TPTP Problem File: SLH0703^1.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    : FOL_Seq_Calc3/0011_Completeness/prob_00186_007623__12350062_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1668 ( 594 unt; 381 typ;   0 def)
%            Number of atoms       : 3425 (1239 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11366 ( 442   ~;  94   |; 228   &;9160   @)
%                                         (   0 <=>;1442  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   6 avg)
%            Number of types       :   74 (  73 usr)
%            Number of type conns  :  774 ( 774   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  311 ( 308 usr;  24 con; 0-3 aty)
%            Number of variables   : 3622 ( 192   ^;3345   !;  85   ?;3622   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:28:44.514
%------------------------------------------------------------------------------
% Could-be-implicit typings (73)
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    fset_o3706400737857578983ist_fm: list_P5616295576739893671ist_fm > fset_P661503646757059847ist_fm ).

thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_FSet_Ofset__of__list_001t__Syntax__Ofm,type,
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thf(sy_c_FSet_Ofthe__elem_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_FSet_Olinorder__class_OfMin_001t__Extended____Nat__Oenat,type,
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thf(sy_c_FSet_Olinorder__class_OfMin_001t__Nat__Onat,type,
    linorder_fMin_nat: fset_nat > nat ).

thf(sy_c_FSet_Osemilattice__order__fset_001t__Extended____Nat__Oenat,type,
    semila7129917228889371579d_enat: ( extended_enat > extended_enat > extended_enat ) > ( extended_enat > extended_enat > $o ) > ( extended_enat > extended_enat > $o ) > $o ).

thf(sy_c_FSet_Osemilattice__order__fset_001t__Nat__Onat,type,
    semila6541227223090250811et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    minus_8437252545254675822ist_fm: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Syntax__Ofm_J,type,
    minus_minus_set_fm: set_fm > set_fm > set_fm ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Extended____Nat__Oenat,type,
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thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_If_001t__List__Olist_It__Syntax__Ofm_J,type,
    if_list_fm: $o > list_fm > list_fm > list_fm ).

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

thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Set__Oset_It__Syntax__Ofm_J,type,
    if_set_fm: $o > set_fm > set_fm > set_fm ).

thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    if_str8948254419368749791m_rule: $o > stream727092118206550309m_rule > stream727092118206550309m_rule > stream727092118206550309m_rule ).

thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Obutlast_001t__Syntax__Ofm,type,
    butlast_fm: list_fm > list_fm ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Ocoset_001t__Syntax__Ofm,type,
    coset_fm: list_fm > set_fm ).

thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__Syntax__Ofm,type,
    enumerate_fm: nat > list_fm > list_P1547346012799288711nat_fm ).

thf(sy_c_List_Ogen__length_001t__Syntax__Ofm,type,
    gen_length_fm: nat > list_fm > nat ).

thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Oinsert_001t__Syntax__Ofm,type,
    insert_fm: fm > list_fm > list_fm ).

thf(sy_c_List_Olenlex_001t__List__Olist_It__Syntax__Ofm_J,type,
    lenlex_list_fm: set_Pr7058068377845519745ist_fm > set_Pr441346426823788833ist_fm ).

thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
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thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olenlex_001t__Syntax__Ofm,type,
    lenlex_fm: set_Pr4463079037648049377_fm_fm > set_Pr7058068377845519745ist_fm ).

thf(sy_c_List_Olex_001t__List__Olist_It__Syntax__Ofm_J,type,
    lex_list_fm: set_Pr7058068377845519745ist_fm > set_Pr441346426823788833ist_fm ).

thf(sy_c_List_Olex_001t__Nat__Onat,type,
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thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olex_001t__Syntax__Ofm,type,
    lex_fm: set_Pr4463079037648049377_fm_fm > set_Pr7058068377845519745ist_fm ).

thf(sy_c_List_Olexord_001t__List__Olist_It__Syntax__Ofm_J,type,
    lexord_list_fm: set_Pr7058068377845519745ist_fm > set_Pr441346426823788833ist_fm ).

thf(sy_c_List_Olexord_001t__Nat__Onat,type,
    lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lexord5831005462426227802at_nat: set_Pr553994874890374343at_nat > set_Pr4087777274317423175at_nat ).

thf(sy_c_List_Olexord_001t__Syntax__Ofm,type,
    lexord_fm: set_Pr4463079037648049377_fm_fm > set_Pr7058068377845519745ist_fm ).

thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_It__Syntax__Ofm_J_Mt__Nat__Onat_J,type,
    cons_list_fm_nat: ( list_fm > nat ) > list_list_fm_nat > list_list_fm_nat ).

thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    cons_P4861729644591583992at_nat: ( product_prod_nat_nat > nat ) > list_P9162950289778280392at_nat > list_P9162950289778280392at_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Syntax__Ofm_J,type,
    cons_list_fm: list_fm > list_list_fm > list_list_fm ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Syntax__Ofm_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    cons_P768676612401224159m_rule: produc164195504107695125m_rule > list_P8332459875677254181m_rule > list_P8332459875677254181m_rule ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    cons_P8732206157123786781at_nat: produc859450856879609959at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
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thf(sy_c_List_Olist_OCons_001t__Syntax__Ofm,type,
    cons_fm: fm > list_fm > list_fm ).

thf(sy_c_List_Olist_ONil_001_062_It__List__Olist_It__Syntax__Ofm_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_List_Olist_ONil_001t__Syntax__Ofm,type,
    nil_fm: list_fm ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Syntax__Ofm_J,type,
    set_list_fm2: list_list_fm > set_list_fm ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    set_Pr8505323785428441536m_rule: list_P8332459875677254181m_rule > set_Pr1008144964186165195m_rule ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr1322963821424748924at_nat: list_P4765738968734470637at_nat > set_Pr553994874890374343at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Syntax__Ofm_J,type,
    set_Pr3075457875423956113nat_fm: list_P4746041832035913334nat_fm > set_Pr4875362410256910876nat_fm ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr657957747041424951at_nat: list_P5089989197324543260at_nat > set_Pr5219309775545540802at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
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thf(sy_c_List_Olist_Oset_001t__Syntax__Ofm,type,
    set_fm2: list_fm > set_fm ).

thf(sy_c_List_Olistrel_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    listre6567353182834750131ist_fm: set_Pr7058068377845519745ist_fm > set_Pr441346426823788833ist_fm ).

thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
    listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre818007680106770737at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olistrel_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
    listrel_fm_fm: set_Pr4463079037648049377_fm_fm > set_Pr7058068377845519745ist_fm ).

thf(sy_c_List_Omeasures_001t__List__Olist_It__Syntax__Ofm_J,type,
    measures_list_fm: list_list_fm_nat > set_Pr7058068377845519745ist_fm ).

thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
    measures_nat: list_nat_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Omeasures_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    measur2679027848233739777at_nat: list_P9162950289778280392at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Syntax__Ofm_J,type,
    nth_list_fm: list_list_fm > nat > list_fm ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    nth_Pr4548706283066508072ist_fm: list_P5616295576739893671ist_fm > nat > produc1996495991257130529ist_fm ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).

thf(sy_c_List_Onth_001t__Syntax__Ofm,type,
    nth_fm: list_fm > nat > fm ).

thf(sy_c_List_Onth_001t__Syntax__Orule,type,
    nth_rule: list_rule > nat > rule ).

thf(sy_c_List_Onull_001t__Syntax__Ofm,type,
    null_fm: list_fm > $o ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    remove2181804207701385843at_nat: produc859450856879609959at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_OremoveAll_001t__Syntax__Ofm,type,
    removeAll_fm: fm > list_fm > list_fm ).

