F8 @ S8 )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F8 @ S8 @ A42 @ B8 @ A62 @ A72 @ A82 ) ) )
         => ( ! [B7: state > $o,A10: aassertion_state] :
                ( ( A52
                  = ( aImp_state @ B7 @ A10 ) )
               => ~ ( packag3171892606229212990_state @ Plus @ Core @ Stable @ A1 @ A22 @ A32 @ ( packag1330488657391168505_state @ A42 @ B7 ) @ A10 @ A62 @ A72 @ A82 ) )
           => ( ! [Witness2: produc7272280185086996969_state > state,B8: state > $o] :
                  ( ( A52
                    = ( aSem_state @ B8 ) )
                 => ( ( A62 = A1 )
                   => ( ( A72 = A22 )
                     => ( ! [A5: state,U7: state,T5: state > state] :
                            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) @ A32 )
                           => ( ( A42 @ A5 )
                             => ( ( sep_greater_state @ Plus @ A5 @ ( Witness2 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) )
                                & ( B8 @ ( Witness2 @ ( produc5886875603596089627_state @ A5 @ ( produc8632743443909064678_state @ U7 @ T5 ) ) ) ) ) ) )
                       => ( A82
                         != ( sup_su3837536599209510219_state
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A4: state,U6: state,T4: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ~ ( A42 @ A4 ) ) )
                            @ ( collec906845016635587262_state
                              @ ^ [Uu3: produc7272280185086996969_state] :
                                ? [A4: state,U6: state,T4: state > state] :
                                  ( ( Uu3
                                    = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A4 @ ( Witness2 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ ( Witness2 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) @ T4 ) ) )
                                  & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                  & ( A42 @ A4 ) ) ) ) ) ) ) ) )
             => ~ ! [Phi6: state,M5: state] :
                    ( ( ( some_state @ A1 )
                      = ( Plus @ Phi6 @ M5 ) )
                   => ! [F8: state,S8: set_Pr4231240917283394591_state] :
                        ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi6 @ F8 @ S8 @ A42 @ A52 @ A62 @ A72 @ A82 )
                       => ( ( Stable @ M5 )
                         => ( ( ( some_state @ F8 )
                              = ( Plus @ A22 @ M5 ) )
                           => ( S8
                             != ( collec906845016635587262_state
                                @ ^ [Uu3: produc7272280185086996969_state] :
                                  ? [A4: state,U6: state,T4: state > state,R2: state] :
                                    ( ( Uu3
                                      = ( produc5886875603596089627_state @ R2 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ A32 )
                                    & ( ( some_state @ R2 )
                                      = ( Plus @ A4 @ ( sep_minus_state @ Plus @ Core @ ( T4 @ F8 ) @ ( T4 @ A22 ) ) ) )
                                    & ( sep_defined_state @ Plus @ R2 @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% package_logic.package_rhs.cases
thf(fact_780_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 )
     => ( ! [A3: state,U: state,T3: state > state,B7: state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( Pc @ A3 )
             => ( ( B7
                  = ( Witness3 @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) ) )
               => ( ( sep_greater_state @ Plus @ A3 @ B7 )
                  & ( B2 @ B7 ) ) ) ) )
       => ( ( S3
            = ( sup_su3837536599209510219_state
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state,U6: state,T4: state > state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                    & ~ ( Pc @ A4 ) ) )
              @ ( collec906845016635587262_state
                @ ^ [Uu3: produc7272280185086996969_state] :
                  ? [A4: state,U6: state,T4: state > state,B3: state] :
                    ( ( Uu3
                      = ( produc5886875603596089627_state @ ( sep_minus_state @ Plus @ Core @ A4 @ B3 ) @ ( produc8632743443909064678_state @ ( the_state @ ( Plus @ U6 @ B3 ) ) @ T4 ) ) )
                    & ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                    & ( Pc @ A4 )
                    & ( B3
                      = ( Witness3 @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) ) ) ) ) ) )
         => ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ ( aSem_state @ B2 ) @ Phi @ F @ S3 ) ) ) ) ).

% package_logic.ASem
thf(fact_781_sep__algebra_Osucc__antisym,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( sep_greater_state @ Plus @ B @ A )
         => ( A = B ) ) ) ) ).

% sep_algebra.succ_antisym
thf(fact_782_sep__algebra_Osucc__trans,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( sep_greater_state @ Plus @ B @ C )
         => ( sep_greater_state @ Plus @ A @ C ) ) ) ) ).

% sep_algebra.succ_trans
thf(fact_783_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_784_sep__algebra_Ogreater_Ocong,axiom,
    sep_greater_state = sep_greater_state ).

% sep_algebra.greater.cong
thf(fact_785_sep__algebra_Obigger__sum__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,C: val,A: val,B: val,A6: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ C )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_val @ Plus @ A @ A6 )
         => ? [B5: val] :
              ( ( sep_greater_val @ Plus @ B5 @ B )
              & ( ( some_val @ C )
                = ( Plus @ A6 @ B5 ) ) ) ) ) ) ).

% sep_algebra.bigger_sum_smaller
thf(fact_786_sep__algebra_Obigger__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,C: state,A: state,B: state,A6: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ C )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_state @ Plus @ A @ A6 )
         => ? [B5: state] :
              ( ( sep_greater_state @ Plus @ B5 @ B )
              & ( ( some_state @ C )
                = ( Plus @ A6 @ B5 ) ) ) ) ) ) ).

% sep_algebra.bigger_sum_smaller
thf(fact_787_sep__algebra_Osmaller__than__core,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Y: val,X3: val,Z2: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ Y @ X3 )
       => ( ( ( some_val @ Z2 )
            = ( Plus @ X3 @ ( Core @ Y ) ) )
         => ( ( Core @ Z2 )
            = ( Core @ Y ) ) ) ) ) ).

% sep_algebra.smaller_than_core
thf(fact_788_sep__algebra_Osmaller__than__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X3: state,Z2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X3 )
       => ( ( ( some_state @ Z2 )
            = ( Plus @ X3 @ ( Core @ Y ) ) )
         => ( ( Core @ Z2 )
            = ( Core @ Y ) ) ) ) ) ).

% sep_algebra.smaller_than_core
thf(fact_789_sep__algebra_Oaddition__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A6: val,A: val,X6: val,B: val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A6 @ A )
       => ( ( ( some_val @ X6 )
            = ( Plus @ A6 @ B ) )
         => ( ( ( some_val @ X3 )
              = ( Plus @ A @ B ) )
           => ( sep_greater_val @ Plus @ X6 @ X3 ) ) ) ) ) ).

% sep_algebra.addition_bigger
thf(fact_790_sep__algebra_Oaddition__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A6: state,A: state,X6: state,B: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A6 @ A )
       => ( ( ( some_state @ X6 )
            = ( Plus @ A6 @ B ) )
         => ( ( ( some_state @ X3 )
              = ( Plus @ A @ B ) )
           => ( sep_greater_state @ Plus @ X6 @ X3 ) ) ) ) ) ).

% sep_algebra.addition_bigger
thf(fact_791_sep__algebra_Ogreater__equiv,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A @ B )
        = ( ? [C3: val] :
              ( ( some_val @ A )
              = ( Plus @ C3 @ B ) ) ) ) ) ).

% sep_algebra.greater_equiv
thf(fact_792_sep__algebra_Ogreater__equiv,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B )
        = ( ? [C3: state] :
              ( ( some_state @ A )
              = ( Plus @ C3 @ B ) ) ) ) ) ).

% sep_algebra.greater_equiv
thf(fact_793_sep__algebra_Ominus__unique,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X3: val,Y: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( some_val @ B )
            = ( Plus @ A @ X3 ) )
          & ( sep_greater_val @ Plus @ X3 @ ( Core @ B ) ) )
       => ( ( ( ( some_val @ B )
              = ( Plus @ A @ Y ) )
            & ( sep_greater_val @ Plus @ Y @ ( Core @ B ) ) )
         => ( X3 = Y ) ) ) ) ).

% sep_algebra.minus_unique
thf(fact_794_sep__algebra_Ominus__unique,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X3: state,Y: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B )
            = ( Plus @ A @ X3 ) )
          & ( sep_greater_state @ Plus @ X3 @ ( Core @ B ) ) )
       => ( ( ( ( some_state @ B )
              = ( Plus @ A @ Y ) )
            & ( sep_greater_state @ Plus @ Y @ ( Core @ B ) ) )
         => ( X3 = Y ) ) ) ) ).

% sep_algebra.minus_unique
thf(fact_795_sep__algebra_Ominus__exists,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ B @ A )
       => ? [X: val] :
            ( ( ( some_val @ B )
              = ( Plus @ A @ X ) )
            & ( sep_greater_val @ Plus @ X @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_exists
thf(fact_796_sep__algebra_Ominus__exists,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ B @ A )
       => ? [X: state] :
            ( ( ( some_state @ B )
              = ( Plus @ A @ X ) )
            & ( sep_greater_state @ Plus @ X @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_exists
thf(fact_797_sep__algebra_Oextract__core,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( ( some_val @ B )
            = ( Plus @ A @ X3 ) )
          & ( sep_greater_val @ Plus @ X3 @ ( Core @ B ) ) )
       => ( ( Core @ X3 )
          = ( Core @ B ) ) ) ) ).

% sep_algebra.extract_core
thf(fact_798_sep__algebra_Oextract__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( ( some_state @ B )
            = ( Plus @ A @ X3 ) )
          & ( sep_greater_state @ Plus @ X3 @ ( Core @ B ) ) )
       => ( ( Core @ X3 )
          = ( Core @ B ) ) ) ) ).

% sep_algebra.extract_core
thf(fact_799_sep__algebra_Ogreater__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ A @ B )
        = ( ? [C3: val] :
              ( ( some_val @ A )
              = ( Plus @ B @ C3 ) ) ) ) ) ).

% sep_algebra.greater_def
thf(fact_800_sep__algebra_Ogreater__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ A @ B )
        = ( ? [C3: state] :
              ( ( some_state @ A )
              = ( Plus @ B @ C3 ) ) ) ) ) ).

% sep_algebra.greater_def
thf(fact_801_greater__def,axiom,
    ( greater
    = ( sep_greater_state @ plus ) ) ).

% greater_def
thf(fact_802_package__logic_Ogreater__than__sum__exists,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val,B1: val,B22: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A @ B )
       => ( ( ( some_val @ B )
            = ( Plus @ B1 @ B22 ) )
         => ? [R3: val] :
              ( ( ( some_val @ A )
                = ( Plus @ R3 @ B22 ) )
              & ( sep_greater_val @ Plus @ ( Core @ R3 ) @ ( Core @ A ) )
              & ( sep_greater_val @ Plus @ R3 @ B1 ) ) ) ) ) ).

% package_logic.greater_than_sum_exists
thf(fact_803_package__logic_Ogreater__than__sum__exists,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state,B1: state,B22: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( ( some_state @ B )
            = ( 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_804_package__logic_Obigger__core__sum__defined,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ ( Core @ A ) @ B )
       => ( ( some_val @ A )
          = ( Plus @ A @ B ) ) ) ) ).

% package_logic.bigger_core_sum_defined
thf(fact_805_package__logic_Obigger__core__sum__defined,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ ( Core @ A ) @ B )
       => ( ( some_state @ A )
          = ( Plus @ A @ B ) ) ) ) ).

% package_logic.bigger_core_sum_defined
thf(fact_806_package__logic_Obigger__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,A: val,B: val,Phi2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ Phi )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_val @ Plus @ Phi2 @ Phi )
         => ? [B5: val] :
              ( ( sep_greater_val @ Plus @ B5 @ B )
              & ( ( some_val @ Phi2 )
                = ( Plus @ A @ B5 ) ) ) ) ) ) ).

% package_logic.bigger_sum
thf(fact_807_package__logic_Obigger__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,A: state,B: state,Phi2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ Phi )
          = ( Plus @ A @ B ) )
       => ( ( sep_greater_state @ Plus @ Phi2 @ Phi )
         => ? [B5: state] :
              ( ( sep_greater_state @ Plus @ B5 @ B )
              & ( ( some_state @ Phi2 )
                = ( Plus @ A @ B5 ) ) ) ) ) ) ).

% package_logic.bigger_sum
thf(fact_808_sep__algebra_Ominus__default,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ~ ( sep_greater_state @ Plus @ B @ A )
       => ( ( sep_minus_state @ Plus @ Core @ B @ A )
          = B ) ) ) ).

% sep_algebra.minus_default
thf(fact_809_sep__algebra_Ominus__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ X3 @ A )
       => ( sep_greater_state @ Plus @ X3 @ ( sep_minus_state @ Plus @ Core @ X3 @ A ) ) ) ) ).

% sep_algebra.minus_smaller
thf(fact_810_sep__algebra_Ogreater__minus__trans,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X3: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X3 )
       => ( ( sep_greater_state @ Plus @ X3 @ A )
         => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ Y @ A ) @ ( sep_minus_state @ Plus @ Core @ X3 @ A ) ) ) ) ) ).

% sep_algebra.greater_minus_trans
thf(fact_811_sep__algebra_Opure__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_pure_state @ Plus @ A )
       => ( ( sep_greater_state @ Plus @ A @ B )
         => ( sep_pure_state @ Plus @ B ) ) ) ) ).

% sep_algebra.pure_smaller
thf(fact_812_sep__algebra_Osmaller__compatible,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A6: state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_defined_state @ Plus @ A6 @ B )
       => ( ( sep_greater_state @ Plus @ A6 @ A )
         => ( sep_defined_state @ Plus @ A @ B ) ) ) ) ).

% sep_algebra.smaller_compatible
thf(fact_813_sep__algebra_Osmaller__compatible__core,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ X3 )
       => ( sep_defined_state @ Plus @ X3 @ ( Core @ Y ) ) ) ) ).

% sep_algebra.smaller_compatible_core
thf(fact_814_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 )
        = ( ! [X5: state] :
              ( ( sep_greater_state @ Plus @ X5 @ ( F @ X5 ) )
              & ( P @ ( F @ X5 ) )
              & ! [P3: state] :
                  ( ( ( P @ P3 )
                    & ( sep_greater_state @ Plus @ X5 @ P3 ) )
                 => ( sep_greater_state @ Plus @ ( F @ X5 ) @ P3 ) ) ) ) ) ) ).

% sep_algebra.max_projection_prop_def
thf(fact_815_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 )
     => ( ! [X: state] : ( sep_greater_state @ Plus @ X @ ( F @ X ) )
       => ( ! [X: state] : ( P @ ( F @ X ) )
         => ( ! [X: state,P4: state] :
                ( ( ( P @ P4 )
                  & ( sep_greater_state @ Plus @ X @ P4 ) )
               => ( sep_greater_state @ Plus @ ( F @ X ) @ P4 ) )
           => ( sep_ma8214210560313151521_state @ Plus @ P @ F ) ) ) ) ) ).

% sep_algebra.max_projection_propI
thf(fact_816_sep__algebra_Ompp__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( sep_greater_state @ Plus @ X3 @ ( F @ X3 ) ) ) ) ).

% sep_algebra.mpp_smaller
thf(fact_817_sep__algebra_Ompp__mono,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_greater_state @ Plus @ A @ B )
         => ( sep_greater_state @ Plus @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% sep_algebra.mpp_mono
thf(fact_818_sep__algebra_OmppI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,A: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( sep_greater_state @ Plus @ A @ X3 )
         => ( ( P @ X3 )
           => ( ( sep_greater_state @ Plus @ X3 @ ( F @ A ) )
             => ( X3
                = ( F @ A ) ) ) ) ) ) ) ).

% sep_algebra.mppI
thf(fact_819_sep__algebra_Omax__projection__propE_I3_J,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o,F: state > state,P2: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_ma8214210560313151521_state @ Plus @ P @ F )
       => ( ( ( P @ P2 )
            & ( sep_greater_state @ Plus @ X3 @ P2 ) )
         => ( sep_greater_state @ Plus @ ( F @ X3 ) @ P2 ) ) ) ) ).

% sep_algebra.max_projection_propE(3)
thf(fact_820_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ! [Phi6: produc7272280185086996969_state,Phi4: produc7272280185086996969_state] :
            ( ( ( sep_gr1233117237951779860_state @ Plus @ Phi6 @ Phi4 )
              & ( member7461423308196221312_state @ Phi4 @ A2 ) )
           => ( member7461423308196221312_state @ Phi6 @ A2 ) )
       => ( sep_up5671800011468993857_state @ Plus @ A2 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_821_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ! [Phi6: produc9003781588162623004on_val,Phi4: produc9003781588162623004on_val] :
            ( ( ( sep_gr8124549141218502215on_val @ Plus @ Phi6 @ Phi4 )
              & ( member2879030624002546611on_val @ Phi4 @ A2 ) )
           => ( member2879030624002546611on_val @ Phi6 @ A2 ) )
       => ( sep_up3467202520194525556on_val @ Plus @ A2 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_822_sep__algebra_Oup__closedI,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ! [Phi6: produc3882902684156697774_state,Phi4: produc3882902684156697774_state] :
            ( ( ( sep_gr6114829331755084611_state @ Plus @ Phi6 @ Phi4 )
              & ( member6715010751053357271_state @ Phi4 @ A2 ) )
           => ( member6715010751053357271_state @ Phi6 @ A2 ) )
       => ( sep_up2356903840689987414_state @ Plus @ A2 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_823_sep__algebra_Oup__closedI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [Phi6: state,Phi4: state] :
            ( ( ( sep_greater_state @ Plus @ Phi6 @ Phi4 )
              & ( member_state @ Phi4 @ A2 ) )
           => ( member_state @ Phi6 @ A2 ) )
       => ( sep_up_closed_state @ Plus @ A2 ) ) ) ).

% sep_algebra.up_closedI
thf(fact_824_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_up5671800011468993857_state @ Plus @ A2 )
        = ( ! [Phi5: produc7272280185086996969_state] :
              ( ? [X5: produc7272280185086996969_state] :
                  ( ( member7461423308196221312_state @ X5 @ A2 )
                  & ( sep_gr1233117237951779860_state @ Plus @ Phi5 @ X5 ) )
             => ( member7461423308196221312_state @ Phi5 @ A2 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_825_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_up3467202520194525556on_val @ Plus @ A2 )
        = ( ! [Phi5: produc9003781588162623004on_val] :
              ( ? [X5: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ X5 @ A2 )
                  & ( sep_gr8124549141218502215on_val @ Plus @ Phi5 @ X5 ) )
             => ( member2879030624002546611on_val @ Phi5 @ A2 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_826_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_up2356903840689987414_state @ Plus @ A2 )
        = ( ! [Phi5: produc3882902684156697774_state] :
              ( ? [X5: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X5 @ A2 )
                  & ( sep_gr6114829331755084611_state @ Plus @ Phi5 @ X5 ) )
             => ( member6715010751053357271_state @ Phi5 @ A2 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_827_sep__algebra_Oup__closed__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ A2 )
        = ( ! [Phi5: state] :
              ( ? [X5: state] :
                  ( ( member_state @ X5 @ A2 )
                  & ( sep_greater_state @ Plus @ Phi5 @ X5 ) )
             => ( member_state @ Phi5 @ A2 ) ) ) ) ) ).

% sep_algebra.up_closed_def
thf(fact_828_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 )
        = ( ! [X5: state,Y4: state] :
              ( ( ( sep_greater_state @ Plus @ Y4 @ X5 )
                & ( P @ X5 ) )
             => ( P @ Y4 ) ) ) ) ) ).

% sep_algebra.mono_prop_def
thf(fact_829_sep__algebra_Omono__propI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [X: state,Y3: state] :
            ( ( ( sep_greater_state @ Plus @ Y3 @ X )
              & ( P @ X ) )
           => ( P @ Y3 ) )
       => ( sep_mono_prop_state @ Plus @ P ) ) ) ).

% sep_algebra.mono_propI
thf(fact_830_package__logic_Obigger__the,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,X6: val,Y: val,X3: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_val @ A )
          = ( Plus @ X6 @ Y ) )
       => ( ( sep_greater_val @ Plus @ X6 @ X3 )
         => ( sep_greater_val @ Plus @ ( the_val @ ( Plus @ ( Core @ A ) @ X6 ) ) @ ( the_val @ ( Plus @ ( Core @ A ) @ X3 ) ) ) ) ) ) ).

% package_logic.bigger_the
thf(fact_831_package__logic_Obigger__the,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,X6: state,Y: state,X3: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( ( some_state @ A )
          = ( Plus @ X6 @ Y ) )
       => ( ( sep_greater_state @ Plus @ X6 @ X3 )
         => ( sep_greater_state @ Plus @ ( the_state @ ( Plus @ ( Core @ A ) @ X6 ) ) @ ( the_state @ ( Plus @ ( Core @ A ) @ X3 ) ) ) ) ) ) ).

% package_logic.bigger_the
thf(fact_832_sep__algebra_Ominus__equiv__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ B @ A )
       => ( ( ( some_val @ B )
            = ( Plus @ A @ ( sep_minus_val @ Plus @ Core @ B @ A ) ) )
          & ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ B @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_equiv_def
thf(fact_833_sep__algebra_Ominus__equiv__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ B @ A )
       => ( ( ( some_state @ B )
            = ( Plus @ A @ ( sep_minus_state @ Plus @ Core @ B @ A ) ) )
          & ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ B @ A ) @ ( Core @ B ) ) ) ) ) ).