thf(sy_c_List_Oreplicate_001t__List__Olist_It__Syntax__Ofm_J,type,
    replicate_list_fm: nat > list_fm > list_list_fm ).

thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: nat > nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    replic4812330464126889441ist_fm: nat > produc1996495991257130529ist_fm > list_P5616295576739893671ist_fm ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    replic6713244433751818279at_nat: nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Oreplicate_001t__Syntax__Ofm,type,
    replicate_fm: nat > fm > list_fm ).

thf(sy_c_List_Orotate1_001t__Syntax__Ofm,type,
    rotate1_fm: list_fm > list_fm ).

thf(sy_c_List_Otake_001t__List__Olist_It__Syntax__Ofm_J,type,
    take_list_fm: nat > list_list_fm > list_list_fm ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    take_P2173866234530122223at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    take_P5254422574997664853at_nat: nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Otake_001t__Syntax__Ofm,type,
    take_fm: nat > list_fm > list_fm ).

thf(sy_c_List_Ozip_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    zip_list_fm_list_fm: list_list_fm > list_list_fm > list_P5616295576739893671ist_fm ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
    zip_Pr2138331976196620527m_rule: list_P5616295576739893671ist_fm > list_rule > list_P8332459875677254181m_rule ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    zip_Pr1780431989191363299m_rule: list_P8332459875677254181m_rule > list_P8332459875677254181m_rule > list_P8801850819564216977m_rule ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    zip_Pr935030979083031159at_nat: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > list_P4765738968734470637at_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Syntax__Ofm,type,
    zip_Pr9151751282906081600nat_fm: list_P8469869581646625389at_nat > list_fm > list_P4746041832035913334nat_fm ).

thf(sy_c_List_Ozip_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    zip_fm7413572573862219166at_nat: list_fm > list_P8469869581646625389at_nat > list_P5089989197324543260at_nat ).

thf(sy_c_List_Ozip_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
    zip_fm_fm: list_fm > list_fm > list_P6934817935912052487_fm_fm ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J,type,
    size_s4563186235979089028ist_fm: list_list_fm > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    size_s2722402132374190611ist_fm: list_P5616295576739893671ist_fm > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s3679842834875189465at_nat: list_P8469869581646625389at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Syntax__Ofm_J,type,
    size_size_list_fm: list_fm > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Syntax__Orule_J,type,
    size_size_list_rule: list_rule > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Ofm,type,
    size_size_fm: fm > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Orule,type,
    size_size_rule: rule > nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    bot_bo7573314457883560170_nat_o: produc859450856879609959at_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Syntax__Ofm_M_Eo_J,type,
    bot_bot_fm_o: fm > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Nat__Onat_J,type,
    bot_bot_fset_nat: fset_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    bot_bo2367426573206113139ist_fm: fset_P661503646757059847ist_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Ofm_J,type,
    bot_bot_set_fm: set_fm ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    ord_le8344328325585982387ist_fm: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Syntax__Ofm_J,type,
    ord_less_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
    ord_le2787558655864224659d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
    ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Syntax__Ofm_J_J,type,
    ord_less_eq_o_set_fm: ( $o > set_fm ) > ( $o > set_fm ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    ord_le2064643713053750439ist_fm: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Syntax__Ofm_J,type,
    ord_less_eq_fset_fm: fset_fm > fset_fm > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    ord_le2055072402069232929ist_fm: set_Pr7058068377845519745ist_fm > set_Pr7058068377845519745ist_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ofm_J,type,
    ord_less_eq_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    ord_mi3151844185881495662ist_fm: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Syntax__Ofm_J,type,
    ord_min_set_fm: set_fm > set_fm > set_fm ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
    order_2428742583041560895d_enat: ( extended_enat > $o ) > extended_enat ).

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

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Syntax__Ofm_J,type,
    order_6179083242224974829set_fm: ( set_fm > $o ) > set_fm ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J_001t__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J,type,
    produc8668676954508811513ist_fm: list_list_fm > list_list_fm > produc8337847990479660737ist_fm ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5943733680697469783at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > produc6392793444374437607at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc1338542795132623831at_nat: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > produc289266582803401575at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    produc381145313068854617ist_fm: list_fm > list_fm > produc1996495991257130529ist_fm ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Syntax__Ofm,type,
    product_Pair_nat_fm: nat > fm > product_prod_nat_fm ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
    produc491467635432902671m_rule: produc1996495991257130529ist_fm > rule > produc164195504107695125m_rule ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    produc5927390650430071747m_rule: produc164195504107695125m_rule > produc164195504107695125m_rule > produc3793571666044074251m_rule ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4662710985925991255at_nat: produc859450856879609959at_nat > produc859450856879609959at_nat > produc349518998152878311at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Syntax__Ofm,type,
    produc6439020385818810464nat_fm: produc859450856879609959at_nat > fm > produc5183350815473539046nat_fm ).

thf(sy_c_Product__Type_OPair_001t__Syntax__Ofm_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4700841676774948030at_nat: fm > produc859450856879609959at_nat > produc2765850687091007884at_nat ).

thf(sy_c_Product__Type_OPair_001t__Syntax__Ofm_001t__Syntax__Ofm,type,
    product_Pair_fm_fm: fm > fm > product_prod_fm_fm ).

thf(sy_c_Product__Type_OPair_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    produc7063755442767259331ist_fm: rule > produc1996495991257130529ist_fm > produc4630866025709511057ist_fm ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    produc1501393135466168645ist_fm: produc1996495991257130529ist_fm > list_fm ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
    produc6879501374131015971m_rule: produc164195504107695125m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc3213797794245857475at_nat: produc859450856879609959at_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    produc4228417144610596823ist_fm: produc4630866025709511057ist_fm > rule ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    produc4588648349897876871ist_fm: produc1996495991257130529ist_fm > list_fm ).

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thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    stl_Pr2200982372225425851m_rule: stream6494289010434245521m_rule > stream6494289010434245521m_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Syntax__Ofm,type,
    stl_fm: stream_fm > stream_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__Syntax__Orule,type,
    stl_rule: stream_rule > stream_rule ).

thf(sy_c_Stream_Oszip_001t__List__Olist_It__Syntax__Ofm_J_001t__List__Olist_It__Syntax__Ofm_J,type,
    szip_list_fm_list_fm: stream_list_fm > stream_list_fm > stream8299795917829157543ist_fm ).

thf(sy_c_Stream_Oszip_001t__Nat__Onat_001t__Nat__Onat,type,
    szip_nat_nat: stream_nat > stream_nat > stream6724221391990029191at_nat ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    szip_P6003557782763710490ist_fm: stream8299795917829157543ist_fm > stream8299795917829157543ist_fm > stream3409308193418444653ist_fm ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    szip_P5379742314337339356m_rule: stream8299795917829157543ist_fm > stream727092118206550309m_rule > stream642806101564963573m_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Syntax__Orule,type,
    szip_P1977448745965526924m_rule: stream8299795917829157543ist_fm > stream_rule > stream727092118206550309m_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    szip_P4314639285670189082at_nat: stream6724221391990029191at_nat > stream6724221391990029191at_nat > stream8372878641218411373at_nat ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    szip_P5077708527527104496ist_fm: stream727092118206550309m_rule > stream8299795917829157543ist_fm > stream8936951515322355153ist_fm ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    szip_P2499414959592755846m_rule: stream727092118206550309m_rule > stream727092118206550309m_rule > stream6494289010434245521m_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule,type,
    szip_P553291425819358518e_rule: stream727092118206550309m_rule > stream_rule > stream2312013117288958913e_rule ).

thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    szip_r8549736553299883584ist_fm: stream_rule > stream8299795917829157543ist_fm > stream1960312990768957601ist_fm ).

thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    szip_r2254114822203693110m_rule: stream_rule > stream727092118206550309m_rule > stream1878305877988316353m_rule ).

thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Syntax__Orule,type,
    szip_rule_rule: stream_rule > stream_rule > stream3588992565182678257e_rule ).

thf(sy_c_Syntax_Ofm_OImp,type,
    imp: fm > fm > fm ).

thf(sy_c_Syntax_Ofm_Osize__fm,type,
    size_fm: fm > nat ).

thf(sy_c_Syntax_Orule_OImpR,type,
    impR: fm > fm > rule ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Syntax__Ofm_J,type,
    member_list_fm: list_fm > set_list_fm > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J_Mt__List__Olist_It__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member197921522497012330ist_fm: produc8337847990479660737ist_fm > set_Pr441346426823788833ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4574794575480667280at_nat: produc289266582803401575at_nat > set_Pr4087777274317423175at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    member8102475879199740618ist_fm: produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    member4220325220686508332m_rule: produc164195504107695125m_rule > set_Pr1008144964186165195m_rule > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Syntax__Ofm_J,type,
    member9198537383846477821nat_fm: produc5183350815473539046nat_fm > set_Pr4875362410256910876nat_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6781037255463946659at_nat: produc2765850687091007884at_nat > set_Pr5219309775545540802at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Ofm_Mt__Syntax__Ofm_J,type,
    member8474499337054950954_fm_fm: product_prod_fm_fm > set_Pr4463079037648049377_fm_fm > $o ).

thf(sy_c_member_001t__Syntax__Ofm,type,
    member_fm: fm > set_fm > $o ).

thf(sy_v_j____,type,
    j: nat ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_q____,type,
    q: fm ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

% Relevant facts (1267)
thf(fact_0__092_060open_062p_A_091_092_060in_062_093_Alhsd_A_Istl_A_Isdrop_Aj_Asteps_J_J_A_092_060and_062_Aq_A_091_092_060in_062_093_Arhsd_A_Istl_A_Isdrop_Aj_Asteps_J_J_092_060close_062,axiom,
    ( ( member_fm @ p @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) ) ) )
    & ( member_fm @ q @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) ) ) ) ) ).

% \<open>p [\<in>] lhsd (stl (sdrop j steps)) \<and> q [\<in>] rhsd (stl (sdrop j steps))\<close>
thf(fact_1__092_060open_062p_A_092_060_094bold_062_092_060longrightarrow_062_Aq_A_092_060in_062_AtreeB_Asteps_092_060close_062,axiom,
    member_fm @ ( imp @ p @ q ) @ ( treeB @ steps ) ).

% \<open>p \<^bold>\<longrightarrow> q \<in> treeB steps\<close>
thf(fact_2__092_060open_062snd_A_Ishd_A_Isdrop_Aj_Asteps_J_J_A_061_AImpR_Ap_Aq_092_060close_062,axiom,
    ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) )
    = ( impR @ p @ q ) ) ).

% \<open>snd (shd (sdrop j steps)) = ImpR p q\<close>
thf(fact_3_sdrop__stl,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( stl_Pr5027778045650968273ist_fm @ S ) )
      = ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_4_sdrop__stl,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_5_sdrop__stl,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( stl_Pr2200982372225425851m_rule @ S ) )
      = ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_6_sdrop__stl,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062j_O_Asnd_A_Ishd_A_Isdrop_Aj_Asteps_J_J_A_061_AImpR_Ap_Aq_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [J: nat] :
        ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ J @ steps ) ) )
       != ( impR @ p @ q ) ) ).

% \<open>\<And>thesis. (\<And>j. snd (shd (sdrop j steps)) = ImpR p q \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_treeA__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeA @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeA @ Steps ) ) ).

% treeA_sdrop
thf(fact_9_treeB__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeB @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeB @ Steps ) ) ).

% treeB_sdrop
thf(fact_10_treeA__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeA @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeA_snth
thf(fact_11_treeB__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeB @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeB_snth
thf(fact_12_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_14_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_15_sdrop__szip,axiom,
    ! [N: nat,S1: stream_list_fm,S2: stream_list_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_16_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_17_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_18_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_19_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_1261538545438237392ist_fm @ N @ ( szip_P6003557782763710490ist_fm @ S1 @ S2 ) )
      = ( szip_P6003557782763710490ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_20_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_3792585760419738520ist_fm @ N @ ( szip_P5077708527527104496ist_fm @ S1 @ S2 ) )
      = ( szip_P5077708527527104496ist_fm @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_21_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream727092118206550309m_rule] :
      ( ( sdrop_9094843686636319420m_rule @ N @ ( szip_P5379742314337339356m_rule @ S1 @ S2 ) )
      = ( szip_P5379742314337339356m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_22_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_23_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_24_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) )
      = ( sdrop_4442373711808556042ist_fm @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_25_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N @ S ) )
      = ( sdrop_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_26_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) )
      = ( sdrop_7373388980841566196m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_27_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_28_stream_Oexpand,axiom,
    ! [Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( ( ( shd_Pr7235097944458474089m_rule @ Stream )
          = ( shd_Pr7235097944458474089m_rule @ Stream2 ) )
        & ( ( stl_Pr950425576149878629m_rule @ Stream )
          = ( stl_Pr950425576149878629m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_29_stream_Ocoinduct,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_30_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ ( suc @ N ) @ S )
      = ( sdrop_4442373711808556042ist_fm @ N @ ( stl_Pr5027778045650968273ist_fm @ S ) ) ) ).

% sdrop.simps(2)
thf(fact_31_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( sdrop_rule @ ( suc @ N ) @ S )
      = ( sdrop_rule @ N @ ( stl_rule @ S ) ) ) ).

% sdrop.simps(2)
thf(fact_32_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ ( suc @ N ) @ S )
      = ( sdrop_7373388980841566196m_rule @ N @ ( stl_Pr2200982372225425851m_rule @ S ) ) ) ).

% sdrop.simps(2)
thf(fact_33_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S )
      = ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S ) ) ) ).

% sdrop.simps(2)
thf(fact_34_stream_Ocoinduct__strong,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_35_rule_Oinject_I3_J,axiom,
    ! [X61: fm,X62: fm,Y61: fm,Y62: fm] :
      ( ( ( impR @ X61 @ X62 )
        = ( impR @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% rule.inject(3)
thf(fact_36_fm_Oinject_I2_J,axiom,
    ! [X31: fm,X32: fm,Y31: fm,Y32: fm] :
      ( ( ( imp @ X31 @ X32 )
        = ( imp @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% fm.inject(2)
thf(fact_37_subsetI,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A )
         => ( member8206827879206165904at_nat @ X @ B ) )
     => ( ord_le3000389064537975527at_nat @ A @ B ) ) ).

% subsetI
thf(fact_38_subsetI,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ A )
         => ( member_fm @ X @ B ) )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% subsetI
thf(fact_39_subset__antisym,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_40_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_41_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_42_order__refl,axiom,
    ! [X3: set_fm] : ( ord_less_eq_set_fm @ X3 @ X3 ) ).

% order_refl
thf(fact_43_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_44_order__refl,axiom,
    ! [X3: extended_enat] : ( ord_le2932123472753598470d_enat @ X3 @ X3 ) ).