% sep_algebra.minus_equiv_def
thf(fact_834_sep__algebra_Ominus__and__plus,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Omega: val,Omega2: val,R4: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ Omega )
          = ( Plus @ Omega2 @ R4 ) )
       => ( ( sep_greater_val @ Plus @ Omega2 @ A )
         => ( ( some_val @ ( sep_minus_val @ Plus @ Core @ Omega @ A ) )
            = ( Plus @ ( sep_minus_val @ Plus @ Core @ Omega2 @ A ) @ R4 ) ) ) ) ) ).

% sep_algebra.minus_and_plus
thf(fact_835_sep__algebra_Ominus__and__plus,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Omega: state,Omega2: state,R4: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ Omega )
          = ( Plus @ Omega2 @ R4 ) )
       => ( ( sep_greater_state @ Plus @ Omega2 @ A )
         => ( ( some_state @ ( sep_minus_state @ Plus @ Core @ Omega @ A ) )
            = ( Plus @ ( sep_minus_state @ Plus @ Core @ Omega2 @ A ) @ R4 ) ) ) ) ) ).

% sep_algebra.minus_and_plus
thf(fact_836_sep__algebra_Ominus__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,X3: val,A: val,B: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ X3 )
          = ( Plus @ A @ B ) )
       => ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ X3 @ A ) @ B ) ) ) ).

% sep_algebra.minus_bigger
thf(fact_837_sep__algebra_Ominus__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,X3: state,A: state,B: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ X3 )
          = ( Plus @ A @ B ) )
       => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ X3 @ A ) @ B ) ) ) ).

% sep_algebra.minus_bigger
thf(fact_838_sep__algebra_Ominus__sum,axiom,
    ! [Plus: val > val > option_val,Core: val > val,A: val,B: val,C: val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ A )
          = ( Plus @ B @ C ) )
       => ( ( sep_greater_val @ Plus @ X3 @ A )
         => ( ( sep_minus_val @ Plus @ Core @ X3 @ A )
            = ( sep_minus_val @ Plus @ Core @ ( sep_minus_val @ Plus @ Core @ X3 @ B ) @ C ) ) ) ) ) ).

% sep_algebra.minus_sum
thf(fact_839_sep__algebra_Ominus__sum,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A: state,B: state,C: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ A )
          = ( Plus @ B @ C ) )
       => ( ( sep_greater_state @ Plus @ X3 @ A )
         => ( ( sep_minus_state @ Plus @ Core @ X3 @ A )
            = ( sep_minus_state @ Plus @ Core @ ( sep_minus_state @ Plus @ Core @ X3 @ B ) @ C ) ) ) ) ) ).

% sep_algebra.minus_sum
thf(fact_840_sep__algebra_OminusI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( ( some_val @ B )
          = ( Plus @ A @ X3 ) )
       => ( ( sep_greater_val @ Plus @ X3 @ ( Core @ B ) )
         => ( X3
            = ( sep_minus_val @ Plus @ Core @ B @ A ) ) ) ) ) ).

% sep_algebra.minusI
thf(fact_841_sep__algebra_OminusI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ( some_state @ B )
          = ( Plus @ A @ X3 ) )
       => ( ( sep_greater_state @ Plus @ X3 @ ( Core @ B ) )
         => ( X3
            = ( sep_minus_state @ Plus @ Core @ B @ A ) ) ) ) ) ).

% sep_algebra.minusI
thf(fact_842_package__logic_Ominus__some,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A: val,B: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A @ B )
       => ( ( some_val @ A )
          = ( Plus @ B @ ( sep_minus_val @ Plus @ Core @ A @ B ) ) ) ) ) ).

% package_logic.minus_some
thf(fact_843_package__logic_Ominus__some,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A: state,B: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A @ B )
       => ( ( some_state @ A )
          = ( Plus @ B @ ( sep_minus_state @ Plus @ Core @ A @ B ) ) ) ) ) ).

% package_logic.minus_some
thf(fact_844_package__logic_Oprove__last__completeness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A6: val,A: val,Nf1: val,F2: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_val @ Plus @ A6 @ A )
       => ( ( ( some_val @ A )
            = ( Plus @ Nf1 @ F2 ) )
         => ( sep_greater_val @ Plus @ ( sep_minus_val @ Plus @ Core @ A6 @ Nf1 ) @ F2 ) ) ) ) ).

% package_logic.prove_last_completeness
thf(fact_845_package__logic_Oprove__last__completeness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A6: state,A: state,Nf1: state,F2: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( sep_greater_state @ Plus @ A6 @ A )
       => ( ( ( some_state @ A )
            = ( Plus @ Nf1 @ F2 ) )
         => ( sep_greater_state @ Plus @ ( sep_minus_state @ Plus @ Core @ A6 @ Nf1 ) @ F2 ) ) ) ) ).

% package_logic.prove_last_completeness
thf(fact_846_sep__algebra_Osmaller__pure__sum__smaller,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Y: val,A: val,B: val,X3: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_greater_val @ Plus @ Y @ A )
       => ( ( sep_greater_val @ Plus @ Y @ B )
         => ( ( ( some_val @ X3 )
              = ( Plus @ A @ B ) )
           => ( ( sep_pure_val @ Plus @ B )
             => ( sep_greater_val @ Plus @ Y @ X3 ) ) ) ) ) ) ).

% sep_algebra.smaller_pure_sum_smaller
thf(fact_847_sep__algebra_Osmaller__pure__sum__smaller,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Y: state,A: state,B: state,X3: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_greater_state @ Plus @ Y @ A )
       => ( ( sep_greater_state @ Plus @ Y @ B )
         => ( ( ( some_state @ X3 )
              = ( Plus @ A @ B ) )
           => ( ( sep_pure_state @ Plus @ B )
             => ( sep_greater_state @ Plus @ Y @ X3 ) ) ) ) ) ) ).

% sep_algebra.smaller_pure_sum_smaller
thf(fact_848_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_up6452048470549419908_state @ Plus @ A2 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Phi5: produc3882902684156697774_state,Phi3: produc3882902684156697774_state] :
              ( ( Uu3 = Phi5 )
              & ( sep_gr6114829331755084611_state @ Plus @ Phi5 @ Phi3 )
              & ( member6715010751053357271_state @ Phi3 @ A2 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_849_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_up6054414863798424979_state @ Plus @ A2 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Phi5: produc7272280185086996969_state,Phi3: produc7272280185086996969_state] :
              ( ( Uu3 = Phi5 )
              & ( sep_gr1233117237951779860_state @ Plus @ Phi5 @ Phi3 )
              & ( member7461423308196221312_state @ Phi3 @ A2 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_850_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_up7144661806627585478on_val @ Plus @ A2 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi5: produc9003781588162623004on_val,Phi3: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi5 )
              & ( sep_gr8124549141218502215on_val @ Plus @ Phi5 @ Phi3 )
              & ( member2879030624002546611on_val @ Phi3 @ A2 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_851_sep__algebra_Oupper__closure__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up1246176804924251236_state @ Plus @ A2 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi5: state,Phi3: state] :
              ( ( Uu3 = Phi5 )
              & ( sep_greater_state @ Plus @ Phi5 @ Phi3 )
              & ( member_state @ Phi3 @ A2 ) ) ) ) ) ).

% sep_algebra.upper_closure_def
thf(fact_852_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,Omega2: produc3882902684156697774_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ( sep_un1732284790778891013_state @ Plus @ A2 @ Omega2 )
        = ( collec394671971739345689_state
          @ ^ [Uu3: produc3882902684156697774_state] :
            ? [Omega3: produc3882902684156697774_state] :
              ( ( Uu3 = Omega3 )
              & ( member6715010751053357271_state @ Omega3 @ A2 )
              & ( sep_gr6114829331755084611_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_853_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,Omega2: produc7272280185086996969_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ( sep_un1299216510214111186_state @ Plus @ A2 @ Omega2 )
        = ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [Omega3: produc7272280185086996969_state] :
              ( ( Uu3 = Omega3 )
              & ( member7461423308196221312_state @ Omega3 @ A2 )
              & ( sep_gr1233117237951779860_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_854_sep__algebra_Ounder__def,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,Omega2: produc9003781588162623004on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ( sep_un3765025543224000005on_val @ Plus @ A2 @ Omega2 )
        = ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Omega3: produc9003781588162623004on_val] :
              ( ( Uu3 = Omega3 )
              & ( member2879030624002546611on_val @ Omega3 @ A2 )
              & ( sep_gr8124549141218502215on_val @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_855_sep__algebra_Ounder__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,Omega2: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_under_state @ Plus @ A2 @ Omega2 )
        = ( collect_state
          @ ^ [Uu3: state] :
            ? [Omega3: state] :
              ( ( Uu3 = Omega3 )
              & ( member_state @ Omega3 @ A2 )
              & ( sep_greater_state @ Plus @ Omega2 @ Omega3 ) ) ) ) ) ).

% sep_algebra.under_def
thf(fact_856_sep__algebra_Ominus__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,B: val,A: val] :
      ( ( sep_algebra_val @ Plus @ Core )
     => ( ( sep_minus_val @ Plus @ Core @ B @ A )
        = ( fun_THE_default_val @ B
          @ ^ [X5: val] :
              ( ( ( some_val @ B )
                = ( Plus @ A @ X5 ) )
              & ( sep_greater_val @ Plus @ X5 @ ( Core @ B ) ) ) ) ) ) ).

% sep_algebra.minus_def
thf(fact_857_sep__algebra_Ominus__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B: state,A: state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_minus_state @ Plus @ Core @ B @ A )
        = ( fun_TH8221542053827138557_state @ B
          @ ^ [X5: state] :
              ( ( ( some_state @ B )
                = ( Plus @ A @ X5 ) )
              & ( sep_greater_state @ Plus @ X5 @ ( Core @ B ) ) ) ) ) ) ).

% sep_algebra.minus_def
thf(fact_858_package__logic_Osat__star__exists__bigger,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,A2: aassertion_val,B2: aassertion_val,Phi: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_val @ Plus @ ( aStar_val @ A2 @ B2 ) @ Phi )
       => ( ( packag5732365195153136185on_val @ Plus @ Core @ A2 )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ B2 )
           => ? [A3: val,B7: val] :
                ( ( sep_greater_val @ Plus @ ( Core @ A3 ) @ ( Core @ Phi ) )
                & ( sep_greater_val @ Plus @ ( Core @ B7 ) @ ( Core @ Phi ) )
                & ( ( some_val @ Phi )
                  = ( Plus @ A3 @ B7 ) )
                & ( package_sat_val @ Plus @ A2 @ A3 )
                & ( package_sat_val @ Plus @ B2 @ B7 ) ) ) ) ) ) ).

% package_logic.sat_star_exists_bigger
thf(fact_859_package__logic_Osat__star__exists__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,A2: aassertion_state,B2: aassertion_state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ( package_sat_state @ Plus @ ( aStar_state @ A2 @ B2 ) @ Phi )
       => ( ( packag7298478212905792770_state @ Plus @ Core @ A2 )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ B2 )
           => ? [A3: state,B7: state] :
                ( ( sep_greater_state @ Plus @ ( Core @ A3 ) @ ( Core @ Phi ) )
                & ( sep_greater_state @ Plus @ ( Core @ B7 ) @ ( Core @ Phi ) )
                & ( ( some_state @ Phi )
                  = ( Plus @ A3 @ B7 ) )
                & ( package_sat_state @ Plus @ A2 @ A3 )
                & ( package_sat_state @ Plus @ B2 @ B7 ) ) ) ) ) ) ).

% package_logic.sat_star_exists_bigger
thf(fact_860_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 )
     => ( ! [A3: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( sep_defined_state @ Plus @ A3 @ U )
              & ( sep_greater_state @ Plus @ ( Core @ A3 ) @ ( Core @ U ) )
              & ( packag4817583915586052229_state @ Plus @ Unit @ T3 )
              & ( sep_greater_state @ Plus @ A3 @ ( Core @ ( T3 @ F ) ) ) ) )
       => ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F ) ) ) ).

% package_logic.valid_package_setI
thf(fact_861_package__logic_Opackage__rhs__connection__instantiate,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,F: val,S2: set_Pr4723104092121316667al_val,Pc: val > $o,A2: aassertion_val,Phi2: val,F3: val,S3: set_Pr4723104092121316667al_val,A: val,U3: val,T2: val > val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
       => ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A @ ( produc303288909735261391al_val @ U3 @ T2 ) ) @ S2 )
         => ~ ! [Au2: val] :
                ( ( ( some_val @ Au2 )
                  = ( Plus @ A @ U3 ) )
               => ~ ( ( sep_defined_val @ Plus @ Au2 @ ( sep_minus_val @ Plus @ Core @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                   => ? [A9: val,U4: val] :
                        ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A9 @ ( produc303288909735261391al_val @ U4 @ T2 ) ) @ S3 )
                        & ( sep_greater_val @ Plus @ ( Core @ A9 ) @ ( Core @ A ) )
                        & ( ( Plus @ Au2 @ ( sep_minus_val @ Plus @ Core @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                          = ( Plus @ A9 @ U4 ) )
                        & ( sep_greater_val @ Plus @ U4 @ U3 )
                        & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A2 @ A9 @ U4 @ U3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connection_instantiate
thf(fact_862_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,A2: 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 @ A2 @ 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 ) ) )
                   => ? [A9: state,U4: state] :
                        ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A9 @ ( produc8632743443909064678_state @ U4 @ T2 ) ) @ S3 )
                        & ( sep_greater_state @ Plus @ ( Core @ A9 ) @ ( Core @ A ) )
                        & ( ( Plus @ Au2 @ ( sep_minus_state @ Plus @ Core @ ( T2 @ F3 ) @ ( T2 @ F ) ) )
                          = ( Plus @ A9 @ U4 ) )
                        & ( sep_greater_state @ Plus @ U4 @ U3 )
                        & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A2 @ A9 @ U4 @ U3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connection_instantiate
thf(fact_863_package__logic_Opackage__rhs__connectionI,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,F: val,Phi2: val,F3: val,S2: set_Pr4723104092121316667al_val,S3: set_Pr4723104092121316667al_val,Pc: val > $o,A2: aassertion_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( ( Plus @ Phi @ F )
          = ( Plus @ Phi2 @ F3 ) )
       => ( ( Stable @ F3 )
         => ( ( sep_defined_val @ Plus @ Phi @ F )
           => ( ( sep_greater_val @ Plus @ F3 @ F )
             => ( ! [A3: val,U: val,T3: val > val] :
                    ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A3 @ ( produc303288909735261391al_val @ U @ T3 ) ) @ S2 )
                   => ? [Au: val] :
                        ( ( ( some_val @ Au )
                          = ( Plus @ A3 @ U ) )
                        & ( ( sep_defined_val @ Plus @ Au @ ( sep_minus_val @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                         => ? [A8: val,U2: val] :
                              ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A8 @ ( produc303288909735261391al_val @ U2 @ T3 ) ) @ S3 )
                              & ( sep_greater_val @ Plus @ ( Core @ A8 ) @ ( Core @ A3 ) )
                              & ( ( Plus @ Au @ ( sep_minus_val @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                                = ( Plus @ A8 @ U2 ) )
                              & ( sep_greater_val @ Plus @ U2 @ U )
                              & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A2 @ A8 @ U2 @ U ) ) ) ) )
               => ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connectionI
thf(fact_864_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,A2: 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 )
             => ( ! [A3: state,U: state,T3: state > state] :
                    ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
                   => ? [Au: state] :
                        ( ( ( some_state @ Au )
                          = ( Plus @ A3 @ U ) )
                        & ( ( sep_defined_state @ Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                         => ? [A8: state,U2: state] :
                              ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A8 @ ( produc8632743443909064678_state @ U2 @ T3 ) ) @ S3 )
                              & ( sep_greater_state @ Plus @ ( Core @ A8 ) @ ( Core @ A3 ) )
                              & ( ( Plus @ Au @ ( sep_minus_state @ Plus @ Core @ ( T3 @ F3 ) @ ( T3 @ F ) ) )
                                = ( Plus @ A8 @ U2 ) )
                              & ( sep_greater_state @ Plus @ U2 @ U )
                              & ( packag8070727096299822599_state @ Plus @ Core @ Pc @ A2 @ A8 @ U2 @ U ) ) ) ) )
               => ( packag2054414653763826108_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 ) ) ) ) ) ) ) ).

% package_logic.package_rhs_connectionI
thf(fact_865_PartialSA_Ogeneral__soundness,axiom,
    ! [Phi: state,S2: set_Pr2391701600042252046_state,A2: aassertion_state,Phi2: state,F: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ unit
        @ ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A4: state,T4: state > state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ unit @ T4 ) ) )
              & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A4 @ T4 ) @ S2 ) ) )
        @ ^ [Uu3: state] : $true
        @ A2
        @ Phi2
        @ F
        @ S3 )
     => ( ! [A3: state,T3: state > state] :
            ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A3 @ T3 ) @ S2 )
           => ( packag4817583915586052229_state @ plus @ unit @ T3 ) )
       => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
         => ( ( ( packag8361946002163212404_state @ plus @ ( package_sat_state @ plus @ A2 ) )
              | ( packag4152378406607043437_state @ plus @ S3 ) )
           => ( ( ( some_state @ Phi )
                = ( plus @ Phi2 @ F ) )
              & ( stable @ F )
              & ! [X4: produc3882902684156697774_state] :
                  ( ( member6715010751053357271_state @ X4 @ S2 )
                 => ( produc3645877548210629341tate_o
                    @ ^ [A4: state,T4: state > state] :
                        ( ( defined @ A4 @ ( T4 @ F ) )
                       => ( package_sat_state @ plus @ A2 @ ( the_state @ ( plus @ A4 @ ( T4 @ F ) ) ) ) )
                    @ X4 ) ) ) ) ) ) ) ).

% PartialSA.general_soundness
thf(fact_866_package__logic_Ogeneral__soundness,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,S2: set_Pr6546617632949187467al_val,A2: aassertion_val,Phi2: val,F: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi @ Unit
          @ ( collec8300311316619386842al_val
            @ ^ [Uu3: produc2072440392815185925al_val] :
              ? [A4: val,T4: val > val] :
                ( ( Uu3
                  = ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ Unit @ T4 ) ) )
                & ( member331055627401653996al_val @ ( produc303288909735261391al_val @ A4 @ T4 ) @ S2 ) ) )
          @ ^ [Uu3: val] : $true
          @ A2
          @ Phi2
          @ F
          @ S3 )
       => ( ! [A3: val,T3: val > val] :
              ( ( member331055627401653996al_val @ ( produc303288909735261391al_val @ A3 @ T3 ) @ S2 )
             => ( packag4175576287423129206er_val @ Plus @ Unit @ T3 ) )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ A2 )
           => ( ( ( packag2085742556458808647ic_val @ Plus @ ( package_sat_val @ Plus @ A2 ) )
                | ( packag3202507263771882441al_val @ Plus @ S3 ) )
             => ( ( ( some_val @ Phi )
                  = ( Plus @ Phi2 @ F ) )
                & ( Stable @ F )
                & ! [X4: produc5503543764249810261al_val] :
                    ( ( member331055627401653996al_val @ X4 @ S2 )
                   => ( produc7666680587469814684_val_o
                      @ ^ [A4: val,T4: val > val] :
                          ( ( sep_defined_val @ Plus @ A4 @ ( T4 @ F ) )
                         => ( package_sat_val @ Plus @ A2 @ ( the_val @ ( Plus @ A4 @ ( T4 @ F ) ) ) ) )
                      @ X4 ) ) ) ) ) ) ) ) ).

% package_logic.general_soundness
thf(fact_867_package__logic_Ogeneral__soundness,axiom,
    ! [Plus: state > state > option_state,Core: state > state,Unit: state,Stable: state > $o,Phi: state,S2: set_Pr2391701600042252046_state,A2: 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] :
              ? [A4: state,T4: state > state] :
                ( ( Uu3
                  = ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ Unit @ T4 ) ) )
                & ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A4 @ T4 ) @ S2 ) ) )
          @ ^ [Uu3: state] : $true
          @ A2
          @ Phi2
          @ F
          @ S3 )
       => ( ! [A3: state,T3: state > state] :
              ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A3 @ T3 ) @ S2 )
             => ( packag4817583915586052229_state @ Plus @ Unit @ T3 ) )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A2 )
           => ( ( ( packag8361946002163212404_state @ Plus @ ( package_sat_state @ Plus @ A2 ) )
                | ( packag4152378406607043437_state @ Plus @ S3 ) )
             => ( ( ( some_state @ Phi )
                  = ( Plus @ Phi2 @ F ) )
                & ( Stable @ F )
                & ! [X4: produc3882902684156697774_state] :
                    ( ( member6715010751053357271_state @ X4 @ S2 )
                   => ( produc3645877548210629341tate_o
                      @ ^ [A4: state,T4: state > state] :
                          ( ( sep_defined_state @ Plus @ A4 @ ( T4 @ F ) )
                         => ( package_sat_state @ Plus @ A2 @ ( the_state @ ( Plus @ A4 @ ( T4 @ F ) ) ) ) )
                      @ X4 ) ) ) ) ) ) ) ) ).