% order_refl
thf(fact_45_dual__order_Orefl,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_46_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_47_dual__order_Orefl,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_48_prod_Oexpand,axiom,
    ! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
      ( ( ( ( produc1501393135466168645ist_fm @ Prod )
          = ( produc1501393135466168645ist_fm @ Prod2 ) )
        & ( ( produc4588648349897876871ist_fm @ Prod )
          = ( produc4588648349897876871ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_49_prod_Oexpand,axiom,
    ! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
      ( ( ( ( produc6879501374131015971m_rule @ Prod )
          = ( produc6879501374131015971m_rule @ Prod2 ) )
        & ( ( produc7165828336582415457m_rule @ Prod )
          = ( produc7165828336582415457m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_50_prod__eqI,axiom,
    ! [P: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P )
        = ( produc1501393135466168645ist_fm @ Q ) )
     => ( ( ( produc4588648349897876871ist_fm @ P )
          = ( produc4588648349897876871ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_51_prod__eqI,axiom,
    ! [P: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P )
        = ( produc6879501374131015971m_rule @ Q ) )
     => ( ( ( produc7165828336582415457m_rule @ P )
          = ( produc7165828336582415457m_rule @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_52_exE__realizer_H,axiom,
    ! [P2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4588648349897876871ist_fm @ P ) @ ( produc1501393135466168645ist_fm @ P ) )
     => ~ ! [X: list_fm,Y: list_fm] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_53_exE__realizer_H,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,P: produc164195504107695125m_rule] :
      ( ( P2 @ ( produc7165828336582415457m_rule @ P ) @ ( produc6879501374131015971m_rule @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y: rule] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_54_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_55_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R @ X @ X )
       => ( ! [X: nat,Y: nat,Z: nat] :
              ( ( R @ X @ Y )
             => ( ( R @ Y @ Z )
               => ( R @ X @ Z ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_56_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P2 @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_57_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P2 @ M2 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_58_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_59_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_60_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_61_Suc__le__D,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
     => ? [M4: nat] :
          ( M3
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_62_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_63_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_64_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_65_order__antisym__conv,axiom,
    ! [Y3: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y3 @ X3 )
     => ( ( ord_less_eq_set_fm @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_66_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_67_order__antisym__conv,axiom,
    ! [Y3: extended_enat,X3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
     => ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_68_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_69_linorder__le__cases,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ord_le2932123472753598470d_enat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_70_ord__le__eq__subst,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > set_fm,C: set_fm] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_79_ord__eq__le__subst,axiom,
    ! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: set_fm > extended_enat,B2: set_fm,C: set_fm] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
    ! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
    ! [A2: set_fm,F: extended_enat > set_fm,B2: extended_enat,C: extended_enat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
    ! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_88_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_89_linorder__linear,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
      | ( ord_le2932123472753598470d_enat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_90_order__eq__refl,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_fm @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_91_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_92_order__eq__refl,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( X3 = Y3 )
     => ( ord_le2932123472753598470d_enat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_93_mem__Collect__eq,axiom,
    ! [A2: fm,P2: fm > $o] :
      ( ( member_fm @ A2 @ ( collect_fm @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A2: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A2 @ ( collec7088162979684241874at_nat @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A: set_fm] :
      ( ( collect_fm
        @ ^ [X4: fm] : ( member_fm @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_97_order__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_98_order__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_99_order__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_100_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_101_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_102_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_103_order__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > set_fm,C: set_fm] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_104_order__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_105_order__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_106_order__subst1,axiom,
    ! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_107_order__subst1,axiom,
    ! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_108_order__subst1,axiom,
    ! [A2: set_fm,F: extended_enat > set_fm,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_109_order__subst1,axiom,
    ! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_110_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_111_order__subst1,axiom,
    ! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_112_order__subst1,axiom,
    ! [A2: extended_enat,F: set_fm > extended_enat,B2: set_fm,C: set_fm] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_113_order__subst1,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_114_order__subst1,axiom,
    ! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_115_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A3 @ B3 )
          & ( ord_less_eq_set_fm @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_116_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_118_antisym,axiom,
    ! [A2: set_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_119_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_120_antisym,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_121_dual__order_Otrans,axiom,
    ! [B2: set_fm,A2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ B2 @ A2 )
     => ( ( ord_less_eq_set_fm @ C @ B2 )
       => ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_122_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_123_dual__order_Otrans,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ B2 )
       => ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_124_dual__order_Oantisym,axiom,
    ! [B2: set_fm,A2: set_fm] :
      ( ( ord_less_eq_set_fm @ B2 @ A2 )
     => ( ( ord_less_eq_set_fm @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_125_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_126_dual__order_Oantisym,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_127_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A3 )
          & ( ord_less_eq_set_fm @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_128_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_129_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ( ord_le2932123472753598470d_enat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_130_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
         => ( P2 @ A4 @ B4 ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( P2 @ B4 @ A4 )
           => ( P2 @ A4 @ B4 ) )
       => ( P2 @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_131_linorder__wlog,axiom,
    ! [P2: extended_enat > extended_enat > $o,A2: extended_enat,B2: extended_enat] :
      ( ! [A4: extended_enat,B4: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A4 @ B4 )
         => ( P2 @ A4 @ B4 ) )
     => ( ! [A4: extended_enat,B4: extended_enat] :
            ( ( P2 @ B4 @ A4 )
           => ( P2 @ A4 @ B4 ) )
       => ( P2 @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_132_order__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ Z3 )
       => ( ord_less_eq_set_fm @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_133_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z3 )
       => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_134_order__trans,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_le2932123472753598470d_enat @ Y3 @ Z3 )
       => ( ord_le2932123472753598470d_enat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_135_order_Otrans,axiom,
    ! [A2: set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% order.trans
thf(fact_136_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_137_order_Otrans,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_138_order__antisym,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_139_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_140_order__antisym,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_141_ord__le__eq__trans,axiom,
    ! [A2: set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_142_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_143_ord__le__eq__trans,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_144_ord__eq__le__trans,axiom,
    ! [A2: set_fm,B2: set_fm,C: set_fm] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_145_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_146_ord__eq__le__trans,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( A2 = B2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_147_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [X4: set_fm,Y5: set_fm] :
          ( ( ord_less_eq_set_fm @ X4 @ Y5 )
          & ( ord_less_eq_set_fm @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_148_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_149_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X4 @ Y5 )
          & ( ord_le2932123472753598470d_enat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_150_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_151_le__cases3,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z3: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
       => ~ ( ord_le2932123472753598470d_enat @ Y3 @ Z3 ) )
     => ( ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
         => ~ ( ord_le2932123472753598470d_enat @ X3 @ Z3 ) )
       => ( ( ( ord_le2932123472753598470d_enat @ X3 @ Z3 )
           => ~ ( ord_le2932123472753598470d_enat @ Z3 @ Y3 ) )
         => ( ( ( ord_le2932123472753598470d_enat @ Z3 @ Y3 )
             => ~ ( ord_le2932123472753598470d_enat @ Y3 @ X3 ) )
           => ( ( ( ord_le2932123472753598470d_enat @ Y3 @ Z3 )
               => ~ ( ord_le2932123472753598470d_enat @ Z3 @ X3 ) )
             => ~ ( ( ord_le2932123472753598470d_enat @ Z3 @ X3 )
                 => ~ ( ord_le2932123472753598470d_enat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_152_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_153_nle__le,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) )
      = ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_154_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_155_Suc__inject,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Suc_inject
thf(fact_156_Collect__mono__iff,axiom,
    ! [P2: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) )
      = ( ! [X4: fm] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_157_set__eq__subset,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_fm,B5: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B5 )
          & ( ord_less_eq_set_fm @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_158_subset__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_159_Collect__mono,axiom,
    ! [P2: fm > $o,Q2: fm > $o] :
      ( ! [X: fm] :
          ( ( P2 @ X )
         => ( Q2 @ X ) )
     => ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q2 ) ) ) ).

% Collect_mono
thf(fact_160_subset__refl,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% subset_refl
thf(fact_161_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
        ! [T: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T @ A5 )
         => ( member8206827879206165904at_nat @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_162_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
        ! [T: fm] :
          ( ( member_fm @ T @ A5 )
         => ( member_fm @ T @ B5 ) ) ) ) ).

% subset_iff
thf(fact_163_equalityD2,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A = B )
     => ( ord_less_eq_set_fm @ B @ A ) ) ).

% equalityD2
thf(fact_164_equalityD1,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A = B )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% equalityD1
thf(fact_165_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
        ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A5 )
         => ( member8206827879206165904at_nat @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_166_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
        ! [X4: fm] :
          ( ( member_fm @ X4 @ A5 )
         => ( member_fm @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_167_equalityE,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_fm @ A @ B )
         => ~ ( ord_less_eq_set_fm @ B @ A ) ) ) ).

% equalityE
thf(fact_168_subsetD,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ B )
     => ( ( member8206827879206165904at_nat @ C @ A )
       => ( member8206827879206165904at_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_169_subsetD,axiom,
    ! [A: set_fm,B: set_fm,C: fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( member_fm @ C @ A )
       => ( member_fm @ C @ B ) ) ) ).

% subsetD
thf(fact_170_in__mono,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ B )
     => ( ( member8206827879206165904at_nat @ X3 @ A )
       => ( member8206827879206165904at_nat @ X3 @ B ) ) ) ).

% in_mono
thf(fact_171_in__mono,axiom,
    ! [A: set_fm,B: set_fm,X3: fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( member_fm @ X3 @ A )
       => ( member_fm @ X3 @ B ) ) ) ).

% in_mono
thf(fact_172_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_fm,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_fm @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_fm @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_173_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_174_lift__Suc__antimono__le,axiom,
    ! [F: nat > extended_enat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_175_lift__Suc__mono__le,axiom,
    ! [F: nat > set_fm,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_fm @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_fm @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_176_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_177_lift__Suc__mono__le,axiom,
    ! [F: nat > extended_enat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_le2932123472753598470d_enat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_178_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S3 )
            = ( produc1501393135466168645ist_fm @ T ) )
          & ( ( produc4588648349897876871ist_fm @ S3 )
            = ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_179_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc164195504107695125m_rule,Z2: produc164195504107695125m_rule] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S3 )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S3 )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_180_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B )
      = ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_181_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
      = ( ! [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X4 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_182_snd__swap,axiom,
    ! [X3: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X3 ) )
      = ( produc6879501374131015971m_rule @ X3 ) ) ).

% snd_swap
thf(fact_183_snd__swap,axiom,
    ! [X3: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X3 ) )
      = ( produc1501393135466168645ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_184_snd__swap,axiom,
    ! [X3: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X3 ) )
      = ( produc4228417144610596823ist_fm @ X3 ) ) ).

% snd_swap
thf(fact_185_fst__swap,axiom,
    ! [X3: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X3 ) )
      = ( produc4514744107061996309ist_fm @ X3 ) ) ).

% fst_swap
thf(fact_186_fst__swap,axiom,
    ! [X3: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X3 ) )
      = ( produc4588648349897876871ist_fm @ X3 ) ) ).

% fst_swap
thf(fact_187_fst__swap,axiom,
    ! [X3: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X3 ) )
      = ( produc7165828336582415457m_rule @ X3 ) ) ).

% fst_swap
thf(fact_188_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_189_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S2: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S2 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_190_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_191_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_192_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_193_Greatest__equality,axiom,
    ! [P2: set_fm > $o,X3: set_fm] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_fm] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_fm @ Y @ X3 ) )
       => ( ( order_6179083242224974829set_fm @ P2 )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_194_Greatest__equality,axiom,
    ! [P2: extended_enat > $o,X3: extended_enat] :
      ( ( P2 @ X3 )
     => ( ! [Y: extended_enat] :
            ( ( P2 @ Y )
           => ( ord_le2932123472753598470d_enat @ Y @ X3 ) )
       => ( ( order_2428742583041560895d_enat @ P2 )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_195_Greatest__equality,axiom,
    ! [P2: nat > $o,X3: nat] :
      ( ( P2 @ X3 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ X3 ) )
       => ( ( order_Greatest_nat @ P2 )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_196_GreatestI2__order,axiom,
    ! [P2: set_fm > $o,X3: set_fm,Q2: set_fm > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: set_fm] :
            ( ( P2 @ Y )
           => ( ord_less_eq_set_fm @ Y @ X3 ) )
       => ( ! [X: set_fm] :
              ( ( P2 @ X )
             => ( ! [Y6: set_fm] :
                    ( ( P2 @ Y6 )
                   => ( ord_less_eq_set_fm @ Y6 @ X ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( order_6179083242224974829set_fm @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_197_GreatestI2__order,axiom,
    ! [P2: extended_enat > $o,X3: extended_enat,Q2: extended_enat > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: extended_enat] :
            ( ( P2 @ Y )
           => ( ord_le2932123472753598470d_enat @ Y @ X3 ) )
       => ( ! [X: extended_enat] :
              ( ( P2 @ X )
             => ( ! [Y6: extended_enat] :
                    ( ( P2 @ Y6 )
                   => ( ord_le2932123472753598470d_enat @ Y6 @ X ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( order_2428742583041560895d_enat @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_198_GreatestI2__order,axiom,
    ! [P2: nat > $o,X3: nat,Q2: nat > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ X3 ) )
       => ( ! [X: nat] :
              ( ( P2 @ X )
             => ( ! [Y6: nat] :
                    ( ( P2 @ Y6 )
                   => ( ord_less_eq_nat @ Y6 @ X ) )
               => ( Q2 @ X ) ) )
         => ( Q2 @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_199_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P3 @ ( shd_Pr7235097944458474089m_rule @ S3 ) ) @ ( sdrop_5541336384294084785m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ S3 ) ) @ S3 ) ) ) ).

% sdrop_while.simps
thf(fact_200__092_060open_062fst_A_Ishd_A_Istl_A_Isdrop_Aj_Asteps_J_J_J_A_124_092_060in_062_124_A_123_124_Ip_A_D_Alhsd_A_Isdrop_Aj_Asteps_J_M_Aq_A_D_Arhsd_A_Isdrop_Aj_Asteps_J_A_091_092_060div_062_093_A_Ip_A_092_060_094bold_062_092_060longrightarrow_062_Aq_J_J_124_125_092_060close_062,axiom,
    fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ p @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) ) @ ( cons_fm @ q @ ( removeAll_fm @ ( imp @ p @ q ) @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) ) ) ) @ bot_bo2367426573206113139ist_fm ) ).