% package_logic.general_soundness
thf(fact_868_PartialSA_Omono__prop__set,axiom,
    ! [A2: set_state,B2: set_state,P: state > $o] :
      ( ( greater_set @ A2 @ B2 )
     => ( ( sep_setify_state @ P @ B2 )
       => ( ( sep_mono_prop_state @ plus @ P )
         => ( sep_setify_state @ P @ A2 ) ) ) ) ).

% PartialSA.mono_prop_set
thf(fact_869_PartialSA_Oup__closed__bigger__subset,axiom,
    ! [B2: set_state,A2: set_state] :
      ( ( sep_up_closed_state @ plus @ B2 )
     => ( ( greater_set @ A2 @ B2 )
       => ( ord_le2494988322063910608_state @ A2 @ B2 ) ) ) ).

% PartialSA.up_closed_bigger_subset
thf(fact_870_split__part,axiom,
    ! [P: $o,Q: state > ( state > state ) > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [A4: state,B3: state > state] :
            ( P
            & ( Q @ A4 @ B3 ) ) )
      = ( ^ [Ab2: produc3882902684156697774_state] :
            ( P
            & ( produc3645877548210629341tate_o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_871_split__part,axiom,
    ! [P: $o,Q: state > produc3882902684156697774_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [A4: state,B3: produc3882902684156697774_state] :
            ( P
            & ( Q @ A4 @ B3 ) ) )
      = ( ^ [Ab2: produc7272280185086996969_state] :
            ( P
            & ( produc1745419677418529872tate_o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_872_case__prod__conv,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_873_case__prod__conv,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_874_case__prodI2,axiom,
    ! [P2: produc1317171644805857433on_val,C: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ! [A3: produc9003781588162623004on_val,B7: produc9003781588162623004on_val] :
          ( ( P2
            = ( produc4958730763592889809on_val @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc1845727980445322802_val_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_875_case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,C: option_state > option_state > $o] :
      ( ! [A3: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc6082990154846732246tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_876_case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,C: prat > state > $o] :
      ( ! [A3: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc1228703470279370976tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_877_case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,C: prat > set_state > $o] :
      ( ! [A3: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc7925652287191295402tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_878_case__prodI2,axiom,
    ! [P2: produc9003781588162623004on_val,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ! [A3: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( ( P2
            = ( produc74620467203698198on_val @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc5248837549992124053_val_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_879_case__prodI2,axiom,
    ! [P2: produc3882902684156697774_state,C: state > ( state > state ) > $o] :
      ( ! [A3: state,B7: state > state] :
          ( ( P2
            = ( produc8632743443909064678_state @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc3645877548210629341tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_880_case__prodI2,axiom,
    ! [P2: produc7272280185086996969_state,C: state > produc3882902684156697774_state > $o] :
      ( ! [A3: state,B7: produc3882902684156697774_state] :
          ( ( P2
            = ( produc5886875603596089627_state @ A3 @ B7 ) )
         => ( C @ A3 @ B7 ) )
     => ( produc1745419677418529872tate_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_881_case__prodI,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( F @ A @ B )
     => ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% case_prodI
thf(fact_882_case__prodI,axiom,
    ! [F: option_state > option_state > $o,A: option_state,B: option_state] :
      ( ( F @ A @ B )
     => ( produc6082990154846732246tate_o @ F @ ( produc9160152616361873709_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_883_case__prodI,axiom,
    ! [F: prat > state > $o,A: prat,B: state] :
      ( ( F @ A @ B )
     => ( produc1228703470279370976tate_o @ F @ ( produc6205719116230455971_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_884_case__prodI,axiom,
    ! [F: prat > set_state > $o,A: prat,B: set_state] :
      ( ( F @ A @ B )
     => ( produc7925652287191295402tate_o @ F @ ( produc9080284787853255769_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_885_case__prodI,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( F @ A @ B )
     => ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% case_prodI
thf(fact_886_case__prodI,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( F @ A @ B )
     => ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_887_case__prodI,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( F @ A @ B )
     => ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B ) ) ) ).

% case_prodI
thf(fact_888_case__prodE2,axiom,
    ! [Q: $o > $o,P: state > ( state > state ) > $o,Z2: produc3882902684156697774_state] :
      ( ( Q @ ( produc3645877548210629341tate_o @ P @ Z2 ) )
     => ~ ! [X: state,Y3: state > state] :
            ( ( Z2
              = ( produc8632743443909064678_state @ X @ Y3 ) )
           => ~ ( Q @ ( P @ X @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_889_case__prodE2,axiom,
    ! [Q: $o > $o,P: state > produc3882902684156697774_state > $o,Z2: produc7272280185086996969_state] :
      ( ( Q @ ( produc1745419677418529872tate_o @ P @ Z2 ) )
     => ~ ! [X: state,Y3: produc3882902684156697774_state] :
            ( ( Z2
              = ( produc5886875603596089627_state @ X @ Y3 ) )
           => ~ ( Q @ ( P @ X @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_890_case__prod__eta,axiom,
    ! [F: produc3882902684156697774_state > $o] :
      ( ( produc3645877548210629341tate_o
        @ ^ [X5: state,Y4: state > state] : ( F @ ( produc8632743443909064678_state @ X5 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_891_case__prod__eta,axiom,
    ! [F: produc7272280185086996969_state > $o] :
      ( ( produc1745419677418529872tate_o
        @ ^ [X5: state,Y4: produc3882902684156697774_state] : ( F @ ( produc5886875603596089627_state @ X5 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_892_cond__case__prod__eta,axiom,
    ! [F: state > ( state > state ) > $o,G: produc3882902684156697774_state > $o] :
      ( ! [X: state,Y3: state > state] :
          ( ( F @ X @ Y3 )
          = ( G @ ( produc8632743443909064678_state @ X @ Y3 ) ) )
     => ( ( produc3645877548210629341tate_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_893_cond__case__prod__eta,axiom,
    ! [F: state > produc3882902684156697774_state > $o,G: produc7272280185086996969_state > $o] :
      ( ! [X: state,Y3: produc3882902684156697774_state] :
          ( ( F @ X @ Y3 )
          = ( G @ ( produc5886875603596089627_state @ X @ Y3 ) ) )
     => ( ( produc1745419677418529872tate_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_894_old_Oprod_Ocase,axiom,
    ! [F: state > ( state > state ) > $o,X1: state,X2: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_895_old_Oprod_Ocase,axiom,
    ! [F: state > produc3882902684156697774_state > $o,X1: state,X2: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_896_case__prodD,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_897_case__prodD,axiom,
    ! [F: option_state > option_state > $o,A: option_state,B: option_state] :
      ( ( produc6082990154846732246tate_o @ F @ ( produc9160152616361873709_state @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_898_case__prodD,axiom,
    ! [F: prat > state > $o,A: prat,B: state] :
      ( ( produc1228703470279370976tate_o @ F @ ( produc6205719116230455971_state @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_899_case__prodD,axiom,
    ! [F: prat > set_state > $o,A: prat,B: set_state] :
      ( ( produc7925652287191295402tate_o @ F @ ( produc9080284787853255769_state @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_900_case__prodD,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_901_case__prodD,axiom,
    ! [F: state > ( state > state ) > $o,A: state,B: state > state] :
      ( ( produc3645877548210629341tate_o @ F @ ( produc8632743443909064678_state @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_902_case__prodD,axiom,
    ! [F: state > produc3882902684156697774_state > $o,A: state,B: produc3882902684156697774_state] :
      ( ( produc1745419677418529872tate_o @ F @ ( produc5886875603596089627_state @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_903_case__prodE,axiom,
    ! [C: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,P2: produc1317171644805857433on_val] :
      ( ( produc1845727980445322802_val_o @ C @ P2 )
     => ~ ! [X: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( P2
              = ( produc4958730763592889809on_val @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_904_case__prodE,axiom,
    ! [C: option_state > option_state > $o,P2: produc3142500478612311029_state] :
      ( ( produc6082990154846732246tate_o @ C @ P2 )
     => ~ ! [X: option_state,Y3: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_905_case__prodE,axiom,
    ! [C: prat > state > $o,P2: produc4844402304972151915_state] :
      ( ( produc1228703470279370976tate_o @ C @ P2 )
     => ~ ! [X: prat,Y3: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_906_case__prodE,axiom,
    ! [C: prat > set_state > $o,P2: produc6251377953716854561_state] :
      ( ( produc7925652287191295402tate_o @ C @ P2 )
     => ~ ! [X: prat,Y3: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_907_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: produc9003781588162623004on_val] :
      ( ( produc5248837549992124053_val_o @ C @ P2 )
     => ~ ! [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P2
              = ( produc74620467203698198on_val @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_908_case__prodE,axiom,
    ! [C: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ( produc3645877548210629341tate_o @ C @ P2 )
     => ~ ! [X: state,Y3: state > state] :
            ( ( P2
              = ( produc8632743443909064678_state @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_909_case__prodE,axiom,
    ! [C: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ( produc1745419677418529872tate_o @ C @ P2 )
     => ~ ! [X: state,Y3: produc3882902684156697774_state] :
            ( ( P2
              = ( produc5886875603596089627_state @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_910_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3882902684156697774_state] :
      ( produc3645877548210629341tate_o
      @ ^ [Uu3: state,Uv3: state > state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_911_prod_Odisc__eq__case,axiom,
    ! [Prod: produc7272280185086996969_state] :
      ( produc1745419677418529872tate_o
      @ ^ [Uu3: state,Uv3: produc3882902684156697774_state] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_912_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F: state > ( state > state ) > $o,Prod: produc3882902684156697774_state] :
      ( ( H3 @ ( produc3645877548210629341tate_o @ F @ Prod ) )
      = ( produc3645877548210629341tate_o
        @ ^ [X12: state,X23: state > state] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_913_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F: state > produc3882902684156697774_state > $o,Prod: produc7272280185086996969_state] :
      ( ( H3 @ ( produc1745419677418529872tate_o @ F @ Prod ) )
      = ( produc1745419677418529872tate_o
        @ ^ [X12: state,X23: produc3882902684156697774_state] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_914_PartialSA_Olarger__set__refl,axiom,
    ! [A2: set_state] : ( greater_set @ A2 @ A2 ) ).

% PartialSA.larger_set_refl
thf(fact_915_PartialSA_Osucc__set__trans,axiom,
    ! [A2: set_state,B2: set_state,C2: set_state] :
      ( ( greater_set @ A2 @ B2 )
     => ( ( greater_set @ B2 @ C2 )
       => ( greater_set @ A2 @ C2 ) ) ) ).

% PartialSA.succ_set_trans
thf(fact_916_Ball__def,axiom,
    ( ball_state
    = ( ^ [A7: set_state,P6: state > $o] :
        ! [X5: state] :
          ( ( member_state @ X5 @ A7 )
         => ( P6 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_917_Ball__def,axiom,
    ( ball_P7403102246598991807_state
    = ( ^ [A7: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] :
        ! [X5: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X5 @ A7 )
         => ( P6 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_918_Ball__def,axiom,
    ( ball_P2871712132005019634on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] :
        ! [X5: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X5 @ A7 )
         => ( P6 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_919_Ball__def,axiom,
    ( ball_P1066487999220904024_state
    = ( ^ [A7: set_Pr2391701600042252046_state,P6: produc3882902684156697774_state > $o] :
        ! [X5: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X5 @ A7 )
         => ( P6 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_920_PartialSA_Ogreater__set__def,axiom,
    ( greater_set
    = ( ^ [A7: set_state,B4: set_state] :
        ! [X5: state] :
          ( ( member_state @ X5 @ A7 )
         => ? [Y4: state] :
              ( ( member_state @ Y4 @ B4 )
              & ( greater @ X5 @ Y4 ) ) ) ) ) ).

% PartialSA.greater_set_def
thf(fact_921_PartialSA_Ogreater__setI,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ! [A3: state] :
          ( ( member_state @ A3 @ A2 )
         => ? [X4: state] :
              ( ( member_state @ X4 @ B2 )
              & ( greater @ A3 @ X4 ) ) )
     => ( greater_set @ A2 @ B2 ) ) ).

% PartialSA.greater_setI
thf(fact_922_Collect__case__prod__mono,axiom,
    ! [A2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,B2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( ord_le3852884270828607079_val_o @ A2 @ B2 )
     => ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ A2 ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_923_Collect__case__prod__mono,axiom,
    ! [A2: state > ( state > state ) > $o,B2: state > ( state > state ) > $o] :
      ( ( ord_le1106678366802930969tate_o @ A2 @ B2 )
     => ( ord_le6483787752022044334_state @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ A2 ) ) @ ( collec394671971739345689_state @ ( produc3645877548210629341tate_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_924_Collect__case__prod__mono,axiom,
    ! [A2: state > produc3882902684156697774_state > $o,B2: state > produc3882902684156697774_state > $o] :
      ( ( ord_le6704609105599929480tate_o @ A2 @ B2 )
     => ( ord_le4032880777312351359_state @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ A2 ) ) @ ( collec906845016635587262_state @ ( produc1745419677418529872tate_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_925_PartialSA_Obigger__set,axiom,
    ! [A12: set_state,A2: set_state,B2: set_state] :
      ( ( greater_set @ A12 @ A2 )
     => ( greater_set @ ( add_set @ A12 @ B2 ) @ ( add_set @ A2 @ B2 ) ) ) ).

% PartialSA.bigger_set
thf(fact_926_internal__case__prod__def,axiom,
    produc1225464857956169863tate_o = produc3645877548210629341tate_o ).

% internal_case_prod_def
thf(fact_927_internal__case__prod__def,axiom,
    produc3899216655821101094tate_o = produc1745419677418529872tate_o ).

% internal_case_prod_def
thf(fact_928_PartialSA_Osub__bigger,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( greater_set @ A2 @ B2 ) ) ).

% PartialSA.sub_bigger
thf(fact_929_PartialSA_OequivI,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( greater_set @ A2 @ B2 )
     => ( ( greater_set @ B2 @ A2 )
       => ( sep_equiv_state @ plus @ A2 @ B2 ) ) ) ).

% PartialSA.equivI
thf(fact_930_PartialSA_Oequiv__def,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ( sep_equiv_state @ plus @ A2 @ B2 )
      = ( ( greater_set @ A2 @ B2 )
        & ( greater_set @ B2 @ A2 ) ) ) ).

% PartialSA.equiv_def
thf(fact_931_PartialSA_Ocompleteness__aux,axiom,
    ! [S2: set_Pr4231240917283394591_state,F1: state > state > ( state > state ) > state,F2: state > state > ( state > state ) > state,Pc: state > $o,A2: aassertion_state,F: state,Phi: state] :
      ( ! [A3: state,U: state,T3: state > state] :
          ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
         => ( ( greater @ ( core @ ( F1 @ A3 @ U @ T3 ) ) @ ( core @ A3 ) )
            & ( ( some_state @ A3 )
              = ( plus @ ( F1 @ A3 @ U @ T3 ) @ ( F2 @ A3 @ U @ T3 ) ) )
            & ( ( Pc @ ( core @ A3 ) )
             => ( package_sat_state @ plus @ A2 @ ( the_state @ ( plus @ ( core @ A3 ) @ ( F1 @ A3 @ U @ T3 ) ) ) ) ) ) )
     => ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
       => ( ( packag7298478212905792770_state @ plus @ core @ A2 )
         => ( ( packag6595153354952283300_state @ plus @ core @ Pc )
           => ( ( ( stable @ F )
                & ( defined @ Phi @ F ) )
             => ? [S8: set_Pr4231240917283394591_state] :
                  ( ( packag3171892606229212990_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi @ F @ S8 )
                  & ! [X4: produc7272280185086996969_state] :
                      ( ( member7461423308196221312_state @ X4 @ S8 )
                     => ( produc1745419677418529872tate_o
                        @ ^ [A13: state] :
                            ( produc3645877548210629341tate_o
                            @ ^ [U8: state,T4: state > state] :
                              ? [A4: state,U6: state] :
                                ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                                & ( greater @ A13 @ ( F2 @ A4 @ U6 @ T4 ) )
                                & ( ( core @ A13 )
                                  = ( core @ A4 ) ) ) )
                        @ X4 ) ) ) ) ) ) ) ) ).

% PartialSA.completeness_aux
thf(fact_932_PartialSA_Opackage__rhs__connection__def,axiom,
    ! [Phi: state,F: state,S2: set_Pr4231240917283394591_state,Pc: state > $o,A2: aassertion_state,Phi2: state,F3: state,S3: set_Pr4231240917283394591_state] :
      ( ( packag2054414653763826108_state @ plus @ core @ stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
      = ( ( greater @ F3 @ F )
        & ( defined @ Phi @ F )
        & ( ( plus @ Phi @ F )
          = ( plus @ Phi2 @ F3 ) )
        & ( stable @ F3 )
        & ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A4: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T4: state > state] :
                    ? [Au3: state] :
                      ( ( ( some_state @ Au3 )
                        = ( plus @ A4 @ 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 @ A4 ) )
                            & ( ( plus @ Au3 @ ( minus @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                              = ( plus @ A13 @ U8 ) )
                            & ( greater @ U8 @ U6 )
                            & ( packag8070727096299822599_state @ plus @ core @ Pc @ A2 @ A13 @ U8 @ U6 ) ) ) ) )
              @ X5 ) ) ) ) ).

% PartialSA.package_rhs_connection_def
thf(fact_933_package__logic_Ocompleteness__aux,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,S2: set_Pr4723104092121316667al_val,F1: val > val > ( val > val ) > val,F2: val > val > ( val > val ) > val,Pc: val > $o,A2: aassertion_val,F: val,Phi: val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: val,U: val,T3: val > val] :
            ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A3 @ ( produc303288909735261391al_val @ U @ T3 ) ) @ S2 )
           => ( ( sep_greater_val @ Plus @ ( Core @ ( F1 @ A3 @ U @ T3 ) ) @ ( Core @ A3 ) )
              & ( ( some_val @ A3 )
                = ( Plus @ ( F1 @ A3 @ U @ T3 ) @ ( F2 @ A3 @ U @ T3 ) ) )
              & ( ( Pc @ ( Core @ A3 ) )
               => ( package_sat_val @ Plus @ A2 @ ( the_val @ ( Plus @ ( Core @ A3 ) @ ( F1 @ A3 @ U @ T3 ) ) ) ) ) ) )
       => ( ( packag7517340229427718612et_val @ Plus @ Core @ Unit @ S2 @ F )
         => ( ( packag5732365195153136185on_val @ Plus @ Core @ A2 )
           => ( ( packag6714868506247462167nd_val @ Plus @ Core @ Pc )
             => ( ( ( Stable @ F )
                  & ( sep_defined_val @ Plus @ Phi @ F ) )
               => ? [S8: set_Pr4723104092121316667al_val] :
                    ( ( packag2280635229005757181hs_val @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi @ F @ S8 )
                    & ! [X4: produc2072440392815185925al_val] :
                        ( ( member5640714103519202460al_val @ X4 @ S8 )
                       => ( produc6995717860488419308_val_o
                          @ ^ [A13: val] :
                              ( produc7666680587469814684_val_o
                              @ ^ [U8: val,T4: val > val] :
                                ? [A4: val,U6: val] :
                                  ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A4 @ ( produc303288909735261391al_val @ U6 @ T4 ) ) @ S2 )
                                  & ( sep_greater_val @ Plus @ A13 @ ( F2 @ A4 @ U6 @ T4 ) )
                                  & ( ( Core @ A13 )
                                    = ( Core @ A4 ) ) ) )
                          @ X4 ) ) ) ) ) ) ) ) ) ).

% package_logic.completeness_aux
thf(fact_934_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,A2: aassertion_state,F: state,Phi: state] :
      ( ( package_logic_state @ Plus @ Core @ Unit @ Stable )
     => ( ! [A3: state,U: state,T3: state > state] :
            ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A3 @ ( produc8632743443909064678_state @ U @ T3 ) ) @ S2 )
           => ( ( sep_greater_state @ Plus @ ( Core @ ( F1 @ A3 @ U @ T3 ) ) @ ( Core @ A3 ) )
              & ( ( some_state @ A3 )
                = ( Plus @ ( F1 @ A3 @ U @ T3 ) @ ( F2 @ A3 @ U @ T3 ) ) )
              & ( ( Pc @ ( Core @ A3 ) )
               => ( package_sat_state @ Plus @ A2 @ ( the_state @ ( Plus @ ( Core @ A3 ) @ ( F1 @ A3 @ U @ T3 ) ) ) ) ) ) )
       => ( ( packag3877571396724326311_state @ Plus @ Core @ Unit @ S2 @ F )
         => ( ( packag7298478212905792770_state @ Plus @ Core @ A2 )
           => ( ( packag6595153354952283300_state @ Plus @ Core @ Pc )
             => ( ( ( Stable @ F )
                  & ( sep_defined_state @ Plus @ Phi @ F ) )
               => ? [S8: set_Pr4231240917283394591_state] :
                    ( ( packag3171892606229212990_state @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi @ F @ S8 )
                    & ! [X4: produc7272280185086996969_state] :
                        ( ( member7461423308196221312_state @ X4 @ S8 )
                       => ( produc1745419677418529872tate_o
                          @ ^ [A13: state] :
                              ( produc3645877548210629341tate_o
                              @ ^ [U8: state,T4: state > state] :
                                ? [A4: state,U6: state] :
                                  ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A4 @ ( produc8632743443909064678_state @ U6 @ T4 ) ) @ S2 )
                                  & ( sep_greater_state @ Plus @ A13 @ ( F2 @ A4 @ U6 @ T4 ) )
                                  & ( ( Core @ A13 )
                                    = ( Core @ A4 ) ) ) )
                          @ X4 ) ) ) ) ) ) ) ) ) ).

% package_logic.completeness_aux
thf(fact_935_case__prod__Pair,axiom,
    ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val )
    = id_Pro2088550933252328842on_val ) ).

% case_prod_Pair
thf(fact_936_case__prod__Pair,axiom,
    ( ( produc803252138055838771_state @ produc9160152616361873709_state )
    = id_Pro8715877757508063718_state ) ).

% case_prod_Pair
thf(fact_937_case__prod__Pair,axiom,
    ( ( produc1922103714420728243_state @ produc6205719116230455971_state )
    = id_Pro5916511905602125660_state ) ).

% case_prod_Pair
thf(fact_938_case__prod__Pair,axiom,
    ( ( produc897481912382009139_state @ produc9080284787853255769_state )
    = id_Pro8860473219607035410_state ) ).

% case_prod_Pair
thf(fact_939_case__prod__Pair,axiom,
    ( ( produc4192255817509754207on_val @ produc74620467203698198on_val )
    = id_Pro1402184692240358763on_val ) ).

% case_prod_Pair
thf(fact_940_mem__case__prodI,axiom,
    ! [Z2: state,C: prat > state > set_state,A: prat,B: state] :
      ( ( member_state @ Z2 @ ( C @ A @ B ) )
     => ( member_state @ Z2 @ ( produc2825319096956678952_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_941_mem__case__prodI,axiom,
    ! [Z2: state,C: prat > set_state > set_state,A: prat,B: set_state] :
      ( ( member_state @ Z2 @ ( C @ A @ B ) )
     => ( member_state @ Z2 @ ( produc5034060725774254814_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_942_mem__case__prodI,axiom,
    ! [Z2: state,C: option_state > option_state > set_state,A: option_state,B: option_state] :
      ( ( member_state @ Z2 @ ( C @ A @ B ) )
     => ( member_state @ Z2 @ ( produc3424155258940669874_state @ C @ ( produc9160152616361873709_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_943_mem__case__prodI,axiom,
    ! [Z2: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state,A: prat,B: state] :
      ( ( member6715010751053357271_state @ Z2 @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc4369308447073708118_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_944_mem__case__prodI,axiom,
    ! [Z2: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state,A: prat,B: set_state] :
      ( ( member6715010751053357271_state @ Z2 @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc9043967962556017184_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_945_mem__case__prodI,axiom,
    ! [Z2: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state,A: option_state,B: option_state] :
      ( ( member6715010751053357271_state @ Z2 @ ( C @ A @ B ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc2209043727768083020_state @ C @ ( produc9160152616361873709_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_946_mem__case__prodI,axiom,
    ! [Z2: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state,A: prat,B: state] :
      ( ( member7461423308196221312_state @ Z2 @ ( C @ A @ B ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc569869127943504087_state @ C @ ( produc6205719116230455971_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_947_mem__case__prodI,axiom,
    ! [Z2: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state,A: prat,B: set_state] :
      ( ( member7461423308196221312_state @ Z2 @ ( C @ A @ B ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc7793558081207154829_state @ C @ ( produc9080284787853255769_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_948_mem__case__prodI,axiom,
    ! [Z2: produc7272280185086996969_state,C: option_state > option_state > set_Pr4231240917283394591_state,A: option_state,B: option_state] :
      ( ( member7461423308196221312_state @ Z2 @ ( C @ A @ B ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc6752009811597927777_state @ C @ ( produc9160152616361873709_state @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_949_mem__case__prodI,axiom,
    ! [Z2: state,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( member_state @ Z2 @ ( C @ A @ B ) )
     => ( member_state @ Z2 @ ( produc7498597735380428723_state @ C @ ( produc74620467203698198on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_950_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z2: state,C: prat > state > set_state] :
      ( ! [A3: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A3 @ B7 ) )
         => ( member_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member_state @ Z2 @ ( produc2825319096956678952_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_951_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z2: state,C: prat > set_state > set_state] :
      ( ! [A3: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A3 @ B7 ) )
         => ( member_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member_state @ Z2 @ ( produc5034060725774254814_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_952_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z2: state,C: option_state > option_state > set_state] :
      ( ! [A3: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A3 @ B7 ) )
         => ( member_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member_state @ Z2 @ ( produc3424155258940669874_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_953_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z2: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state] :
      ( ! [A3: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A3 @ B7 ) )
         => ( member6715010751053357271_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc4369308447073708118_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_954_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z2: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state] :
      ( ! [A3: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A3 @ B7 ) )
         => ( member6715010751053357271_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc9043967962556017184_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_955_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z2: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state] :
      ( ! [A3: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A3 @ B7 ) )
         => ( member6715010751053357271_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member6715010751053357271_state @ Z2 @ ( produc2209043727768083020_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_956_mem__case__prodI2,axiom,
    ! [P2: produc4844402304972151915_state,Z2: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state] :
      ( ! [A3: prat,B7: state] :
          ( ( P2
            = ( produc6205719116230455971_state @ A3 @ B7 ) )
         => ( member7461423308196221312_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc569869127943504087_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_957_mem__case__prodI2,axiom,
    ! [P2: produc6251377953716854561_state,Z2: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state] :
      ( ! [A3: prat,B7: set_state] :
          ( ( P2
            = ( produc9080284787853255769_state @ A3 @ B7 ) )
         => ( member7461423308196221312_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc7793558081207154829_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_958_mem__case__prodI2,axiom,
    ! [P2: produc3142500478612311029_state,Z2: produc7272280185086996969_state,C: option_state > option_state > set_Pr4231240917283394591_state] :
      ( ! [A3: option_state,B7: option_state] :
          ( ( P2
            = ( produc9160152616361873709_state @ A3 @ B7 ) )
         => ( member7461423308196221312_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member7461423308196221312_state @ Z2 @ ( produc6752009811597927777_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_959_mem__case__prodI2,axiom,
    ! [P2: produc9003781588162623004on_val,Z2: state,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state] :
      ( ! [A3: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > option_val] :
          ( ( P2
            = ( produc74620467203698198on_val @ A3 @ B7 ) )
         => ( member_state @ Z2 @ ( C @ A3 @ B7 ) ) )
     => ( member_state @ Z2 @ ( produc7498597735380428723_state @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_960_mem__case__prodE,axiom,
    ! [Z2: state,C: prat > state > set_state,P2: produc4844402304972151915_state] :
      ( ( member_state @ Z2 @ ( produc2825319096956678952_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X @ Y3 ) )
           => ~ ( member_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_961_mem__case__prodE,axiom,
    ! [Z2: state,C: prat > set_state > set_state,P2: produc6251377953716854561_state] :
      ( ( member_state @ Z2 @ ( produc5034060725774254814_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X @ Y3 ) )
           => ~ ( member_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_962_mem__case__prodE,axiom,
    ! [Z2: state,C: option_state > option_state > set_state,P2: produc3142500478612311029_state] :
      ( ( member_state @ Z2 @ ( produc3424155258940669874_state @ C @ P2 ) )
     => ~ ! [X: option_state,Y3: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X @ Y3 ) )
           => ~ ( member_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_963_mem__case__prodE,axiom,
    ! [Z2: produc3882902684156697774_state,C: prat > state > set_Pr2391701600042252046_state,P2: produc4844402304972151915_state] :
      ( ( member6715010751053357271_state @ Z2 @ ( produc4369308447073708118_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X @ Y3 ) )
           => ~ ( member6715010751053357271_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_964_mem__case__prodE,axiom,
    ! [Z2: produc3882902684156697774_state,C: prat > set_state > set_Pr2391701600042252046_state,P2: produc6251377953716854561_state] :
      ( ( member6715010751053357271_state @ Z2 @ ( produc9043967962556017184_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X @ Y3 ) )
           => ~ ( member6715010751053357271_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_965_mem__case__prodE,axiom,
    ! [Z2: produc3882902684156697774_state,C: option_state > option_state > set_Pr2391701600042252046_state,P2: produc3142500478612311029_state] :
      ( ( member6715010751053357271_state @ Z2 @ ( produc2209043727768083020_state @ C @ P2 ) )
     => ~ ! [X: option_state,Y3: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X @ Y3 ) )
           => ~ ( member6715010751053357271_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_966_mem__case__prodE,axiom,
    ! [Z2: produc7272280185086996969_state,C: prat > state > set_Pr4231240917283394591_state,P2: produc4844402304972151915_state] :
      ( ( member7461423308196221312_state @ Z2 @ ( produc569869127943504087_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: state] :
            ( ( P2
              = ( produc6205719116230455971_state @ X @ Y3 ) )
           => ~ ( member7461423308196221312_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_967_mem__case__prodE,axiom,
    ! [Z2: produc7272280185086996969_state,C: prat > set_state > set_Pr4231240917283394591_state,P2: produc6251377953716854561_state] :
      ( ( member7461423308196221312_state @ Z2 @ ( produc7793558081207154829_state @ C @ P2 ) )
     => ~ ! [X: prat,Y3: set_state] :
            ( ( P2
              = ( produc9080284787853255769_state @ X @ Y3 ) )
           => ~ ( member7461423308196221312_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_968_mem__case__prodE,axiom,
    ! [Z2: produc7272280185086996969_state,C: option_state > option_state > set_Pr4231240917283394591_state,P2: produc3142500478612311029_state] :
      ( ( member7461423308196221312_state @ Z2 @ ( produc6752009811597927777_state @ C @ P2 ) )
     => ~ ! [X: option_state,Y3: option_state] :
            ( ( P2
              = ( produc9160152616361873709_state @ X @ Y3 ) )
           => ~ ( member7461423308196221312_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_969_mem__case__prodE,axiom,
    ! [Z2: state,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_state,P2: produc9003781588162623004on_val] :
      ( ( member_state @ Z2 @ ( produc7498597735380428723_state @ C @ P2 ) )
     => ~ ! [X: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P2
              = ( produc74620467203698198on_val @ X @ Y3 ) )
           => ~ ( member_state @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_970_same__fst__def,axiom,
    ( same_f3145481655561538747_state
    = ( ^ [P6: state > $o,R6: state > set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [X8: state,Y8: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [X5: state,Y4: state > state] :
                      ( ( X8 = X5 )
                      & ( P6 @ X5 )
                      & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ Y8 @ Y4 ) @ ( R6 @ X5 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_971_same__fst__def,axiom,
    ( same_f8765775015391913222_state
    = ( ^ [P6: state > $o,R6: state > set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [X8: state,Y8: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [X5: state,Y4: produc3882902684156697774_state] :
                      ( ( X8 = X5 )
                      & ( P6 @ X5 )
                      & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ Y8 @ Y4 ) @ ( R6 @ X5 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_972_PartialSA_Ovalid__package__set__def,axiom,
    ! [S2: set_Pr4231240917283394591_state,F: state] :
      ( ( packag3877571396724326311_state @ plus @ core @ unit @ S2 @ F )
      = ( ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A4: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,T4: state > state] :
                      ( ( defined @ A4 @ U6 )
                      & ( greater @ ( core @ A4 ) @ ( core @ U6 ) )
                      & ( packag4817583915586052229_state @ plus @ unit @ T4 )
                      & ( greater @ A4 @ ( core @ ( T4 @ F ) ) ) ) )
              @ X5 ) ) ) ) ).

% PartialSA.valid_package_set_def
thf(fact_973_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr523392378426063460_state] :
      ( ( packag9013619817833129428_state @ plus @ S2 )
      = ( ! [X5: produc4357056439811976324_state] :
            ( ( member5752234768399382829_state @ X5 @ S2 )
           => ( produc8801812494870056711tate_o
              @ ^ [A4: state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [U6: state,P3: produc3882902684156697774_state] : ( sep_pure_state @ plus @ A4 ) )
              @ X5 ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_974_PartialSA_Opure__remains__def,axiom,
    ! [S2: set_Pr4231240917283394591_state] :
      ( ( packag4152378406607043437_state @ plus @ S2 )
      = ( ! [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ S2 )
           => ( produc1745419677418529872tate_o
              @ ^ [A4: state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [U6: state,P3: state > state] : ( sep_pure_state @ plus @ A4 ) )
              @ X5 ) ) ) ) ).

% PartialSA.pure_remains_def
thf(fact_975_package__logic_Opackage__rhs__connection__def,axiom,
    ! [Plus: val > val > option_val,Core: val > val,Unit: val,Stable: val > $o,Phi: val,F: val,S2: set_Pr4723104092121316667al_val,Pc: val > $o,A2: aassertion_val,Phi2: val,F3: val,S3: set_Pr4723104092121316667al_val] :
      ( ( package_logic_val @ Plus @ Core @ Unit @ Stable )
     => ( ( packag1944724486178758143on_val @ Plus @ Core @ Stable @ Phi @ F @ S2 @ Pc @ A2 @ Phi2 @ F3 @ S3 )
        = ( ( sep_greater_val @ Plus @ F3 @ F )
          & ( sep_defined_val @ Plus @ Phi @ F )
          & ( ( Plus @ Phi @ F )
            = ( Plus @ Phi2 @ F3 ) )
          & ( Stable @ F3 )
          & ! [X5: produc2072440392815185925al_val] :
              ( ( member5640714103519202460al_val @ X5 @ S2 )
             => ( produc6995717860488419308_val_o
                @ ^ [A4: val] :
                    ( produc7666680587469814684_val_o
                    @ ^ [U6: val,T4: val > val] :
                      ? [Au3: val] :
                        ( ( ( some_val @ Au3 )
                          = ( Plus @ A4 @ U6 ) )
                        & ( ( sep_defined_val @ Plus @ Au3 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                         => ? [A13: val,U8: val] :
                              ( ( member5640714103519202460al_val @ ( produc7187740677493149823al_val @ A13 @ ( produc303288909735261391al_val @ U8 @ T4 ) ) @ S3 )
                              & ( sep_greater_val @ Plus @ ( Core @ A13 ) @ ( Core @ A4 ) )
                              & ( ( Plus @ Au3 @ ( sep_minus_val @ Plus @ Core @ ( T4 @ F3 ) @ ( T4 @ F ) ) )
                                = ( Plus @ A13 @ U8 ) )
                              & ( sep_greater_val @ Plus @ U8 @ U6 )
                              & ( packag5057539306968662900at_val @ Plus @ Core @ Pc @ A2 @ A13 @ U8 @ U6 ) ) ) ) )
                @ X5 ) ) ) ) ) ).

% package_logic.package_rhs_connection_def
thf(fact_976_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,A2: 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 @ A2 @ Phi2 @ F3 @ S3 )
        = ( ( sep_greater_state @ Plus @ F3 @ F )
          & ( sep_defined_state @ Plus @ Phi @ F )
          & ( ( Plus @ Phi @ F )
            = ( Plus @ Phi2 @ F3 ) )
          & ( Stable @ F3 )
          & ! [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ S2 )
             => ( produc1745419677418529872tate_o
                @ ^ [A4: state] :
                    ( produc3645877548210629341tate_o
                    @ ^ [U6: state,T4: state > state] :
                      ? [Au3: state] :
                        ( ( ( some_state @ Au3 )
                          = ( Plus @ A4 @ 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 @ A4 ) )
                              & ( ( 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 @ A2 @ A13 @ U8 @ U6 ) ) ) ) )
                @ X5 ) ) ) ) ) ).

% package_logic.package_rhs_connection_def
thf(fact_977_Ball__Collect,axiom,
    ( ball_P7403102246598991807_state
    = ( ^ [A7: set_Pr4231240917283394591_state,P6: produc7272280185086996969_state > $o] : ( ord_le4032880777312351359_state @ A7 @ ( collec906845016635587262_state @ P6 ) ) ) ) ).

% Ball_Collect
thf(fact_978_Ball__Collect,axiom,
    ( ball_P2871712132005019634on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,P6: produc9003781588162623004on_val > $o] : ( ord_le6900622908513426738on_val @ A7 @ ( collec7475294502966882545on_val @ P6 ) ) ) ) ).

% Ball_Collect
thf(fact_979_Ball__Collect,axiom,
    ( ball_state
    = ( ^ [A7: set_state,P6: state > $o] : ( ord_le2494988322063910608_state @ A7 @ ( collect_state @ P6 ) ) ) ) ).

% Ball_Collect
thf(fact_980_split__cong,axiom,
    ! [Q2: produc3882902684156697774_state,F: state > ( state > state ) > $o,G: state > ( state > state ) > $o,P2: produc3882902684156697774_state] :
      ( ! [X: state,Y3: state > state] :
          ( ( ( produc8632743443909064678_state @ X @ Y3 )
            = Q2 )
         => ( ( F @ X @ Y3 )
            = ( G @ X @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc3645877548210629341tate_o @ F @ P2 )
          = ( produc3645877548210629341tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_981_split__cong,axiom,
    ! [Q2: produc7272280185086996969_state,F: state > produc3882902684156697774_state > $o,G: state > produc3882902684156697774_state > $o,P2: produc7272280185086996969_state] :
      ( ! [X: state,Y3: produc3882902684156697774_state] :
          ( ( ( produc5886875603596089627_state @ X @ Y3 )
            = Q2 )
         => ( ( F @ X @ Y3 )
            = ( G @ X @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc1745419677418529872tate_o @ F @ P2 )
          = ( produc1745419677418529872tate_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_982_cwand__def,axiom,
    ( cwand
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A4: state,X5: state] :
                  ( ( ( member_state @ A4 @ A7 )
                    & ( ( some_state @ X5 )
                      = ( plus @ ( r @ A4 @ W2 ) @ A4 ) ) )
                 => ( member_state @ X5 @ B4 ) ) ) ) ) ) ).

% cwand_def
thf(fact_983_wand__def,axiom,
    ( wand
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [W2: state] :
              ( ( Uu3 = W2 )
              & ! [A4: state,X5: state] :
                  ( ( ( member_state @ A4 @ A7 )
                    & ( ( some_state @ X5 )
                      = ( plus @ W2 @ A4 ) ) )
                 => ( member_state @ X5 @ B4 ) ) ) ) ) ) ).

% wand_def
thf(fact_984_lex__prod__def,axiom,
    ( lex_pr8992908938815505897on_val
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr4305388979797279737on_val] :
          ( collec4536212101028033490on_val
          @ ( produc3489267648137303980_val_o
            @ ( produc7816547785737266186_val_o
              @ ^ [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
                  ( produc1845727980445322802_val_o
                  @ ^ [A13: produc9003781588162623004on_val,B11: produc9003781588162623004on_val] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_985_lex__prod__def,axiom,
    ( lex_pr849482514101377111_state
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr1688445902015331925_state] :
          ( collec3985216178052149586_state
          @ ( produc6773883055741934892tate_o
            @ ( produc6001348999437208714tate_o
              @ ^ [A4: produc9003781588162623004on_val,B3: option_state] :
                  ( produc3589443509783526980tate_o
                  @ ^ [A13: produc9003781588162623004on_val,B11: option_state] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_986_lex__prod__def,axiom,
    ( lex_pr7503510086898530263on_val
    = ( ^ [Ra: set_Pr1688445902015331925_state,Rb: set_Pr4305388979797279737on_val] :
          ( collec5286658484364120402on_val
          @ ( produc6745459475987627308_val_o
            @ ( produc5489745010067198858_val_o
              @ ^ [A4: option_state,B3: produc9003781588162623004on_val] :
                  ( produc8282195775912345284_val_o
                  @ ^ [A13: option_state,B11: produc9003781588162623004on_val] :
                      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_987_lex__prod__def,axiom,
    ( lex_pr8336311929016760645_state
    = ( ^ [Ra: set_Pr1688445902015331925_state,Rb: set_Pr1688445902015331925_state] :
          ( collec6784228404298526930_state
          @ ( produc1935756327278076076tate_o
            @ ( produc5315882979514849802tate_o
              @ ^ [A4: option_state,B3: option_state] :
                  ( produc6082990154846732246tate_o
                  @ ^ [A13: option_state,B11: option_state] :
                      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_988_lex__prod__def,axiom,
    ( lex_pr4041356198452583422_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr5527899070135025255_state] :
          ( collec2083760461196787346_state
          @ ( produc7167296152490707052tate_o
            @ ( produc8588859638187112394tate_o
              @ ^ [A4: state,B3: state > state] :
                  ( produc3645877548210629341tate_o
                  @ ^ [A13: state,B11: state > state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member6978531283771894448_state @ ( produc5362444129537099455_state @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_989_lex__prod__def,axiom,
    ( lex_pr7268379037978345219_state
    = ( ^ [Ra: set_Pr1785066336555260981_state,Rb: set_Pr4846723933414538759_state] :
          ( collec4625684881440051358_state
          @ ( produc3133855996611560728tate_o
            @ ( produc6466221716036377566tate_o
              @ ^ [A4: state,B3: produc3882902684156697774_state] :
                  ( produc1745419677418529872tate_o
                  @ ^ [A13: state,B11: produc3882902684156697774_state] :
                      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A4 @ A13 ) @ Ra )
                      | ( ( A4 = A13 )
                        & ( member3801449062345016400_state @ ( produc2343482328048119831_state @ B3 @ B11 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_990_case__prod__Pair__iden,axiom,
    ! [P2: produc1317171644805857433on_val] :
      ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_991_case__prod__Pair__iden,axiom,
    ! [P2: produc3142500478612311029_state] :
      ( ( produc803252138055838771_state @ produc9160152616361873709_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_992_case__prod__Pair__iden,axiom,
    ! [P2: produc4844402304972151915_state] :
      ( ( produc1922103714420728243_state @ produc6205719116230455971_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_993_case__prod__Pair__iden,axiom,
    ! [P2: produc6251377953716854561_state] :
      ( ( produc897481912382009139_state @ produc9080284787853255769_state @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_994_case__prod__Pair__iden,axiom,
    ! [P2: produc9003781588162623004on_val] :
      ( ( produc4192255817509754207on_val @ produc74620467203698198on_val @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_995_type__definition__state,axiom,
    ( type_d7385716227342696708on_val @ rep_state @ abs_state
    @ ( collec7475294502966882545on_val
      @ ^ [Uu3: produc9003781588162623004on_val] :
        ? [Phi3: produc9003781588162623004on_val] :
          ( ( Uu3 = Phi3 )
          & ( valid_state @ Phi3 ) ) ) ) ).