% \<open>fst (shd (stl (sdrop j steps))) |\<in>| {|(p # lhsd (sdrop j steps), q # rhsd (sdrop j steps) [\<div>] (p \<^bold>\<longrightarrow> q))|}\<close>
thf(fact_201_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_set_fm
    = ( ^ [X5: $o > set_fm,Y7: $o > set_fm] :
          ( ( ord_less_eq_set_fm @ ( X5 @ $false ) @ ( Y7 @ $false ) )
          & ( ord_less_eq_set_fm @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_202_le__rel__bool__arg__iff,axiom,
    ( ord_less_eq_o_nat
    = ( ^ [X5: $o > nat,Y7: $o > nat] :
          ( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
          & ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_203_le__rel__bool__arg__iff,axiom,
    ( ord_le2787558655864224659d_enat
    = ( ^ [X5: $o > extended_enat,Y7: $o > extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
          & ( ord_le2932123472753598470d_enat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_204_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_nat @ A2 @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_205_verit__la__disequality,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2 = B2 )
      | ~ ( ord_le2932123472753598470d_enat @ A2 @ B2 )
      | ~ ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_206_old_Oprod_Oinject,axiom,
    ! [A2: list_fm,B2: list_fm,A6: list_fm,B6: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
        = ( produc381145313068854617ist_fm @ A6 @ B6 ) )
      = ( ( A2 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_207_old_Oprod_Oinject,axiom,
    ! [A2: nat,B2: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B2 )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
      = ( ( A2 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_208_old_Oprod_Oinject,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A6: product_prod_nat_nat,B6: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A2 @ B2 )
        = ( produc6161850002892822231at_nat @ A6 @ B6 ) )
      = ( ( A2 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
thf(fact_209_prod_Oinject,axiom,
    ! [X1: list_fm,X2: list_fm,Y1: list_fm,Y2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X1 @ X2 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_210_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_211_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_212_list_Oinject,axiom,
    ! [X21: fm,X22: list_fm,Y21: fm,Y22: list_fm] :
      ( ( ( cons_fm @ X21 @ X22 )
        = ( cons_fm @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_213_removeAll__id,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( remove2181804207701385843at_nat @ X3 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_214_removeAll__id,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ( removeAll_fm @ X3 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_215_swap__simp,axiom,
    ! [X3: list_fm,Y3: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_216_swap__simp,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
      = ( product_Pair_nat_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_217_swap__simp,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( produc7225117575323628663at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
      = ( produc6161850002892822231at_nat @ Y3 @ X3 ) ) ).

% swap_simp
thf(fact_218_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_219_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_220_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_221_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_222_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_223_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_224_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_225_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_226_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_227_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B2: nat] :
      ( ( P2 @ K )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B2 ) )
       => ? [X: nat] :
            ( ( P2 @ X )
            & ! [Y6: nat] :
                ( ( P2 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_228_prod__induct3,axiom,
    ! [P2: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B4: nat,C3: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ C3 ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_229_prod__cases3,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B4: nat,C3: nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ C3 ) ) ) ).

% prod_cases3
thf(fact_230_Pair__inject,axiom,
    ! [A2: list_fm,B2: list_fm,A6: list_fm,B6: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
        = ( produc381145313068854617ist_fm @ A6 @ B6 ) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_231_Pair__inject,axiom,
    ! [A2: nat,B2: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B2 )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_232_Pair__inject,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A6: product_prod_nat_nat,B6: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A2 @ B2 )
        = ( produc6161850002892822231at_nat @ A6 @ B6 ) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
thf(fact_233_prod__cases,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ! [A4: list_fm,B4: list_fm] : ( P2 @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_234_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A4: nat,B4: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_235_prod__cases,axiom,
    ! [P2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat] :
      ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_236_surj__pair,axiom,
    ! [P: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y: list_fm] :
      ( P
      = ( produc381145313068854617ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_237_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X: nat,Y: nat] :
      ( P
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_238_surj__pair,axiom,
    ! [P: produc859450856879609959at_nat] :
    ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( P
      = ( produc6161850002892822231at_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_239_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1996495991257130529ist_fm] :
      ~ ! [A4: list_fm,B4: list_fm] :
          ( Y3
         != ( produc381145313068854617ist_fm @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_240_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A4: nat,B4: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_241_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_242_removeAll_Osimps_I2_J,axiom,
    ! [X3: fm,Y3: fm,Xs: list_fm] :
      ( ( ( X3 = Y3 )
       => ( ( removeAll_fm @ X3 @ ( cons_fm @ Y3 @ Xs ) )
          = ( removeAll_fm @ X3 @ Xs ) ) )
      & ( ( X3 != Y3 )
       => ( ( removeAll_fm @ X3 @ ( cons_fm @ Y3 @ Xs ) )
          = ( cons_fm @ Y3 @ ( removeAll_fm @ X3 @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_243_not__Cons__self2,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( cons_fm @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_244_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B2: nat] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B2 ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_245_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K: nat,B2: nat] :
      ( ( P2 @ K )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_246_GreatestI__nat,axiom,
    ! [P2: nat > $o,K: nat,B2: nat] :
      ( ( P2 @ K )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B2 ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_247_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat,X21: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ X22 ) )
     => ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_248_list_Oset__intros_I2_J,axiom,
    ! [Y3: fm,X22: list_fm,X21: fm] :
      ( ( member_fm @ Y3 @ ( set_fm2 @ X22 ) )
     => ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_249_list_Oset__intros_I1_J,axiom,
    ! [X21: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat] : ( member8206827879206165904at_nat @ X21 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_250_list_Oset__intros_I1_J,axiom,
    ! [X21: fm,X22: list_fm] : ( member_fm @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_251_list_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A2: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ A2 ) )
     => ( ! [Z22: list_P8469869581646625389at_nat] :
            ( A2
           != ( cons_P8732206157123786781at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: list_P8469869581646625389at_nat] :
              ( ( A2
                = ( cons_P8732206157123786781at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_252_list_Oset__cases,axiom,
    ! [E: fm,A2: list_fm] :
      ( ( member_fm @ E @ ( set_fm2 @ A2 ) )
     => ( ! [Z22: list_fm] :
            ( A2
           != ( cons_fm @ E @ Z22 ) )
       => ~ ! [Z1: fm,Z22: list_fm] :
              ( ( A2
                = ( cons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_253_set__ConsD,axiom,
    ! [Y3: produc859450856879609959at_nat,X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_254_set__ConsD,axiom,
    ! [Y3: fm,X3: fm,Xs: list_fm] :
      ( ( member_fm @ Y3 @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_fm @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_255_bot_Oextremum,axiom,
    ! [A2: fset_P661503646757059847ist_fm] : ( ord_le2064643713053750439ist_fm @ bot_bo2367426573206113139ist_fm @ A2 ) ).

% bot.extremum
thf(fact_256_bot_Oextremum,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).

% bot.extremum
thf(fact_257_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_258_bot_Oextremum,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A2 ) ).

% bot.extremum
thf(fact_259_bot_Oextremum__unique,axiom,
    ! [A2: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ bot_bo2367426573206113139ist_fm )
      = ( A2 = bot_bo2367426573206113139ist_fm ) ) ).

% bot.extremum_unique
thf(fact_260_bot_Oextremum__unique,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
      = ( A2 = bot_bot_set_fm ) ) ).

% bot.extremum_unique
thf(fact_261_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_262_bot_Oextremum__unique,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
      = ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
thf(fact_263_bot_Oextremum__uniqueI,axiom,
    ! [A2: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ bot_bo2367426573206113139ist_fm )
     => ( A2 = bot_bo2367426573206113139ist_fm ) ) ).

% bot.extremum_uniqueI
thf(fact_264_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
     => ( A2 = bot_bot_set_fm ) ) ).

% bot.extremum_uniqueI
thf(fact_265_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_266_bot_Oextremum__uniqueI,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
     => ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_267_fst__eqD,axiom,
    ! [X3: nat,Y3: nat,A2: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
        = A2 )
     => ( X3 = A2 ) ) ).

% fst_eqD
thf(fact_268_fst__eqD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
        = A2 )
     => ( X3 = A2 ) ) ).

% fst_eqD
thf(fact_269_fst__eqD,axiom,
    ! [X3: list_fm,Y3: list_fm,A2: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
        = A2 )
     => ( X3 = A2 ) ) ).

% fst_eqD
thf(fact_270_fst__eqD,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: rule,A2: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
        = A2 )
     => ( X3 = A2 ) ) ).

% fst_eqD
thf(fact_271_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_272_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_273_fst__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_274_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_275_snd__eqD,axiom,
    ! [X3: nat,Y3: nat,A2: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) )
        = A2 )
     => ( Y3 = A2 ) ) ).

% snd_eqD
thf(fact_276_snd__eqD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
        = A2 )
     => ( Y3 = A2 ) ) ).

% snd_eqD
thf(fact_277_snd__eqD,axiom,
    ! [X3: list_fm,Y3: list_fm,A2: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
        = A2 )
     => ( Y3 = A2 ) ) ).

% snd_eqD
thf(fact_278_snd__eqD,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: rule,A2: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) )
        = A2 )
     => ( Y3 = A2 ) ) ).

% snd_eqD
thf(fact_279_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_280_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_281_snd__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_282_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_283_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P4 ) @ ( product_fst_nat_nat @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_284_prod_Oswap__def,axiom,
    ( produc7225117575323628663at_nat
    = ( ^ [P4: produc859450856879609959at_nat] : ( produc6161850002892822231at_nat @ ( produc6408287024330202629at_nat @ P4 ) @ ( produc3213797794245857475at_nat @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_285_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc1501393135466168645ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_286_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P4: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc6879501374131015971m_rule @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_287_set__subset__Cons,axiom,
    ! [Xs: list_fm,X3: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_288_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_289_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_290_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_291_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_292_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y3: nat,X3: nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_293_exI__realizer,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_294_exI__realizer,axiom,
    ! [P2: list_fm > list_fm > $o,Y3: list_fm,X3: list_fm] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_295_exI__realizer,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Y3: rule,X3: produc1996495991257130529ist_fm] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_296_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q2: nat > $o,Q: nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_297_conjI__realizer,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat,Q2: product_prod_nat_nat > $o,Q: product_prod_nat_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_298_conjI__realizer,axiom,
    ! [P2: list_fm > $o,P: list_fm,Q2: list_fm > $o,Q: list_fm] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) )
          & ( Q2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_299_conjI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q2: rule > $o,Q: rule] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) )
          & ( Q2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_300_surjective__pairing,axiom,
    ! [T2: product_prod_nat_nat] :
      ( T2
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_301_surjective__pairing,axiom,
    ! [T2: produc859450856879609959at_nat] :
      ( T2
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T2 ) @ ( produc6408287024330202629at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_302_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_303_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_304_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_305_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_306_verit__comp__simplify1_I2_J,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_307_fempty__iff,axiom,
    ! [C: produc1996495991257130529ist_fm] :
      ~ ( fmembe3381613331217039976ist_fm @ C @ bot_bo2367426573206113139ist_fm ) ).

% fempty_iff
thf(fact_308_all__not__fin__conv,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( ! [X4: produc1996495991257130529ist_fm] :
            ~ ( fmembe3381613331217039976ist_fm @ X4 @ A ) )
      = ( A = bot_bo2367426573206113139ist_fm ) ) ).

% all_not_fin_conv
thf(fact_309_finsert__fsubset,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) @ B )
      = ( ( fmembe3381613331217039976ist_fm @ X3 @ B )
        & ( ord_le2064643713053750439ist_fm @ A @ B ) ) ) ).

% finsert_fsubset
thf(fact_310_finsertCI,axiom,
    ! [A2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm,B2: produc1996495991257130529ist_fm] :
      ( ( ~ ( fmembe3381613331217039976ist_fm @ A2 @ B )
       => ( A2 = B2 ) )
     => ( fmembe3381613331217039976ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ B2 @ B ) ) ) ).

% finsertCI
thf(fact_311_finsert__iff,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( fmembe3381613331217039976ist_fm @ A2 @ A ) ) ) ).

% finsert_iff
thf(fact_312_fset__induct2,axiom,
    ! [P2: fset_P661503646757059847ist_fm > fset_P661503646757059847ist_fm > $o,Xsa: fset_P661503646757059847ist_fm,Ysa: fset_P661503646757059847ist_fm] :
      ( ( P2 @ bot_bo2367426573206113139ist_fm @ bot_bo2367426573206113139ist_fm )
     => ( ! [X: produc1996495991257130529ist_fm,Xs2: fset_P661503646757059847ist_fm] :
            ( ~ ( fmembe3381613331217039976ist_fm @ X @ Xs2 )
           => ( P2 @ ( finser3446675674286072169ist_fm @ X @ Xs2 ) @ bot_bo2367426573206113139ist_fm ) )
       => ( ! [Y: produc1996495991257130529ist_fm,Ys: fset_P661503646757059847ist_fm] :
              ( ~ ( fmembe3381613331217039976ist_fm @ Y @ Ys )
             => ( P2 @ bot_bo2367426573206113139ist_fm @ ( finser3446675674286072169ist_fm @ Y @ Ys ) ) )
         => ( ! [X: produc1996495991257130529ist_fm,Xs2: fset_P661503646757059847ist_fm,Y: produc1996495991257130529ist_fm,Ys: fset_P661503646757059847ist_fm] :
                ( ( P2 @ Xs2 @ Ys )
               => ( ~ ( fmembe3381613331217039976ist_fm @ X @ Xs2 )
                 => ( ~ ( fmembe3381613331217039976ist_fm @ Y @ Ys )
                   => ( P2 @ ( finser3446675674286072169ist_fm @ X @ Xs2 ) @ ( finser3446675674286072169ist_fm @ Y @ Ys ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_313_fsingleton__iff,axiom,
    ! [B2: produc1996495991257130529ist_fm,A2: produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B2 @ ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) )
      = ( B2 = A2 ) ) ).

% fsingleton_iff
thf(fact_314_fset__strong__cases,axiom,
    ! [Xs: fset_P661503646757059847ist_fm] :
      ( ( Xs != bot_bo2367426573206113139ist_fm )
     => ~ ! [Ys: fset_P661503646757059847ist_fm,X: produc1996495991257130529ist_fm] :
            ( ~ ( fmembe3381613331217039976ist_fm @ X @ Ys )
           => ( Xs
             != ( finser3446675674286072169ist_fm @ X @ Ys ) ) ) ) ).

% fset_strong_cases
thf(fact_315_fset__induct__stronger,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,S4: fset_P661503646757059847ist_fm] :
      ( ( P2 @ bot_bo2367426573206113139ist_fm )
     => ( ! [X: produc1996495991257130529ist_fm,S5: fset_P661503646757059847ist_fm] :
            ( ~ ( fmembe3381613331217039976ist_fm @ X @ S5 )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finser3446675674286072169ist_fm @ X @ S5 ) ) ) )
       => ( P2 @ S4 ) ) ) ).

% fset_induct_stronger
thf(fact_316_empty__subsetI,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).

% empty_subsetI
thf(fact_317_subset__empty,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
      = ( A = bot_bot_set_fm ) ) ).

% subset_empty
thf(fact_318_finsert__absorb2,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( finser3446675674286072169ist_fm @ X3 @ ( finser3446675674286072169ist_fm @ X3 @ A ) )
      = ( finser3446675674286072169ist_fm @ X3 @ A ) ) ).