% type_definition_state
thf(fact_996_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: option_state,A6: produc9003781588162623004on_val,B6: option_state,R4: set_Pr4305388979797279737on_val,S: set_Pr1688445902015331925_state] :
      ( ( member7415082542495011600_state @ ( produc1288379658824033111_state @ ( produc9082076722414132287_state @ A @ B ) @ ( produc9082076722414132287_state @ A6 @ B6 ) ) @ ( lex_pr849482514101377111_state @ R4 @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_997_in__lex__prod,axiom,
    ! [A: option_state,B: produc9003781588162623004on_val,A6: option_state,B6: produc9003781588162623004on_val,R4: set_Pr1688445902015331925_state,S: set_Pr4305388979797279737on_val] :
      ( ( member8716524848806982416on_val @ ( produc3994388588539396951on_val @ ( produc6512732258356509631on_val @ A @ B ) @ ( produc6512732258356509631on_val @ A6 @ B6 ) ) @ ( lex_pr7503510086898530263on_val @ R4 @ S ) )
      = ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_998_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A6: produc9003781588162623004on_val,B6: produc9003781588162623004on_val,R4: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4958730763592889809on_val @ A6 @ B6 ) ) @ ( lex_pr8992908938815505897on_val @ R4 @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_999_in__lex__prod,axiom,
    ! [A: option_state,B: option_state,A6: option_state,B6: option_state,R4: set_Pr1688445902015331925_state,S: set_Pr1688445902015331925_state] :
      ( ( member2361837818702807184_state @ ( produc2397313312270560215_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc9160152616361873709_state @ A6 @ B6 ) ) @ ( lex_pr8336311929016760645_state @ R4 @ S ) )
      = ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1000_in__lex__prod,axiom,
    ! [A: prat,B: state,A6: prat,B6: state,R4: set_Pr3298805672376727137t_prat,S: set_Pr1785066336555260981_state] :
      ( ( member14790466581074960_state @ ( produc8801277691740291671_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc6205719116230455971_state @ A6 @ B6 ) ) @ ( lex_prod_prat_state @ R4 @ S ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member753036827967488894_state @ ( produc544653723139640077_state @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1001_in__lex__prod,axiom,
    ! [A: prat,B: set_state,A6: prat,B6: set_state,R4: set_Pr3298805672376727137t_prat,S: set_Pr5618112136499017889_state] :
      ( ( member647386464819295632_state @ ( produc516693425924439767_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc9080284787853255769_state @ A6 @ B6 ) ) @ ( lex_pr1787351029354341489_state @ R4 @ S ) )
      = ( ( member5208769068039259050t_prat @ ( produc7183142570155660857t_prat @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member1976142906092959210_state @ ( produc1834047914073684089_state @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1002_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A6: produc1457211279475724562t_char > prat,B6: produc1457211279475724562t_char > option_val,R4: set_Pr1685704261205283783r_prat,S: set_Pr4698884584343881589on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc74620467203698198on_val @ A6 @ B6 ) ) @ ( lex_pr8780192377291449342on_val @ R4 @ S ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A6 ) @ R4 )
        | ( ( A = A6 )
          & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ B6 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_1003_sep__algebra_Omono__prop__set,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state,P: state > $o] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
       => ( ( sep_setify_state @ P @ B2 )
         => ( ( sep_mono_prop_state @ Plus @ P )
           => ( sep_setify_state @ P @ A2 ) ) ) ) ) ).

% sep_algebra.mono_prop_set
thf(fact_1004_multiply__sem__assertion_Osimps,axiom,
    ( multip8064567061438756306ertion
    = ( ^ [P3: prat,P6: set_state] : ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ P3 ) @ P6 ) ) ) ) ).

% multiply_sem_assertion.simps
thf(fact_1005_multiply__sem__assertion_Oelims,axiom,
    ! [X3: prat,Xa2: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X3 @ Xa2 )
        = Y )
     => ( Y
        = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X3 ) @ Xa2 ) ) ) ) ).

% multiply_sem_assertion.elims
thf(fact_1006_image__eqI,axiom,
    ! [B: state,F: state > state,X3: state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member_state @ X3 @ A2 )
       => ( member_state @ B @ ( image_state_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1007_image__eqI,axiom,
    ! [B: produc3882902684156697774_state,F: state > produc3882902684156697774_state,X3: state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member_state @ X3 @ A2 )
       => ( member6715010751053357271_state @ B @ ( image_5311902671526399277_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1008_image__eqI,axiom,
    ! [B: state,F: produc3882902684156697774_state > state,X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member6715010751053357271_state @ X3 @ A2 )
       => ( member_state @ B @ ( image_6759111139246981411_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1009_image__eqI,axiom,
    ! [B: produc7272280185086996969_state,F: state > produc7272280185086996969_state,X3: state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member_state @ X3 @ A2 )
       => ( member7461423308196221312_state @ B @ ( image_632992669292831658_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1010_image__eqI,axiom,
    ! [B: state,F: produc7272280185086996969_state > state,X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member7461423308196221312_state @ X3 @ A2 )
       => ( member_state @ B @ ( image_2710133271811360426_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1011_image__eqI,axiom,
    ! [B: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member6715010751053357271_state @ X3 @ A2 )
       => ( member6715010751053357271_state @ B @ ( image_508898238547643525_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1012_image__eqI,axiom,
    ! [B: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member7461423308196221312_state @ X3 @ A2 )
       => ( member6715010751053357271_state @ B @ ( image_7466395622266389182_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1013_image__eqI,axiom,
    ! [B: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member6715010751053357271_state @ X3 @ A2 )
       => ( member7461423308196221312_state @ B @ ( image_4364510454419742290_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1014_image__eqI,axiom,
    ! [B: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member7461423308196221312_state @ X3 @ A2 )
       => ( member7461423308196221312_state @ B @ ( image_3064086290059306969_state @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1015_image__eqI,axiom,
    ! [B: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val,X3: state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( member_state @ X3 @ A2 )
       => ( member2879030624002546611on_val @ B @ ( image_4772210114854293981on_val @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_1016_image__ident,axiom,
    ! [Y9: set_state] :
      ( ( image_state_state
        @ ^ [X5: state] : X5
        @ Y9 )
      = Y9 ) ).

% image_ident
thf(fact_1017_sep__algebra_Osucc__set__trans,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state,C2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
       => ( ( sep_gr7105985528888466643_state @ Plus @ B2 @ C2 )
         => ( sep_gr7105985528888466643_state @ Plus @ A2 @ C2 ) ) ) ) ).

% sep_algebra.succ_set_trans
thf(fact_1018_sep__algebra_Olarger__set__refl,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( sep_gr7105985528888466643_state @ Plus @ A2 @ A2 ) ) ).

% sep_algebra.larger_set_refl
thf(fact_1019_image__Un,axiom,
    ! [F: state > state,A2: set_state,B2: set_state] :
      ( ( image_state_state @ F @ ( sup_sup_set_state @ A2 @ B2 ) )
      = ( sup_sup_set_state @ ( image_state_state @ F @ A2 ) @ ( image_state_state @ F @ B2 ) ) ) ).

% image_Un
thf(fact_1020_image__Un,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( image_3064086290059306969_state @ F @ ( sup_su3837536599209510219_state @ A2 @ B2 ) )
      = ( sup_su3837536599209510219_state @ ( image_3064086290059306969_state @ F @ A2 ) @ ( image_3064086290059306969_state @ F @ B2 ) ) ) ).

% image_Un
thf(fact_1021_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_3064086290059306969_state @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1022_setcompr__eq__image,axiom,
    ! [F: state > produc7272280185086996969_state,P: state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_632992669292831658_state @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1023_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc7272280185086996969_state,P: produc9003781588162623004on_val > $o] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_3204238854701872908_state @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1024_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > state,P: produc7272280185086996969_state > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_2710133271811360426_state @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1025_setcompr__eq__image,axiom,
    ! [F: state > state,P: state > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_state_state @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1026_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > state,P: produc9003781588162623004on_val > $o] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_1205529093420515805_state @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1027_setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc9003781588162623004on_val,P: produc7272280185086996969_state > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X5: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_3393013468800240652on_val @ F @ ( collec906845016635587262_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1028_setcompr__eq__image,axiom,
    ! [F: state > produc9003781588162623004on_val,P: state > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_4772210114854293981on_val @ F @ ( collect_state @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1029_setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,P: produc9003781588162623004on_val > $o] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X5: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( P @ X5 ) ) )
      = ( image_4170889944715819327on_val @ F @ ( collec7475294502966882545on_val @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1030_Setcompr__eq__image,axiom,
    ! [F: state > state,A2: set_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member_state @ X5 @ A2 ) ) )
      = ( image_state_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1031_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > state,A2: set_Pr2391701600042252046_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member6715010751053357271_state @ X5 @ A2 ) ) )
      = ( image_6759111139246981411_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1032_Setcompr__eq__image,axiom,
    ! [F: state > produc7272280185086996969_state,A2: set_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member_state @ X5 @ A2 ) ) )
      = ( image_632992669292831658_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1033_Setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > state,A2: set_Pr4231240917283394591_state] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member7461423308196221312_state @ X5 @ A2 ) ) )
      = ( image_2710133271811360426_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1034_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > produc7272280185086996969_state,A2: set_Pr2391701600042252046_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member6715010751053357271_state @ X5 @ A2 ) ) )
      = ( image_4364510454419742290_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1035_Setcompr__eq__image,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: produc7272280185086996969_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member7461423308196221312_state @ X5 @ A2 ) ) )
      = ( image_3064086290059306969_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1036_Setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > state,A2: set_Pr7649655576376376530on_val] :
      ( ( collect_state
        @ ^ [Uu3: state] :
          ? [X5: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member2879030624002546611on_val @ X5 @ A2 ) ) )
      = ( image_1205529093420515805_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1037_Setcompr__eq__image,axiom,
    ! [F: state > produc9003781588162623004on_val,A2: set_state] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X5: state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member_state @ X5 @ A2 ) ) )
      = ( image_4772210114854293981on_val @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1038_Setcompr__eq__image,axiom,
    ! [F: produc3882902684156697774_state > produc9003781588162623004on_val,A2: set_Pr2391701600042252046_state] :
      ( ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [X5: produc3882902684156697774_state] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member6715010751053357271_state @ X5 @ A2 ) ) )
      = ( image_4690842878458256517on_val @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1039_Setcompr__eq__image,axiom,
    ! [F: produc9003781588162623004on_val > produc7272280185086996969_state,A2: set_Pr7649655576376376530on_val] :
      ( ( collec906845016635587262_state
        @ ^ [Uu3: produc7272280185086996969_state] :
          ? [X5: produc9003781588162623004on_val] :
            ( ( Uu3
              = ( F @ X5 ) )
            & ( member2879030624002546611on_val @ X5 @ A2 ) ) )
      = ( image_3204238854701872908_state @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_1040_Compr__image__eq,axiom,
    ! [F: state > state,A2: set_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ ( image_state_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_state_state @ F
        @ ( collect_state
          @ ^ [X5: state] :
              ( ( member_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1041_Compr__image__eq,axiom,
    ! [F: state > produc3882902684156697774_state,A2: set_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X5 @ ( image_5311902671526399277_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_5311902671526399277_state @ F
        @ ( collect_state
          @ ^ [X5: state] :
              ( ( member_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1042_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > state,A2: set_Pr2391701600042252046_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ ( image_6759111139246981411_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_6759111139246981411_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X5: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1043_Compr__image__eq,axiom,
    ! [F: state > produc7272280185086996969_state,A2: set_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ ( image_632992669292831658_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_632992669292831658_state @ F
        @ ( collect_state
          @ ^ [X5: state] :
              ( ( member_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1044_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > state,A2: set_Pr4231240917283394591_state,P: state > $o] :
      ( ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ ( image_2710133271811360426_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_2710133271811360426_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1045_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X5 @ ( image_508898238547643525_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_508898238547643525_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X5: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1046_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > produc3882902684156697774_state,A2: set_Pr4231240917283394591_state,P: produc3882902684156697774_state > $o] :
      ( ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X5 @ ( image_7466395622266389182_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_7466395622266389182_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1047_Compr__image__eq,axiom,
    ! [F: produc3882902684156697774_state > produc7272280185086996969_state,A2: set_Pr2391701600042252046_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ ( image_4364510454419742290_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_4364510454419742290_state @ F
        @ ( collec394671971739345689_state
          @ ^ [X5: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1048_Compr__image__eq,axiom,
    ! [F: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ ( image_3064086290059306969_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_3064086290059306969_state @ F
        @ ( collec906845016635587262_state
          @ ^ [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1049_Compr__image__eq,axiom,
    ! [F: produc9003781588162623004on_val > state,A2: set_Pr7649655576376376530on_val,P: state > $o] :
      ( ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ ( image_1205529093420515805_state @ F @ A2 ) )
            & ( P @ X5 ) ) )
      = ( image_1205529093420515805_state @ F
        @ ( collec7475294502966882545on_val
          @ ^ [X5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X5 @ A2 )
              & ( P @ ( F @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1050_image__image,axiom,
    ! [F: state > state,G: state > state,A2: set_state] :
      ( ( image_state_state @ F @ ( image_state_state @ G @ A2 ) )
      = ( image_state_state
        @ ^ [X5: state] : ( F @ ( G @ X5 ) )
        @ A2 ) ) ).

% image_image
thf(fact_1051_imageE,axiom,
    ! [B: state,F: state > state,A2: set_state] :
      ( ( member_state @ B @ ( image_state_state @ F @ A2 ) )
     => ~ ! [X: state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1052_imageE,axiom,
    ! [B: state,F: produc3882902684156697774_state > state,A2: set_Pr2391701600042252046_state] :
      ( ( member_state @ B @ ( image_6759111139246981411_state @ F @ A2 ) )
     => ~ ! [X: produc3882902684156697774_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member6715010751053357271_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1053_imageE,axiom,
    ! [B: produc3882902684156697774_state,F: state > produc3882902684156697774_state,A2: set_state] :
      ( ( member6715010751053357271_state @ B @ ( image_5311902671526399277_state @ F @ A2 ) )
     => ~ ! [X: state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1054_imageE,axiom,
    ! [B: state,F: produc7272280185086996969_state > state,A2: set_Pr4231240917283394591_state] :
      ( ( member_state @ B @ ( image_2710133271811360426_state @ F @ A2 ) )
     => ~ ! [X: produc7272280185086996969_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member7461423308196221312_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1055_imageE,axiom,
    ! [B: produc7272280185086996969_state,F: state > produc7272280185086996969_state,A2: set_state] :
      ( ( member7461423308196221312_state @ B @ ( image_632992669292831658_state @ F @ A2 ) )
     => ~ ! [X: state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1056_imageE,axiom,
    ! [B: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ B @ ( image_508898238547643525_state @ F @ A2 ) )
     => ~ ! [X: produc3882902684156697774_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member6715010751053357271_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1057_imageE,axiom,
    ! [B: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state,A2: set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ B @ ( image_4364510454419742290_state @ F @ A2 ) )
     => ~ ! [X: produc3882902684156697774_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member6715010751053357271_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1058_imageE,axiom,
    ! [B: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state,A2: set_Pr4231240917283394591_state] :
      ( ( member6715010751053357271_state @ B @ ( image_7466395622266389182_state @ F @ A2 ) )
     => ~ ! [X: produc7272280185086996969_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member7461423308196221312_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1059_imageE,axiom,
    ! [B: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state] :
      ( ( member7461423308196221312_state @ B @ ( image_3064086290059306969_state @ F @ A2 ) )
     => ~ ! [X: produc7272280185086996969_state] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member7461423308196221312_state @ X @ A2 ) ) ) ).

% imageE
thf(fact_1060_imageE,axiom,
    ! [B: state,F: produc9003781588162623004on_val > state,A2: set_Pr7649655576376376530on_val] :
      ( ( member_state @ B @ ( image_1205529093420515805_state @ F @ A2 ) )
     => ~ ! [X: produc9003781588162623004on_val] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member2879030624002546611on_val @ X @ A2 ) ) ) ).

% imageE
thf(fact_1061_image__mono,axiom,
    ! [A2: set_state,B2: set_state,F: state > state] :
      ( ( ord_le2494988322063910608_state @ A2 @ B2 )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A2 ) @ ( image_state_state @ F @ B2 ) ) ) ).

% image_mono
thf(fact_1062_image__subsetI,axiom,
    ! [A2: set_state,F: state > state,B2: set_state] :
      ( ! [X: state] :
          ( ( member_state @ X @ A2 )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1063_image__subsetI,axiom,
    ! [A2: set_state,F: state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: state] :
          ( ( member_state @ X @ A2 )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1064_image__subsetI,axiom,
    ! [A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state,B2: set_state] :
      ( ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A2 )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_6759111139246981411_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1065_image__subsetI,axiom,
    ! [A2: set_state,F: state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: state] :
          ( ( member_state @ X @ A2 )
         => ( member7461423308196221312_state @ ( F @ X ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1066_image__subsetI,axiom,
    ! [A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state,B2: set_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A2 )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1067_image__subsetI,axiom,
    ! [A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A2 )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_508898238547643525_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1068_image__subsetI,axiom,
    ! [A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A2 )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1069_image__subsetI,axiom,
    ! [A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: produc3882902684156697774_state] :
          ( ( member6715010751053357271_state @ X @ A2 )
         => ( member7461423308196221312_state @ ( F @ X ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_4364510454419742290_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1070_image__subsetI,axiom,
    ! [A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( member7461423308196221312_state @ X @ A2 )
         => ( member7461423308196221312_state @ ( F @ X ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1071_image__subsetI,axiom,
    ! [A2: set_state,F: state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X: state] :
          ( ( member_state @ X @ A2 )
         => ( member2879030624002546611on_val @ ( F @ X ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ A2 ) @ B2 ) ) ).

% image_subsetI
thf(fact_1072_subset__imageE,axiom,
    ! [B2: set_state,F: state > state,A2: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A2 ) )
     => ~ ! [C5: set_state] :
            ( ( ord_le2494988322063910608_state @ C5 @ A2 )
           => ( B2
             != ( image_state_state @ F @ C5 ) ) ) ) ).

% subset_imageE
thf(fact_1073_image__subset__iff,axiom,
    ! [F: state > state,A2: set_state,B2: set_state] :
      ( ( ord_le2494988322063910608_state @ ( image_state_state @ F @ A2 ) @ B2 )
      = ( ! [X5: state] :
            ( ( member_state @ X5 @ A2 )
           => ( member_state @ ( F @ X5 ) @ B2 ) ) ) ) ).

% image_subset_iff
thf(fact_1074_subset__image__iff,axiom,
    ! [B2: set_state,F: state > state,A2: set_state] :
      ( ( ord_le2494988322063910608_state @ B2 @ ( image_state_state @ F @ A2 ) )
      = ( ? [AA: set_state] :
            ( ( ord_le2494988322063910608_state @ AA @ A2 )
            & ( B2
              = ( image_state_state @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1075_rev__image__eqI,axiom,
    ! [X3: state,A2: set_state,B: state,F: state > state] :
      ( ( member_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member_state @ B @ ( image_state_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1076_rev__image__eqI,axiom,
    ! [X3: state,A2: set_state,B: produc3882902684156697774_state,F: state > produc3882902684156697774_state] :
      ( ( member_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member6715010751053357271_state @ B @ ( image_5311902671526399277_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1077_rev__image__eqI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: state,F: produc3882902684156697774_state > state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member_state @ B @ ( image_6759111139246981411_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1078_rev__image__eqI,axiom,
    ! [X3: state,A2: set_state,B: produc7272280185086996969_state,F: state > produc7272280185086996969_state] :
      ( ( member_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member7461423308196221312_state @ B @ ( image_632992669292831658_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1079_rev__image__eqI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B: state,F: produc7272280185086996969_state > state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member_state @ B @ ( image_2710133271811360426_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1080_rev__image__eqI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: produc3882902684156697774_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member6715010751053357271_state @ B @ ( image_508898238547643525_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1081_rev__image__eqI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B: produc3882902684156697774_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member6715010751053357271_state @ B @ ( image_7466395622266389182_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1082_rev__image__eqI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: produc7272280185086996969_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member7461423308196221312_state @ B @ ( image_4364510454419742290_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1083_rev__image__eqI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B: produc7272280185086996969_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member7461423308196221312_state @ B @ ( image_3064086290059306969_state @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1084_rev__image__eqI,axiom,
    ! [X3: state,A2: set_state,B: produc9003781588162623004on_val,F: state > produc9003781588162623004on_val] :
      ( ( member_state @ X3 @ A2 )
     => ( ( B
          = ( F @ X3 ) )
       => ( member2879030624002546611on_val @ B @ ( image_4772210114854293981on_val @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_1085_ball__imageD,axiom,
    ! [F: state > state,A2: set_state,P: state > $o] :
      ( ! [X: state] :
          ( ( member_state @ X @ ( image_state_state @ F @ A2 ) )
         => ( P @ X ) )
     => ! [X4: state] :
          ( ( member_state @ X4 @ A2 )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_1086_image__cong,axiom,
    ! [M6: set_state,N2: set_state,F: state > state,G: state > state] :
      ( ( M6 = N2 )
     => ( ! [X: state] :
            ( ( member_state @ X @ N2 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_state_state @ F @ M6 )
          = ( image_state_state @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_1087_bex__imageD,axiom,
    ! [F: state > state,A2: set_state,P: state > $o] :
      ( ? [X4: state] :
          ( ( member_state @ X4 @ ( image_state_state @ F @ A2 ) )
          & ( P @ X4 ) )
     => ? [X: state] :
          ( ( member_state @ X @ A2 )
          & ( P @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_1088_image__iff,axiom,
    ! [Z2: state,F: state > state,A2: set_state] :
      ( ( member_state @ Z2 @ ( image_state_state @ F @ A2 ) )
      = ( ? [X5: state] :
            ( ( member_state @ X5 @ A2 )
            & ( Z2
              = ( F @ X5 ) ) ) ) ) ).

% image_iff
thf(fact_1089_imageI,axiom,
    ! [X3: state,A2: set_state,F: state > state] :
      ( ( member_state @ X3 @ A2 )
     => ( member_state @ ( F @ X3 ) @ ( image_state_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1090_imageI,axiom,
    ! [X3: state,A2: set_state,F: state > produc3882902684156697774_state] :
      ( ( member_state @ X3 @ A2 )
     => ( member6715010751053357271_state @ ( F @ X3 ) @ ( image_5311902671526399277_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1091_imageI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( member_state @ ( F @ X3 ) @ ( image_6759111139246981411_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1092_imageI,axiom,
    ! [X3: state,A2: set_state,F: state > produc7272280185086996969_state] :
      ( ( member_state @ X3 @ A2 )
     => ( member7461423308196221312_state @ ( F @ X3 ) @ ( image_632992669292831658_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1093_imageI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( member_state @ ( F @ X3 ) @ ( image_2710133271811360426_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1094_imageI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( member6715010751053357271_state @ ( F @ X3 ) @ ( image_508898238547643525_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1095_imageI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( member6715010751053357271_state @ ( F @ X3 ) @ ( image_7466395622266389182_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1096_imageI,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,F: produc3882902684156697774_state > produc7272280185086996969_state] :
      ( ( member6715010751053357271_state @ X3 @ A2 )
     => ( member7461423308196221312_state @ ( F @ X3 ) @ ( image_4364510454419742290_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1097_imageI,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,F: produc7272280185086996969_state > produc7272280185086996969_state] :
      ( ( member7461423308196221312_state @ X3 @ A2 )
     => ( member7461423308196221312_state @ ( F @ X3 ) @ ( image_3064086290059306969_state @ F @ A2 ) ) ) ).

% imageI
thf(fact_1098_imageI,axiom,
    ! [X3: state,A2: set_state,F: state > produc9003781588162623004on_val] :
      ( ( member_state @ X3 @ A2 )
     => ( member2879030624002546611on_val @ ( F @ X3 ) @ ( image_4772210114854293981on_val @ F @ A2 ) ) ) ).

% imageI
thf(fact_1099_sep__algebra_Ogreater__set_Ocong,axiom,
    sep_gr7105985528888466643_state = sep_gr7105985528888466643_state ).

% sep_algebra.greater_set.cong
thf(fact_1100_sep__algebra_Osub__bigger,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( ord_le2494988322063910608_state @ A2 @ B2 )
       => ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 ) ) ) ).

% sep_algebra.sub_bigger
thf(fact_1101_sep__algebra_Ogreater__set__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
        = ( ! [X5: state] :
              ( ( member_state @ X5 @ A2 )
             => ? [Y4: state] :
                  ( ( member_state @ Y4 @ B2 )
                  & ( sep_greater_state @ Plus @ X5 @ Y4 ) ) ) ) ) ) ).

% sep_algebra.greater_set_def
thf(fact_1102_sep__algebra_Ogreater__setI,axiom,
    ! [Plus: produc7272280185086996969_state > produc7272280185086996969_state > option2747342629145793465_state,Core: produc7272280185086996969_state > produc7272280185086996969_state,A2: set_Pr4231240917283394591_state,B2: set_Pr4231240917283394591_state] :
      ( ( sep_al4323785819428540618_state @ Plus @ Core )
     => ( ! [A3: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ A3 @ A2 )
           => ? [X4: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X4 @ B2 )
                & ( sep_gr1233117237951779860_state @ Plus @ A3 @ X4 ) ) )
       => ( sep_gr2254073943460739586_state @ Plus @ A2 @ B2 ) ) ) ).

% sep_algebra.greater_setI
thf(fact_1103_sep__algebra_Ogreater__setI,axiom,
    ! [Plus: produc9003781588162623004on_val > produc9003781588162623004on_val > option8887860577033306220on_val,Core: produc9003781588162623004on_val > produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B2: set_Pr7649655576376376530on_val] :
      ( ( sep_al5697130883215687421on_val @ Plus @ Core )
     => ( ! [A3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ A3 @ A2 )
           => ? [X4: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X4 @ B2 )
                & ( sep_gr8124549141218502215on_val @ Plus @ A3 @ X4 ) ) )
       => ( sep_gr6201018273107925045on_val @ Plus @ A2 @ B2 ) ) ) ).

% sep_algebra.greater_setI
thf(fact_1104_sep__algebra_Ogreater__setI,axiom,
    ! [Plus: produc3882902684156697774_state > produc3882902684156697774_state > option886743748074664436_state,Core: produc3882902684156697774_state > produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: set_Pr2391701600042252046_state] :
      ( ( sep_al8268612531242297357_state @ Plus @ Core )
     => ( ! [A3: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ A3 @ A2 )
           => ? [X4: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X4 @ B2 )
                & ( sep_gr6114829331755084611_state @ Plus @ A3 @ X4 ) ) )
       => ( sep_gr851398640268573909_state @ Plus @ A2 @ B2 ) ) ) ).

% sep_algebra.greater_setI
thf(fact_1105_sep__algebra_Ogreater__setI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ! [A3: state] :
            ( ( member_state @ A3 @ A2 )
           => ? [X4: state] :
                ( ( member_state @ X4 @ B2 )
                & ( sep_greater_state @ Plus @ A3 @ X4 ) ) )
       => ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 ) ) ) ).

% sep_algebra.greater_setI
thf(fact_1106_greater__set__def,axiom,
    ( greater_set
    = ( sep_gr7105985528888466643_state @ plus ) ) ).

% greater_set_def
thf(fact_1107_sep__algebra_Obigger__set,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A12: set_state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A12 @ A2 )
       => ( sep_gr7105985528888466643_state @ Plus @ ( sep_add_set_state @ Plus @ A12 @ B2 ) @ ( sep_add_set_state @ Plus @ A2 @ B2 ) ) ) ) ).

% sep_algebra.bigger_set
thf(fact_1108_sep__algebra_OequivI,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
       => ( ( sep_gr7105985528888466643_state @ Plus @ B2 @ A2 )
         => ( sep_equiv_state @ Plus @ A2 @ B2 ) ) ) ) ).

% sep_algebra.equivI
thf(fact_1109_sep__algebra_Oequiv__def,axiom,
    ! [Plus: state > state > option_state,Core: state > state,A2: set_state,B2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_equiv_state @ Plus @ A2 @ B2 )
        = ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
          & ( sep_gr7105985528888466643_state @ Plus @ B2 @ A2 ) ) ) ) ).

% sep_algebra.equiv_def
thf(fact_1110_sep__algebra_Oup__closed__bigger__subset,axiom,
    ! [Plus: state > state > option_state,Core: state > state,B2: set_state,A2: set_state] :
      ( ( sep_algebra_state @ Plus @ Core )
     => ( ( sep_up_closed_state @ Plus @ B2 )
       => ( ( sep_gr7105985528888466643_state @ Plus @ A2 @ B2 )
         => ( ord_le2494988322063910608_state @ A2 @ B2 ) ) ) ) ).

% sep_algebra.up_closed_bigger_subset
thf(fact_1111_multiply__sem__assertion_Opelims,axiom,
    ! [X3: prat,Xa2: set_state,Y: set_state] :
      ( ( ( multip8064567061438756306ertion @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X3 @ Xa2 ) )
       => ~ ( ( Y
              = ( sep_up1246176804924251236_state @ plus @ ( image_state_state @ ( multiply @ X3 ) @ Xa2 ) ) )
           => ~ ( accp_P6554740477754186410_state @ multip8073008153270241703on_rel @ ( produc9080284787853255769_state @ X3 @ Xa2 ) ) ) ) ) ).

% multiply_sem_assertion.pelims
thf(fact_1112_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > state,B2: set_state] :
      ( ! [X: state] :
          ( ( P @ X )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_state_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1113_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: state] :
          ( ( P @ X )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_5311902671526399277_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1114_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: state] :
          ( ( P @ X )
         => ( member7461423308196221312_state @ ( F @ X ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_632992669292831658_state @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1115_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > state,B2: set_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( P @ X )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_2710133271811360426_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1116_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( P @ X )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_7466395622266389182_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1117_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc7272280185086996969_state,B2: set_Pr4231240917283394591_state] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( P @ X )
         => ( member7461423308196221312_state @ ( F @ X ) @ B2 ) )
     => ( ord_le4032880777312351359_state @ ( image_3064086290059306969_state @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1118_image__Collect__subsetI,axiom,
    ! [P: state > $o,F: state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X: state] :
          ( ( P @ X )
         => ( member2879030624002546611on_val @ ( F @ X ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_4772210114854293981on_val @ F @ ( collect_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1119_image__Collect__subsetI,axiom,
    ! [P: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > state,B2: set_state] :
      ( ! [X: produc9003781588162623004on_val] :
          ( ( P @ X )
         => ( member_state @ ( F @ X ) @ B2 ) )
     => ( ord_le2494988322063910608_state @ ( image_1205529093420515805_state @ F @ ( collec7475294502966882545on_val @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1120_image__Collect__subsetI,axiom,
    ! [P: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > produc3882902684156697774_state,B2: set_Pr2391701600042252046_state] :
      ( ! [X: produc9003781588162623004on_val] :
          ( ( P @ X )
         => ( member6715010751053357271_state @ ( F @ X ) @ B2 ) )
     => ( ord_le6483787752022044334_state @ ( image_4494208899119068747_state @ F @ ( collec7475294502966882545on_val @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1121_image__Collect__subsetI,axiom,
    ! [P: produc7272280185086996969_state > $o,F: produc7272280185086996969_state > produc9003781588162623004on_val,B2: set_Pr7649655576376376530on_val] :
      ( ! [X: produc7272280185086996969_state] :
          ( ( P @ X )
         => ( member2879030624002546611on_val @ ( F @ X ) @ B2 ) )
     => ( ord_le6900622908513426738on_val @ ( image_3393013468800240652on_val @ F @ ( collec906845016635587262_state @ P ) ) @ B2 ) ) ).

% image_Collect_subsetI
thf(fact_1122_all__subset__image,axiom,
    ! [F: state > state,A2: set_state,P: set_state > $o] :
      ( ( ! [B4: set_state] :
            ( ( ord_le2494988322063910608_state @ B4 @ ( image_state_state @ F @ A2 ) )
           => ( P @ B4 ) ) )
      = ( ! [B4: set_state] :
            ( ( ord_le2494988322063910608_state @ B4 @ A2 )
           => ( P @ ( image_state_state @ F @ B4 ) ) ) ) ) ).

% all_subset_image
thf(fact_1123_pair__imageI,axiom,
    ! [A: prat,B: state,A2: set_Pr6039890582625207499_state,F: prat > state > state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ A2 )
     => ( member_state @ ( F @ A @ B ) @ ( image_6883929567913436896_state @ ( produc5191658985856023922_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1124_pair__imageI,axiom,
    ! [A: prat,B: set_state,A2: set_Pr6847263425858009217_state,F: prat > set_state > state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ A2 )
     => ( member_state @ ( F @ A @ B ) @ ( image_1091725855630477206_state @ ( produc2626907587099862312_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1125_pair__imageI,axiom,
    ! [A: state,B: state > state,A2: set_Pr2391701600042252046_state,F: state > ( state > state ) > state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ A2 )
     => ( member_state @ ( F @ A @ B ) @ ( image_6759111139246981411_state @ ( produc8575939086779714485_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1126_pair__imageI,axiom,
    ! [A: option_state,B: option_state,A2: set_Pr1688445902015331925_state,F: option_state > option_state > state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ A2 )
     => ( member_state @ ( F @ A @ B ) @ ( image_393491784373993066_state @ ( produc7211634496914305532_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1127_pair__imageI,axiom,
    ! [A: state,B: state > state,A2: set_Pr2391701600042252046_state,F: state > ( state > state ) > $o] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_528283217102427119tate_o @ ( produc3645877548210629341tate_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1128_pair__imageI,axiom,
    ! [A: state,B: produc3882902684156697774_state,A2: set_Pr4231240917283394591_state,F: state > produc3882902684156697774_state > state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ A2 )
     => ( member_state @ ( F @ A @ B ) @ ( image_2710133271811360426_state @ ( produc8258182183023037954_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1129_pair__imageI,axiom,
    ! [A: prat,B: state,A2: set_Pr6039890582625207499_state,F: prat > state > produc3882902684156697774_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ A2 )
     => ( member6715010751053357271_state @ ( F @ A @ B ) @ ( image_1979208616201753608_state @ ( produc6727336724318121462_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1130_pair__imageI,axiom,
    ! [A: state,B: produc3882902684156697774_state,A2: set_Pr4231240917283394591_state,F: state > produc3882902684156697774_state > $o] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_6262870063963634088tate_o @ ( produc1745419677418529872tate_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1131_pair__imageI,axiom,
    ! [A: prat,B: set_state,A2: set_Pr6847263425858009217_state,F: prat > set_state > produc3882902684156697774_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ A2 )
     => ( member6715010751053357271_state @ ( F @ A @ B ) @ ( image_7417905060788143314_state @ ( produc8936804542704588736_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1132_pair__imageI,axiom,
    ! [A: state,B: state > state,A2: set_Pr2391701600042252046_state,F: state > ( state > state ) > produc3882902684156697774_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ A2 )
     => ( member6715010751053357271_state @ ( F @ A @ B ) @ ( image_508898238547643525_state @ ( produc5527672907597820531_state @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_1133_multiply__sem__assertion_Ocases,axiom,
    ! [X3: produc6251377953716854561_state] :
      ~ ! [P4: prat,P7: set_state] :
          ( X3
         != ( produc9080284787853255769_state @ P4 @ P7 ) ) ).

% multiply_sem_assertion.cases
thf(fact_1134_None__notin__image__Some,axiom,
    ! [A2: set_state] :
      ~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A2 ) ) ).

% None_notin_image_Some
thf(fact_1135_None__notin__image__Some,axiom,
    ! [A2: set_val] :
      ~ ( member_option_val @ none_val @ ( image_val_option_val @ some_val @ A2 ) ) ).

% None_notin_image_Some
thf(fact_1136_ssubst__Pair__rhs,axiom,
    ! [R4: state,S: produc3882902684156697774_state,R: set_Pr4231240917283394591_state,S9: produc3882902684156697774_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1137_ssubst__Pair__rhs,axiom,
    ! [R4: state,S: state > state,R: set_Pr2391701600042252046_state,S9: state > state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1138_ssubst__Pair__rhs,axiom,
    ! [R4: produc9003781588162623004on_val,S: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S9: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1139_ssubst__Pair__rhs,axiom,
    ! [R4: option_state,S: option_state,R: set_Pr1688445902015331925_state,S9: option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1140_ssubst__Pair__rhs,axiom,
    ! [R4: prat,S: state,R: set_Pr6039890582625207499_state,S9: state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1141_ssubst__Pair__rhs,axiom,
    ! [R4: prat,S: set_state,R: set_Pr6847263425858009217_state,S9: set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1142_ssubst__Pair__rhs,axiom,
    ! [R4: produc1457211279475724562t_char > prat,S: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S9: produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R4 @ S ) @ R )
     => ( ( S9 = S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R4 @ S9 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1143_prop__restrict,axiom,
    ! [X3: produc3882902684156697774_state,Z6: set_Pr2391701600042252046_state,X9: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ( member6715010751053357271_state @ X3 @ Z6 )
     => ( ( ord_le6483787752022044334_state @ Z6
          @ ( collec394671971739345689_state
            @ ^ [X5: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X5 @ X9 )
                & ( P @ X5 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1144_prop__restrict,axiom,
    ! [X3: produc7272280185086996969_state,Z6: set_Pr4231240917283394591_state,X9: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( member7461423308196221312_state @ X3 @ Z6 )
     => ( ( ord_le4032880777312351359_state @ Z6
          @ ( collec906845016635587262_state
            @ ^ [X5: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X5 @ X9 )
                & ( P @ X5 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1145_prop__restrict,axiom,
    ! [X3: produc9003781588162623004on_val,Z6: set_Pr7649655576376376530on_val,X9: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ X3 @ Z6 )
     => ( ( ord_le6900622908513426738on_val @ Z6
          @ ( collec7475294502966882545on_val
            @ ^ [X5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X5 @ X9 )
                & ( P @ X5 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1146_prop__restrict,axiom,
    ! [X3: state,Z6: set_state,X9: set_state,P: state > $o] :
      ( ( member_state @ X3 @ Z6 )
     => ( ( ord_le2494988322063910608_state @ Z6
          @ ( collect_state
            @ ^ [X5: state] :
                ( ( member_state @ X5 @ X9 )
                & ( P @ X5 ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_1147_Collect__restrict,axiom,
    ! [X9: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ord_le6483787752022044334_state
      @ ( collec394671971739345689_state
        @ ^ [X5: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X5 @ X9 )
            & ( P @ X5 ) ) )
      @ X9 ) ).

% Collect_restrict
thf(fact_1148_Collect__restrict,axiom,
    ! [X9: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ord_le4032880777312351359_state
      @ ( collec906845016635587262_state
        @ ^ [X5: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X5 @ X9 )
            & ( P @ X5 ) ) )
      @ X9 ) ).

% Collect_restrict
thf(fact_1149_Collect__restrict,axiom,
    ! [X9: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X5: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X5 @ X9 )
            & ( P @ X5 ) ) )
      @ X9 ) ).

% Collect_restrict
thf(fact_1150_Collect__restrict,axiom,
    ! [X9: set_state,P: state > $o] :
      ( ord_le2494988322063910608_state
      @ ( collect_state
        @ ^ [X5: state] :
            ( ( member_state @ X5 @ X9 )
            & ( P @ X5 ) ) )
      @ X9 ) ).

% Collect_restrict
thf(fact_1151_conj__subset__def,axiom,
    ! [A2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o,Q: produc7272280185086996969_state > $o] :
      ( ( ord_le4032880777312351359_state @ A2
        @ ( collec906845016635587262_state
          @ ^ [X5: produc7272280185086996969_state] :
              ( ( P @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le4032880777312351359_state @ A2 @ ( collec906845016635587262_state @ P ) )
        & ( ord_le4032880777312351359_state @ A2 @ ( collec906845016635587262_state @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1152_conj__subset__def,axiom,
    ! [A2: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ A2
        @ ( collec7475294502966882545on_val
          @ ^ [X5: produc9003781588162623004on_val] :
              ( ( P @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le6900622908513426738on_val @ A2 @ ( collec7475294502966882545on_val @ P ) )
        & ( ord_le6900622908513426738on_val @ A2 @ ( collec7475294502966882545on_val @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1153_conj__subset__def,axiom,
    ! [A2: set_state,P: state > $o,Q: state > $o] :
      ( ( ord_le2494988322063910608_state @ A2
        @ ( collect_state
          @ ^ [X5: state] :
              ( ( P @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le2494988322063910608_state @ A2 @ ( collect_state @ P ) )
        & ( ord_le2494988322063910608_state @ A2 @ ( collect_state @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1154_subset__CollectI,axiom,
    ! [B2: set_Pr2391701600042252046_state,A2: set_Pr2391701600042252046_state,Q: produc3882902684156697774_state > $o,P: produc3882902684156697774_state > $o] :
      ( ( ord_le6483787752022044334_state @ B2 @ A2 )
     => ( ! [X: produc3882902684156697774_state] :
            ( ( member6715010751053357271_state @ X @ B2 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le6483787752022044334_state
          @ ( collec394671971739345689_state
            @ ^ [X5: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X5 @ B2 )
                & ( Q @ X5 ) ) )
          @ ( collec394671971739345689_state
            @ ^ [X5: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1155_subset__CollectI,axiom,
    ! [B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state,Q: produc7272280185086996969_state > $o,P: produc7272280185086996969_state > $o] :
      ( ( ord_le4032880777312351359_state @ B2 @ A2 )
     => ( ! [X: produc7272280185086996969_state] :
            ( ( member7461423308196221312_state @ X @ B2 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le4032880777312351359_state
          @ ( collec906845016635587262_state
            @ ^ [X5: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X5 @ B2 )
                & ( Q @ X5 ) ) )
          @ ( collec906845016635587262_state
            @ ^ [X5: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1156_subset__CollectI,axiom,
    ! [B2: set_Pr7649655576376376530on_val,A2: set_Pr7649655576376376530on_val,Q: produc9003781588162623004on_val > $o,P: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ B2 @ A2 )
     => ( ! [X: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X @ B2 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le6900622908513426738on_val
          @ ( collec7475294502966882545on_val
            @ ^ [X5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X5 @ B2 )
                & ( Q @ X5 ) ) )
          @ ( collec7475294502966882545on_val
            @ ^ [X5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X5 @ A2 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1157_subset__CollectI,axiom,
    ! [B2: set_state,A2: set_state,Q: state > $o,P: state > $o] :
      ( ( ord_le2494988322063910608_state @ B2 @ A2 )
     => ( ! [X: state] :
            ( ( member_state @ X @ B2 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le2494988322063910608_state
          @ ( collect_state
            @ ^ [X5: state] :
                ( ( member_state @ X5 @ B2 )
                & ( Q @ X5 ) ) )
          @ ( collect_state
            @ ^ [X5: state] :
                ( ( member_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1158_subset__Collect__iff,axiom,
    ! [B2: set_Pr2391701600042252046_state,A2: set_Pr2391701600042252046_state,P: produc3882902684156697774_state > $o] :
      ( ( ord_le6483787752022044334_state @ B2 @ A2 )
     => ( ( ord_le6483787752022044334_state @ B2
          @ ( collec394671971739345689_state
            @ ^ [X5: produc3882902684156697774_state] :
                ( ( member6715010751053357271_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: produc3882902684156697774_state] :
              ( ( member6715010751053357271_state @ X5 @ B2 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1159_subset__Collect__iff,axiom,
    ! [B2: set_Pr4231240917283394591_state,A2: set_Pr4231240917283394591_state,P: produc7272280185086996969_state > $o] :
      ( ( ord_le4032880777312351359_state @ B2 @ A2 )
     => ( ( ord_le4032880777312351359_state @ B2
          @ ( collec906845016635587262_state
            @ ^ [X5: produc7272280185086996969_state] :
                ( ( member7461423308196221312_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: produc7272280185086996969_state] :
              ( ( member7461423308196221312_state @ X5 @ B2 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1160_subset__Collect__iff,axiom,
    ! [B2: set_Pr7649655576376376530on_val,A2: set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ B2 @ A2 )
     => ( ( ord_le6900622908513426738on_val @ B2
          @ ( collec7475294502966882545on_val
            @ ^ [X5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X5 @ A2 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X5 @ B2 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1161_subset__Collect__iff,axiom,
    ! [B2: set_state,A2: set_state,P: state > $o] :
      ( ( ord_le2494988322063910608_state @ B2 @ A2 )
     => ( ( ord_le2494988322063910608_state @ B2
          @ ( collect_state
            @ ^ [X5: state] :
                ( ( member_state @ X5 @ A2 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: state] :
              ( ( member_state @ X5 @ B2 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1162_image2__def,axiom,
    ( bNF_Gr1645861816300964896_state
    = ( ^ [A7: set_state,F5: state > prat,G2: state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A4: state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1163_image2__def,axiom,
    ( bNF_Gr8363538844033681878_state
    = ( ^ [A7: set_state,F5: state > prat,G2: state > set_state] :
          ( collec5094083866753559820_state
          @ ^ [Uu3: produc6251377953716854561_state] :
            ? [A4: state] :
              ( ( Uu3
                = ( produc9080284787853255769_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1164_image2__def,axiom,
    ( bNF_Gr8144397171961131434_state
    = ( ^ [A7: set_state,F5: state > option_state,G2: state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A4: state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1165_image2__def,axiom,
    ( bNF_Gr947463461346416350_state
    = ( ^ [A7: set_state,F5: state > state,G2: state > produc3882902684156697774_state] :
          ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A4: state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1166_image2__def,axiom,
    ( bNF_Gr5432878666621363808_state
    = ( ^ [A7: set_Pr2391701600042252046_state,F5: produc3882902684156697774_state > prat,G2: produc3882902684156697774_state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A4: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member6715010751053357271_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1167_image2__def,axiom,
    ( bNF_Gr6818451474134737942_state
    = ( ^ [A7: set_Pr2391701600042252046_state,F5: produc3882902684156697774_state > prat,G2: produc3882902684156697774_state > set_state] :
          ( collec5094083866753559820_state
          @ ^ [Uu3: produc6251377953716854561_state] :
            ? [A4: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc9080284787853255769_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member6715010751053357271_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1168_image2__def,axiom,
    ( bNF_Gr4913868530397957098_state
    = ( ^ [A7: set_Pr2391701600042252046_state,F5: produc3882902684156697774_state > option_state,G2: produc3882902684156697774_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A4: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member6715010751053357271_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1169_image2__def,axiom,
    ( bNF_Gr2250442185837678927_state
    = ( ^ [A7: set_Pr4231240917283394591_state,F5: produc7272280185086996969_state > prat,G2: produc7272280185086996969_state > state] :
          ( collec9096564466166568534_state
          @ ^ [Uu3: produc4844402304972151915_state] :
            ? [A4: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc6205719116230455971_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member7461423308196221312_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1170_image2__def,axiom,
    ( bNF_Gr7237560867856044805_state
    = ( ^ [A7: set_Pr4231240917283394591_state,F5: produc7272280185086996969_state > prat,G2: produc7272280185086996969_state > set_state] :
          ( collec5094083866753559820_state
          @ ^ [Uu3: produc6251377953716854561_state] :
            ? [A4: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc9080284787853255769_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member7461423308196221312_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1171_image2__def,axiom,
    ( bNF_Gr18303635036277918_state
    = ( ^ [A7: set_Pr2391701600042252046_state,F5: produc3882902684156697774_state > state,G2: produc3882902684156697774_state > produc3882902684156697774_state] :
          ( collec906845016635587262_state
          @ ^ [Uu3: produc7272280185086996969_state] :
            ? [A4: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc5886875603596089627_state @ ( F5 @ A4 ) @ ( G2 @ A4 ) ) )
              & ( member6715010751053357271_state @ A4 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_1172_Some__image__these__eq,axiom,
    ! [A2: set_option_state] :
      ( ( image_6076465424260689483_state @ some_state @ ( these_state @ A2 ) )
      = ( collect_option_state
        @ ^ [X5: option_state] :
            ( ( member_option_state @ X5 @ A2 )
            & ( X5 != none_state ) ) ) ) ).

% Some_image_these_eq
thf(fact_1173_Some__image__these__eq,axiom,
    ! [A2: set_option_val] :
      ( ( image_val_option_val @ some_val @ ( these_val @ A2 ) )
      = ( collect_option_val
        @ ^ [X5: option_val] :
            ( ( member_option_val @ X5 @ A2 )
            & ( X5 != none_val ) ) ) ) ).

% Some_image_these_eq
thf(fact_1174_these__image__Some__eq,axiom,
    ! [A2: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_1175_these__image__Some__eq,axiom,
    ! [A2: set_val] :
      ( ( these_val @ ( image_val_option_val @ some_val @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_1176_in__these__eq,axiom,
    ! [X3: produc7272280185086996969_state,A2: set_op8023886964038656367_state] :
      ( ( member7461423308196221312_state @ X3 @ ( these_4347185934205712492_state @ A2 ) )
      = ( member3894984953261475024_state @ ( some_P6297086604321900740_state @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1177_in__these__eq,axiom,
    ! [X3: produc9003781588162623004on_val,A2: set_op6240181410663953058on_val] :
      ( ( member2879030624002546611on_val @ X3 @ ( these_3919539662624406175on_val @ A2 ) )
      = ( member4602090469187516291on_val @ ( some_P2581650809551911671on_val @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1178_in__these__eq,axiom,
    ! [X3: produc3882902684156697774_state,A2: set_op3387789762728285652_state] :
      ( ( member6715010751053357271_state @ X3 @ ( these_7452845017025203435_state @ A2 ) )
      = ( member5919342556599141533_state @ ( some_P5383513176685068947_state @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1179_in__these__eq,axiom,
    ! [X3: state,A2: set_option_state] :
      ( ( member_state @ X3 @ ( these_state @ A2 ) )
      = ( member_option_state @ ( some_state @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1180_in__these__eq,axiom,
    ! [X3: val,A2: set_option_val] :
      ( ( member_val @ X3 @ ( these_val @ A2 ) )
      = ( member_option_val @ ( some_val @ X3 ) @ A2 ) ) ).

% in_these_eq
thf(fact_1181_image2__eqI,axiom,
    ! [B: prat,F: state > prat,X3: state,C: state,G: state > state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_state @ X3 @ A2 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ B @ C ) @ ( bNF_Gr1645861816300964896_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1182_image2__eqI,axiom,
    ! [B: prat,F: state > prat,X3: state,C: set_state,G: state > set_state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_state @ X3 @ A2 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ B @ C ) @ ( bNF_Gr8363538844033681878_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1183_image2__eqI,axiom,
    ! [B: state,F: state > state,X3: state,C: state > state,G: state > state > state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_state @ X3 @ A2 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ B @ C ) @ ( bNF_Gr7364880804673765603_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1184_image2__eqI,axiom,
    ! [B: option_state,F: state > option_state,X3: state,C: option_state,G: state > option_state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_state @ X3 @ A2 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ C ) @ ( bNF_Gr8144397171961131434_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1185_image2__eqI,axiom,
    ! [B: state,F: state > state,X3: state,C: produc3882902684156697774_state,G: state > produc3882902684156697774_state,A2: set_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member_state @ X3 @ A2 )
         => ( member7461423308196221312_state @ ( produc5886875603596089627_state @ B @ C ) @ ( bNF_Gr947463461346416350_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1186_image2__eqI,axiom,
    ! [B: prat,F: produc3882902684156697774_state > prat,X3: produc3882902684156697774_state,C: state,G: produc3882902684156697774_state > state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member6715010751053357271_state @ X3 @ A2 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ B @ C ) @ ( bNF_Gr5432878666621363808_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1187_image2__eqI,axiom,
    ! [B: prat,F: produc3882902684156697774_state > prat,X3: produc3882902684156697774_state,C: set_state,G: produc3882902684156697774_state > set_state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member6715010751053357271_state @ X3 @ A2 )
         => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ B @ C ) @ ( bNF_Gr6818451474134737942_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1188_image2__eqI,axiom,
    ! [B: state,F: produc3882902684156697774_state > state,X3: produc3882902684156697774_state,C: state > state,G: produc3882902684156697774_state > state > state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member6715010751053357271_state @ X3 @ A2 )
         => ( member6715010751053357271_state @ ( produc8632743443909064678_state @ B @ C ) @ ( bNF_Gr5413576055295359779_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1189_image2__eqI,axiom,
    ! [B: option_state,F: produc3882902684156697774_state > option_state,X3: produc3882902684156697774_state,C: option_state,G: produc3882902684156697774_state > option_state,A2: set_Pr2391701600042252046_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member6715010751053357271_state @ X3 @ A2 )
         => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ B @ C ) @ ( bNF_Gr4913868530397957098_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1190_image2__eqI,axiom,
    ! [B: prat,F: produc7272280185086996969_state > prat,X3: produc7272280185086996969_state,C: state,G: produc7272280185086996969_state > state,A2: set_Pr4231240917283394591_state] :
      ( ( B
        = ( F @ X3 ) )
     => ( ( C
          = ( G @ X3 ) )
       => ( ( member7461423308196221312_state @ X3 @ A2 )
         => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ B @ C ) @ ( bNF_Gr2250442185837678927_state @ A2 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_1191_Option_Othese__def,axiom,
    ( these_state
    = ( ^ [A7: set_option_state] :
          ( image_3532137647693456075_state @ the_state
          @ ( collect_option_state
            @ ^ [X5: option_state] :
                ( ( member_option_state @ X5 @ A7 )
                & ( X5 != none_state ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1192_Option_Othese__def,axiom,
    ( these_val
    = ( ^ [A7: set_option_val] :
          ( image_option_val_val @ the_val
          @ ( collect_option_val
            @ ^ [X5: option_val] :
                ( ( member_option_val @ X5 @ A7 )
                & ( X5 != none_val ) ) ) ) ) ) ).

% Option.these_def
thf(fact_1193_relImage__def,axiom,
    ( bNF_Gr4118820077530253598on_val
    = ( ^ [R6: set_Pr4305388979797279737on_val,F5: produc9003781588162623004on_val > produc9003781588162623004on_val] :
          ( collec5166231387257825412on_val
          @ ^ [Uu3: produc1317171644805857433on_val] :
            ? [A14: produc9003781588162623004on_val,A23: produc9003781588162623004on_val] :
              ( ( Uu3
                = ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A14 @ A23 ) @ R6 ) ) ) ) ) ).

% relImage_def
thf(fact_1194_relImage__def,axiom,
    ( bNF_Gr4325946583917791628on_val
    = ( ^ [R6: set_Pr1688445902015331925_state,F5: option_state > produc9003781588162623004on_val] :
          ( collec5166231387257825412on_val
          @ ^ [Uu3: produc1317171644805857433on_val] :
            ? [A14: option_state,A23: option_state] :
              ( ( Uu3
                = ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A14 @ A23 ) @ R6 ) ) ) ) ) ).

% relImage_def
thf(fact_1195_relImage__def,axiom,
    ( bNF_Gr6895291047975414284_state
    = ( ^ [R6: set_Pr4305388979797279737on_val,F5: produc9003781588162623004on_val > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: produc9003781588162623004on_val,A23: produc9003781588162623004on_val] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A14 @ A23 ) @ R6 ) ) ) ) ) ).

% relImage_def
thf(fact_1196_relImage__def,axiom,
    ( bNF_Gr63754628658600314_state
    = ( ^ [R6: set_Pr1688445902015331925_state,F5: option_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: option_state,A23: option_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A14 @ A23 ) @ R6 ) ) ) ) ) ).

% relImage_def
thf(fact_1197_relInvImage__def,axiom,
    ( bNF_Gr4428767212045630361on_val
    = ( ^ [A7: set_state,R6: set_Pr4305388979797279737on_val,F5: state > produc9003781588162623004on_val] :
          ( collec7320380983431419584_state
          @ ^ [Uu3: produc8023240190789890773_state] :
            ? [A14: state,A23: state] :
              ( ( Uu3
                = ( produc544653723139640077_state @ A14 @ A23 ) )
              & ( member_state @ A14 @ A7 )
              & ( member_state @ A23 @ A7 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1198_relInvImage__def,axiom,
    ( bNF_Gr7039482972660942280on_val
    = ( ^ [A7: set_Pr4231240917283394591_state,R6: set_Pr4305388979797279737on_val,F5: produc7272280185086996969_state > produc9003781588162623004on_val] :
          ( collec4625684881440051358_state
          @ ^ [Uu3: produc1362221940583236787_state] :
            ? [A14: produc7272280185086996969_state,A23: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc2330690084503141483_state @ A14 @ A23 ) )
              & ( member7461423308196221312_state @ A14 @ A7 )
              & ( member7461423308196221312_state @ A23 @ A7 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1199_relInvImage__def,axiom,
    ( bNF_Gr1139690717726542281on_val
    = ( ^ [A7: set_Pr2391701600042252046_state,R6: set_Pr4305388979797279737on_val,F5: produc3882902684156697774_state > produc9003781588162623004on_val] :
          ( collec2083760461196787346_state
          @ ^ [Uu3: produc2030649568307348519_state] :
            ? [A14: produc3882902684156697774_state,A23: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc2343482328048119831_state @ A14 @ A23 ) )
              & ( member6715010751053357271_state @ A14 @ A7 )
              & ( member6715010751053357271_state @ A23 @ A7 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1200_relInvImage__def,axiom,
    ( bNF_Gr8759033791985395207_state
    = ( ^ [A7: set_state,R6: set_Pr1688445902015331925_state,F5: state > option_state] :
          ( collec7320380983431419584_state
          @ ^ [Uu3: produc8023240190789890773_state] :
            ? [A14: state,A23: state] :
              ( ( Uu3
                = ( produc544653723139640077_state @ A14 @ A23 ) )
              & ( member_state @ A14 @ A7 )
              & ( member_state @ A23 @ A7 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1201_relInvImage__def,axiom,
    ( bNF_Gr5630821380644201398_state
    = ( ^ [A7: set_Pr4231240917283394591_state,R6: set_Pr1688445902015331925_state,F5: produc7272280185086996969_state > option_state] :
          ( collec4625684881440051358_state
          @ ^ [Uu3: produc1362221940583236787_state] :
            ? [A14: produc7272280185086996969_state,A23: produc7272280185086996969_state] :
              ( ( Uu3
                = ( produc2330690084503141483_state @ A14 @ A23 ) )
              & ( member7461423308196221312_state @ A14 @ A7 )
              & ( member7461423308196221312_state @ A23 @ A7 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1202_relInvImage__def,axiom,
    ( bNF_Gr919661598790814775_state
    = ( ^ [A7: set_Pr2391701600042252046_state,R6: set_Pr1688445902015331925_state,F5: produc3882902684156697774_state > option_state] :
          ( collec2083760461196787346_state
          @ ^ [Uu3: produc2030649568307348519_state] :
            ? [A14: produc3882902684156697774_state,A23: produc3882902684156697774_state] :
              ( ( Uu3
                = ( produc2343482328048119831_state @ A14 @ A23 ) )
              & ( member6715010751053357271_state @ A14 @ A7 )
              & ( member6715010751053357271_state @ A23 @ A7 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1203_relInvImage__def,axiom,
    ( bNF_Gr6662968685559237371on_val
    = ( ^ [A7: set_Pr7649655576376376530on_val,R6: set_Pr4305388979797279737on_val,F5: produc9003781588162623004on_val > produc9003781588162623004on_val] :
          ( collec5166231387257825412on_val
          @ ^ [Uu3: produc1317171644805857433on_val] :
            ? [A14: produc9003781588162623004on_val,A23: produc9003781588162623004on_val] :
              ( ( Uu3
                = ( produc4958730763592889809on_val @ A14 @ A23 ) )
              & ( member2879030624002546611on_val @ A14 @ A7 )
              & ( member2879030624002546611on_val @ A23 @ A7 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1204_relInvImage__def,axiom,
    ( bNF_Gr4364826243083562601_state
    = ( ^ [A7: set_Pr7649655576376376530on_val,R6: set_Pr1688445902015331925_state,F5: produc9003781588162623004on_val > option_state] :
          ( collec5166231387257825412on_val
          @ ^ [Uu3: produc1317171644805857433on_val] :
            ? [A14: produc9003781588162623004on_val,A23: produc9003781588162623004on_val] :
              ( ( Uu3
                = ( produc4958730763592889809on_val @ A14 @ A23 ) )
              & ( member2879030624002546611on_val @ A14 @ A7 )
              & ( member2879030624002546611on_val @ A23 @ A7 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1205_relInvImage__def,axiom,
    ( bNF_Gr1795481779025939945on_val
    = ( ^ [A7: set_option_state,R6: set_Pr4305388979797279737on_val,F5: option_state > produc9003781588162623004on_val] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: option_state,A23: option_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ A14 @ A23 ) )
              & ( member_option_state @ A14 @ A7 )
              & ( member_option_state @ A23 @ A7 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1206_relInvImage__def,axiom,
    ( bNF_Gr621135371174781015_state
    = ( ^ [A7: set_option_state,R6: set_Pr1688445902015331925_state,F5: option_state > option_state] :
          ( collec8144523193623705312_state
          @ ^ [Uu3: produc3142500478612311029_state] :
            ? [A14: option_state,A23: option_state] :
              ( ( Uu3
                = ( produc9160152616361873709_state @ A14 @ A23 ) )
              & ( member_option_state @ A14 @ A7 )
              & ( member_option_state @ A23 @ A7 )
              & ( member3029510603097127326_state @ ( produc9160152616361873709_state @ ( F5 @ A14 ) @ ( F5 @ A23 ) ) @ R6 ) ) ) ) ) ).

% relInvImage_def
thf(fact_1207_apsnd__conv,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,X3: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( produc4354335345776199764on_val @ F @ ( produc4958730763592889809on_val @ X3 @ Y ) )
      = ( produc4958730763592889809on_val @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1208_apsnd__conv,axiom,
    ! [F: option_state > option_state,X3: option_state,Y: option_state] :
      ( ( produc860558495739588510_state @ F @ ( produc9160152616361873709_state @ X3 @ Y ) )
      = ( produc9160152616361873709_state @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1209_apsnd__conv,axiom,
    ! [F: state > state,X3: prat,Y: state] :
      ( ( produc7605673951949977028e_prat @ F @ ( produc6205719116230455971_state @ X3 @ Y ) )
      = ( produc6205719116230455971_state @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1210_apsnd__conv,axiom,
    ! [F: state > set_state,X3: prat,Y: state] :
      ( ( produc5637201501821557114e_prat @ F @ ( produc6205719116230455971_state @ X3 @ Y ) )
      = ( produc9080284787853255769_state @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1211_apsnd__conv,axiom,
    ! [F: set_state > state,X3: prat,Y: set_state] :
      ( ( produc4712232198508907386e_prat @ F @ ( produc9080284787853255769_state @ X3 @ Y ) )
      = ( produc6205719116230455971_state @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1212_apsnd__conv,axiom,
    ! [F: set_state > set_state,X3: prat,Y: set_state] :
      ( ( produc8191592070915954480e_prat @ F @ ( produc9080284787853255769_state @ X3 @ Y ) )
      = ( produc9080284787853255769_state @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1213_apsnd__conv,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X3: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( produc1735415053012026015r_prat @ F @ ( produc74620467203698198on_val @ X3 @ Y ) )
      = ( produc74620467203698198on_val @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1214_relImage__relInvImage,axiom,
    ! [R: set_Pr1785066336555260981_state,F: state > state,A2: set_state] :
      ( ( ord_le2777189432094499797_state @ R
        @ ( produc4572292869282958142_state @ ( image_state_state @ F @ A2 )
          @ ^ [Uu3: state] : ( image_state_state @ F @ A2 ) ) )
     => ( ( bNF_Gr6805951496242791002_state @ ( bNF_Gr4114592414835599415_state @ A2 @ R @ F ) @ F )
        = R ) ) ).

% relImage_relInvImage
thf(fact_1215_UNIV__I,axiom,
    ! [X3: produc7272280185086996969_state] : ( member7461423308196221312_state @ X3 @ top_to174962084833059535_state ) ).

% UNIV_I
thf(fact_1216_UNIV__I,axiom,
    ! [X3: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X3 @ top_to1561438921946393474on_val ) ).

% UNIV_I
thf(fact_1217_UNIV__I,axiom,
    ! [X3: produc3882902684156697774_state] : ( member6715010751053357271_state @ X3 @ top_to1744471583079920734_state ) ).

% UNIV_I
thf(fact_1218_UNIV__I,axiom,
    ! [X3: state] : ( member_state @ X3 @ top_top_set_state ) ).

% UNIV_I
thf(fact_1219_mem__Sigma__iff,axiom,
    ! [A: state,B: state,A2: set_state,B2: state > set_state] :
      ( ( member753036827967488894_state @ ( produc544653723139640077_state @ A @ B ) @ ( produc4572292869282958142_state @ A2 @ B2 ) )
      = ( ( member_state @ A @ A2 )
        & ( member_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1220_mem__Sigma__iff,axiom,
    ! [A: prat,B: state,A2: set_prat,B2: prat > set_state] :
      ( ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A2 @ B2 ) )
      = ( ( member_prat @ A @ A2 )
        & ( member_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1221_mem__Sigma__iff,axiom,
    ! [A: prat,B: set_state,A2: set_prat,B2: prat > set_set_state] :
      ( ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A2 @ B2 ) )
      = ( ( member_prat @ A @ A2 )
        & ( member_set_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1222_mem__Sigma__iff,axiom,
    ! [A: state,B: state > state,A2: set_state,B2: state > set_state_state] :
      ( ( member6715010751053357271_state @ ( produc8632743443909064678_state @ A @ B ) @ ( produc6646226935541082007_state @ A2 @ B2 ) )
      = ( ( member_state @ A @ A2 )
        & ( member_state_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1223_mem__Sigma__iff,axiom,
    ! [A: option_state,B: option_state,A2: set_option_state,B2: option_state > set_option_state] :
      ( ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A2 @ B2 ) )
      = ( ( member_option_state @ A @ A2 )
        & ( member_option_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1224_mem__Sigma__iff,axiom,
    ! [A: produc3882902684156697774_state,B: state,A2: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state > set_state] :
      ( ( member1546683425902591150_state @ ( produc7334084071316671761_state @ A @ B ) @ ( produc4278832915810242336_state @ A2 @ B2 ) )
      = ( ( member6715010751053357271_state @ A @ A2 )
        & ( member_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1225_mem__Sigma__iff,axiom,
    ! [A: state,B: produc3882902684156697774_state,A2: set_state,B2: state > set_Pr2391701600042252046_state] :
      ( ( member7461423308196221312_state @ ( produc5886875603596089627_state @ A @ B ) @ ( produc2831624448089660202_state @ A2 @ B2 ) )
      = ( ( member_state @ A @ A2 )
        & ( member6715010751053357271_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1226_mem__Sigma__iff,axiom,
    ! [A: state,B: produc7272280185086996969_state,A2: set_state,B2: state > set_Pr4231240917283394591_state] :
      ( ( member5752234768399382829_state @ ( produc5194695453622026300_state @ A @ B ) @ ( produc8552965988182875245_state @ A2 @ B2 ) )
      = ( ( member_state @ A @ A2 )
        & ( member7461423308196221312_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1227_mem__Sigma__iff,axiom,
    ! [A: produc7272280185086996969_state,B: state,A2: set_Pr4231240917283394591_state,B2: produc7272280185086996969_state > set_state] :
      ( ( member5722111062937351085_state @ ( produc7271836056140555068_state @ A @ B ) @ ( produc1406734553846628205_state @ A2 @ B2 ) )
      = ( ( member7461423308196221312_state @ A @ A2 )
        & ( member_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1228_mem__Sigma__iff,axiom,
    ! [A: produc3882902684156697774_state,B: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B2: produc3882902684156697774_state > set_Pr2391701600042252046_state] :
      ( ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ B ) @ ( produc5372175771586804680_state @ A2 @ B2 ) )
      = ( ( member6715010751053357271_state @ A @ A2 )
        & ( member6715010751053357271_state @ B @ ( B2 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1229_SigmaI,axiom,
    ! [A: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: produc7272280185086996969_state,B2: produc3882902684156697774_state > set_Pr4231240917283394591_state] :
      ( ( member6715010751053357271_state @ A @ A2 )
     => ( ( member7461423308196221312_state @ B @ ( B2 @ A ) )
       => ( member4364272472968950365_state @ ( produc209405200909791808_state @ A @ B ) @ ( produc4180506523417663567_state @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1230_SigmaI,axiom,
    ! [A: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: produc9003781588162623004on_val,B2: produc3882902684156697774_state > set_Pr7649655576376376530on_val] :
      ( ( member6715010751053357271_state @ A @ A2 )
     => ( ( member2879030624002546611on_val @ B @ ( B2 @ A ) )
       => ( member2392189377969538832on_val @ ( produc8040891007501466739on_val @ A @ B ) @ ( produc8744377254544660610on_val @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1231_SigmaI,axiom,
    ! [A: produc3882902684156697774_state,A2: set_Pr2391701600042252046_state,B: produc3882902684156697774_state,B2: produc3882902684156697774_state > set_Pr2391701600042252046_state] :
      ( ( member6715010751053357271_state @ A @ A2 )
     => ( ( member6715010751053357271_state @ B @ ( B2 @ A ) )
       => ( member3801449062345016400_state @ ( produc2343482328048119831_state @ A @ B ) @ ( produc5372175771586804680_state @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1232_SigmaI,axiom,
    ! [A: produc9003781588162623004on_val,A2: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val,B2: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ A2 )
     => ( ( member2879030624002546611on_val @ B @ ( B2 @ A ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1233_SigmaI,axiom,
    ! [A: option_state,A2: set_option_state,B: option_state,B2: option_state > set_option_state] :
      ( ( member_option_state @ A @ A2 )
     => ( ( member_option_state @ B @ ( B2 @ A ) )
       => ( member3029510603097127326_state @ ( produc9160152616361873709_state @ A @ B ) @ ( produc7708045757989049950_state @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1234_SigmaI,axiom,
    ! [A: prat,A2: set_prat,B: state,B2: prat > set_state] :
      ( ( member_prat @ A @ A2 )
     => ( ( member_state @ B @ ( B2 @ A ) )
       => ( member6226390569306242836_state @ ( produc6205719116230455971_state @ A @ B ) @ ( produc1658785944879207636_state @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1235_SigmaI,axiom,
    ! [A: prat,A2: set_prat,B: set_state,B2: prat > set_set_state] :
      ( ( member_prat @ A @ A2 )
     => ( ( member_set_state @ B @ ( B2 @ A ) )
       => ( member6708919011922255306_state @ ( produc9080284787853255769_state @ A @ B ) @ ( produc4653271056321233546_state @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1236_SigmaI,axiom,
    ! [A: produc1457211279475724562t_char > prat,A2: set_Pr3433046674199179443r_prat,B: produc1457211279475724562t_char > option_val,B2: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2336342582916103932r_prat @ A @ A2 )
     => ( ( member1577473829074448177on_val @ B @ ( B2 @ A ) )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A2 @ B2 ) ) ) ) ).

% SigmaI
thf(fact_1237_PartialSA_Oup__closed__plus__UNIV,axiom,
    ! [A2: set_state] : ( sep_up_closed_state @ plus @ ( add_set @ A2 @ top_top_set_state ) ) ).

% PartialSA.up_closed_plus_UNIV
thf(fact_1238_valid__stateI,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( valid_mask @ Pi3 )
     => ( ! [Hl2: produc1457211279475724562t_char] :
            ( ( ppos @ ( Pi3 @ Hl2 ) )
           => ( ( H3 @ Hl2 )
             != none_val ) )
       => ( valid_state @ ( produc74620467203698198on_val @ Pi3 @ H3 ) ) ) ) ).

% valid_stateI
thf(fact_1239_valid__heap__def,axiom,
    ( valid_heap
    = ( ^ [Pi5: produc1457211279475724562t_char > prat,H5: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi5 @ Hl ) )
         => ( ( H5 @ Hl )
           != none_val ) ) ) ) ).

% valid_heap_def
thf(fact_1240_valid__heapI,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi3 @ Hl2 ) )
         => ( ( H3 @ Hl2 )
           != none_val ) )
     => ( valid_heap @ Pi3 @ H3 ) ) ).

% valid_heapI
thf(fact_1241_PartialHeapSA_Ostable__def,axiom,
    ( stable
    = ( ^ [Phi3: state] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi3 @ Hl ) )
          = ( ( get_h @ Phi3 @ Hl )
           != none_val ) ) ) ) ).

% PartialHeapSA.stable_def
thf(fact_1242_stableI,axiom,
    ! [Phi: state] :
      ( ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi @ Hl2 ) )
          = ( ( get_h @ Phi @ Hl2 )
           != none_val ) )
     => ( stable @ Phi ) ) ).

% stableI
thf(fact_1243_asm0,axiom,
    ( ( ppos @ alpha )
    & ( ppos @ beta )
    & ( ( padd @ alpha @ beta )
      = pwrite ) ) ).

% asm0
thf(fact_1244_calculation_I3_J,axiom,
    ( ( ppos @ p )
    & ( pgte @ pwrite @ p ) ) ).

% calculation(3)
thf(fact_1245_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_1246_PartialSA_Osum__then__singleton,axiom,
    ! [A: state,B: state,C: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ C ) )
      = ( ( insert_state @ A @ bot_bot_set_state )
        = ( add_set @ ( insert_state @ B @ bot_bot_set_state ) @ ( insert_state @ C @ bot_bot_set_state ) ) ) ) ).

% PartialSA.sum_then_singleton
thf(fact_1247_PartialSA_Osetify__sum,axiom,
    ! [P: state > $o,A2: set_state,B2: set_state] :
      ( ( sep_setify_state @ P @ ( add_set @ A2 @ B2 ) )
      = ( ! [X5: state] :
            ( ( member_state @ X5 @ A2 )
           => ( sep_setify_state @ P @ ( add_set @ ( insert_state @ X5 @ bot_bot_set_state ) @ B2 ) ) ) ) ) ).

% PartialSA.setify_sum
thf(fact_1248_PartialSA_Oempty__set__sum,axiom,
    ! [A2: set_state] :
      ( ( add_set @ bot_bot_set_state @ A2 )
      = bot_bot_set_state ) ).

% PartialSA.empty_set_sum
thf(fact_1249_mult__write__mask,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ pwrite @ Pi3 )
      = Pi3 ) ).

% mult_write_mask
thf(fact_1250_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi3: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi3 )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi3 ) @ ( multiply_mask @ B @ Pi3 ) ) ) ).

% mult_add_states
thf(fact_1251_wand__equiv__def,axiom,
    ( wand
    = ( ^ [A7: set_state,B4: set_state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [Phi3: state] :
              ( ( Uu3 = Phi3 )
              & ( ord_le2494988322063910608_state @ ( add_set @ A7 @ ( insert_state @ Phi3 @ bot_bot_set_state ) ) @ B4 ) ) ) ) ) ).

% wand_equiv_def
thf(fact_1252_combinableI,axiom,
    ! [A2: set_state,B2: set_state] :
      ( ! [A3: prat,B7: prat] :
          ( ( ( ppos @ A3 )
            & ( ppos @ B7 )
            & ( ( padd @ A3 @ B7 )
              = pwrite ) )
         => ( ord_le2494988322063910608_state @ ( add_set @ ( multip8064567061438756306ertion @ A3 @ ( cwand @ A2 @ B2 ) ) @ ( multip8064567061438756306ertion @ B7 @ ( cwand @ A2 @ B2 ) ) ) @ ( cwand @ A2 @ B2 ) ) )
     => ( combinable @ ( cwand @ A2 @ B2 ) ) ) ).

% combinableI
thf(fact_1253__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_O_A_092_060lbrakk_062a_A_124_D_124_Amultiply_Ap_Aw_059_Appos_Ap_A_092_060and_062_Apgte_Apwrite_Ap_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P4: prat] :
        ( ( defined @ a2 @ ( multiply @ P4 @ w ) )
       => ~ ( ( ppos @ P4 )
            & ( pgte @ pwrite @ P4 ) ) ) ).

% \<open>\<And>thesis. (\<And>p. \<lbrakk>a |#| multiply p w; ppos p \<and> pgte pwrite p\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1254__092_060open_062pgte_Apwrite_A_092_060alpha_062_A_092_060and_062_Apgte_Apwrite_A_092_060beta_062_092_060close_062,axiom,
    ( ( pgte @ pwrite @ alpha )
    & ( pgte @ pwrite @ beta ) ) ).

% \<open>pgte pwrite \<alpha> \<and> pgte pwrite \<beta>\<close>
thf(fact_1255_larger__implies__greater__mask__hl,axiom,
    ! [A: state,B: state,Hl3: produc1457211279475724562t_char] :
      ( ( greater @ A @ B )
     => ( pgte @ ( get_m @ A @ Hl3 ) @ ( get_m @ B @ Hl3 ) ) ) ).

% larger_implies_greater_mask_hl
thf(fact_1256_multiply__order,axiom,
    ! [P2: prat,A: state,B: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( greater @ A @ B )
       => ( greater @ ( multiply @ P2 @ A ) @ ( multiply @ P2 @ B ) ) ) ) ).

% multiply_order
thf(fact_1257_get__h__multiply,axiom,
    ! [P2: prat,X3: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_h @ ( multiply @ P2 @ X3 ) )
        = ( get_h @ X3 ) ) ) ).

% get_h_multiply
thf(fact_1258_valid__mult,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,P2: prat] :
      ( ( valid_mask @ Pi3 )
     => ( ( pgte @ pwrite @ P2 )
       => ( valid_mask @ ( multiply_mask @ P2 @ Pi3 ) ) ) ) ).

% valid_mult
thf(fact_1259_get__m__smaller__mask,axiom,
    ! [P2: prat,A: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_m @ ( multiply @ P2 @ A ) )
        = ( multiply_mask @ P2 @ ( get_m @ A ) ) ) ) ).

% get_m_smaller_mask
thf(fact_1260_scaled__def,axiom,
    ( scaled
    = ( ^ [Phi3: state] :
          ( collect_state
          @ ^ [Uu3: state] :
            ? [P3: prat] :
              ( ( Uu3
                = ( multiply @ P3 @ Phi3 ) )
              & ( ppos @ P3 )
              & ( pgte @ pwrite @ P3 ) ) ) ) ) ).

% scaled_def
thf(fact_1261_non__scalable__instantiate,axiom,
    ! [W: state,A: state] :
      ( ~ ( scalable @ W @ A )
     => ? [P4: prat] :
          ( ( ppos @ P4 )
          & ( pgte @ pwrite @ P4 )
          & ( defined @ A @ ( multiply @ P4 @ W ) ) ) ) ).

% non_scalable_instantiate
thf(fact_1262_multiply__valid,axiom,
    ! [P2: prat,Phi: state] :
      ( ( pgte @ pwrite @ P2 )
     => ( valid_state @ ( produc74620467203698198on_val @ ( multiply_mask @ P2 @ ( get_m @ Phi ) ) @ ( get_h @ Phi ) ) ) ) ).

% multiply_valid
thf(fact_1263_combinable__def,axiom,
    ( combinable
    = ( ^ [P6: set_state] :
        ! [Alpha: prat,Beta: prat] :
          ( ( ( ppos @ Alpha )
            & ( ppos @ Beta )
            & ( pgte @ pwrite @ ( padd @ Alpha @ Beta ) ) )
         => ( ord_le2494988322063910608_state @ ( add_set @ ( multip8064567061438756306ertion @ Alpha @ P6 ) @ ( multip8064567061438756306ertion @ Beta @ P6 ) ) @ ( multip8064567061438756306ertion @ ( padd @ Alpha @ Beta ) @ P6 ) ) ) ) ) ).

% combinable_def
thf(fact_1264_valid__sum_I2_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( ( upper_bounded @ A @ Ma )
         => ( ( upper_bounded @ B @ Mb )
           => ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
             => ( upper_bounded @ ( add_ma6586698345353345811t_char @ A @ B ) @ ( padd @ Ma @ Mb ) ) ) ) ) ) ) ).

% valid_sum(2)
thf(fact_1265_valid__sum_I1_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,Ma: prat,Mb: prat] :
      ( ( valid_mask @ A )
     => ( ( valid_mask @ B )
       => ( ( upper_bounded @ A @ Ma )
         => ( ( upper_bounded @ B @ Mb )
           => ( ( pgte @ pwrite @ ( padd @ Ma @ Mb ) )
             => ( valid_mask @ ( add_ma6586698345353345811t_char @ A @ B ) ) ) ) ) ) ) ).

% valid_sum(1)
thf(fact_1266_upper__boundedI,axiom,
    ! [P2: prat,Pi3: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi3 @ Hl2 ) )
     => ( upper_bounded @ Pi3 @ P2 ) ) ).

% upper_boundedI
thf(fact_1267_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi5: produc1457211279475724562t_char > prat,P3: prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ P3 @ ( Pi5 @ Hl ) ) ) ) ).

% upper_bounded_def
thf(fact_1268_upper__bounded__bigger,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,A: prat,B: prat] :
      ( ( upper_bounded @ Pi3 @ A )
     => ( ( pgte @ B @ A )
       => ( upper_bounded @ Pi3 @ B ) ) ) ).

% upper_bounded_bigger
thf(fact_1269_valid__multiply,axiom,
    ! [A: produc1457211279475724562t_char > prat,Ma: prat,P2: prat] :
      ( ( valid_mask @ A )
     => ( ( upper_bounded @ A @ Ma )
       => ( ( pgte @ pwrite @ ( pmult @ Ma @ P2 ) )
         => ( valid_mask @ ( multiply_mask @ P2 @ A ) ) ) ) ) ).

% valid_multiply
thf(fact_1270_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P3: prat,Pi5: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( pmult @ P3 @ ( Pi5 @ Hl ) ) ) ) ).

% multiply_mask_def
thf(fact_1271_upper__bounded__smaller,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat,A: prat,P2: prat] :
      ( ( upper_bounded @ Pi3 @ A )
     => ( upper_bounded @ ( multiply_mask @ P2 @ Pi3 ) @ ( pmult @ P2 @ A ) ) ) ).

% upper_bounded_smaller
thf(fact_1272_multiply__twice,axiom,
    ! [A: prat,B: prat,X3: state] :
      ( ( ( pgte @ pwrite @ A )
        & ( pgte @ pwrite @ B ) )
     => ( ( multiply @ A @ ( multiply @ B @ X3 ) )
        = ( multiply @ ( pmult @ A @ B ) @ X3 ) ) ) ).

% multiply_twice
thf(fact_1273_get__m__smaller,axiom,
    ! [P2: prat,A: state,Hl3: produc1457211279475724562t_char] :
      ( ( pgte @ pwrite @ P2 )
     => ( ( get_m @ ( multiply @ P2 @ A ) @ Hl3 )
        = ( pmult @ P2 @ ( get_m @ A @ Hl3 ) ) ) ) ).

% get_m_smaller

% Helper facts (5)
thf(help_If_2_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X3: state,Y: state] :
      ( ( if_state @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PartialHeapSA__Ostate_T,axiom,
    ! [X3: state,Y: state] :
      ( ( if_state @ $true @ X3 @ Y )
      = X3 ) ).

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,
    ! [X3: option_state,Y: option_state] :
      ( ( if_option_state @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X3: option_state,Y: option_state] :
      ( ( if_option_state @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [Ya: state] :
      ( ( ( some_state @ Ya )
        = ( plus @ ( r @ a2 @ wa ) @ a2 ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------