% finsert_absorb2
thf(fact_319_fsubsetI,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
         => ( fmembe3381613331217039976ist_fm @ X @ B ) )
     => ( ord_le2064643713053750439ist_fm @ A @ B ) ) ).

% fsubsetI
thf(fact_320_fempty__fsubsetI,axiom,
    ! [X3: fset_P661503646757059847ist_fm] : ( ord_le2064643713053750439ist_fm @ bot_bo2367426573206113139ist_fm @ X3 ) ).

% fempty_fsubsetI
thf(fact_321_fsubset__fempty,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ bot_bo2367426573206113139ist_fm )
      = ( A = bot_bo2367426573206113139ist_fm ) ) ).

% fsubset_fempty
thf(fact_322_eqfelem__imp__iff,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( X3 = Y3 )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
        = ( fmembe3381613331217039976ist_fm @ Y3 @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_323_if__split__fmem2,axiom,
    ! [A2: produc1996495991257130529ist_fm,Q2: $o,X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ ( if_fse3714977293194272717ist_fm @ Q2 @ X3 @ Y3 ) )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ A2 @ X3 ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ A2 @ Y3 ) ) ) ) ).

% if_split_fmem2
thf(fact_324_if__split__fmem1,axiom,
    ! [Q2: $o,X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,B2: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q2 @ X3 @ Y3 ) @ B2 )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ X3 @ B2 ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ Y3 @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_325_eqfset__imp__iff,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
        = ( fmembe3381613331217039976ist_fm @ X3 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_326_eq__fmem__trans,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( A2 = B2 )
     => ( ( fmembe3381613331217039976ist_fm @ B2 @ A )
       => ( fmembe3381613331217039976ist_fm @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_327_fequalityCE,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( A = B )
     => ( ( ( fmembe3381613331217039976ist_fm @ C @ A )
         => ~ ( fmembe3381613331217039976ist_fm @ C @ B ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A )
           => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_328_fset__eqI,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
          = ( fmembe3381613331217039976ist_fm @ X @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_329_fsubsetD,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A )
       => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% fsubsetD
thf(fact_330_fin__mono,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
       => ( fmembe3381613331217039976ist_fm @ X3 @ B ) ) ) ).

% fin_mono
thf(fact_331_finsert__commute,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( finser3446675674286072169ist_fm @ X3 @ ( finser3446675674286072169ist_fm @ Y3 @ A ) )
      = ( finser3446675674286072169ist_fm @ Y3 @ ( finser3446675674286072169ist_fm @ X3 @ A ) ) ) ).

% finsert_commute
thf(fact_332_fsubset__finsertI2,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,B2: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ord_le2064643713053750439ist_fm @ A @ ( finser3446675674286072169ist_fm @ B2 @ B ) ) ) ).

% fsubset_finsertI2
thf(fact_333_fsubset__finsertI,axiom,
    ! [B: fset_P661503646757059847ist_fm,A2: produc1996495991257130529ist_fm] : ( ord_le2064643713053750439ist_fm @ B @ ( finser3446675674286072169ist_fm @ A2 @ B ) ) ).

% fsubset_finsertI
thf(fact_334_finsert__mono,axiom,
    ! [C2: fset_P661503646757059847ist_fm,D: fset_P661503646757059847ist_fm,A2: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ C2 @ D )
     => ( ord_le2064643713053750439ist_fm @ ( finser3446675674286072169ist_fm @ A2 @ C2 ) @ ( finser3446675674286072169ist_fm @ A2 @ D ) ) ) ).

% finsert_mono
thf(fact_335_mk__disjoint__finsert,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ A )
     => ? [B7: fset_P661503646757059847ist_fm] :
          ( ( A
            = ( finser3446675674286072169ist_fm @ A2 @ B7 ) )
          & ~ ( fmembe3381613331217039976ist_fm @ A2 @ B7 ) ) ) ).

% mk_disjoint_finsert
thf(fact_336_finsert__eq__iff,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ~ ( fmembe3381613331217039976ist_fm @ A2 @ A )
     => ( ~ ( fmembe3381613331217039976ist_fm @ B2 @ B )
       => ( ( ( finser3446675674286072169ist_fm @ A2 @ A )
            = ( finser3446675674286072169ist_fm @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: fset_P661503646757059847ist_fm] :
                  ( ( A
                    = ( finser3446675674286072169ist_fm @ B2 @ C4 ) )
                  & ~ ( fmembe3381613331217039976ist_fm @ B2 @ C4 )
                  & ( B
                    = ( finser3446675674286072169ist_fm @ A2 @ C4 ) )
                  & ~ ( fmembe3381613331217039976ist_fm @ A2 @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_337_finsert__absorb,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ A )
     => ( ( finser3446675674286072169ist_fm @ A2 @ A )
        = A ) ) ).

% finsert_absorb
thf(fact_338_finsert__ident,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ B )
       => ( ( ( finser3446675674286072169ist_fm @ X3 @ A )
            = ( finser3446675674286072169ist_fm @ X3 @ B ) )
          = ( A = B ) ) ) ) ).

% finsert_ident
thf(fact_339_set__finsert,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ~ ! [B7: fset_P661503646757059847ist_fm] :
            ( ( A
              = ( finser3446675674286072169ist_fm @ X3 @ B7 ) )
           => ( fmembe3381613331217039976ist_fm @ X3 @ B7 ) ) ) ).

% set_finsert
thf(fact_340_finsertI2,axiom,
    ! [A2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm,B2: produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ B )
     => ( fmembe3381613331217039976ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ B2 @ B ) ) ) ).

% finsertI2
thf(fact_341_finsertI1,axiom,
    ! [A2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] : ( fmembe3381613331217039976ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ A2 @ B ) ) ).

% finsertI1
thf(fact_342_finsertE,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( fmembe3381613331217039976ist_fm @ A2 @ A ) ) ) ).

% finsertE
thf(fact_343_fsubset__finsert,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( ord_le2064643713053750439ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ B ) )
        = ( ord_le2064643713053750439ist_fm @ A @ B ) ) ) ).

% fsubset_finsert
thf(fact_344_equalsffemptyI,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ! [Y: produc1996495991257130529ist_fm] :
          ~ ( fmembe3381613331217039976ist_fm @ Y @ A )
     => ( A = bot_bo2367426573206113139ist_fm ) ) ).

% equalsffemptyI
thf(fact_345_equalsffemptyD,axiom,
    ! [A: fset_P661503646757059847ist_fm,A2: produc1996495991257130529ist_fm] :
      ( ( A = bot_bo2367426573206113139ist_fm )
     => ~ ( fmembe3381613331217039976ist_fm @ A2 @ A ) ) ).

% equalsffemptyD
thf(fact_346_ex__fin__conv,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( ? [X4: produc1996495991257130529ist_fm] : ( fmembe3381613331217039976ist_fm @ X4 @ A ) )
      = ( A != bot_bo2367426573206113139ist_fm ) ) ).

% ex_fin_conv
thf(fact_347_femptyE,axiom,
    ! [A2: produc1996495991257130529ist_fm] :
      ~ ( fmembe3381613331217039976ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ).

% femptyE
thf(fact_348_finsert__not__fempty,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( finser3446675674286072169ist_fm @ A2 @ A )
     != bot_bo2367426573206113139ist_fm ) ).

% finsert_not_fempty
thf(fact_349_fsingleton__inject,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm] :
      ( ( ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm )
        = ( finser3446675674286072169ist_fm @ B2 @ bot_bo2367426573206113139ist_fm ) )
     => ( A2 = B2 ) ) ).

% fsingleton_inject
thf(fact_350_fdoubleton__eq__iff,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm,D2: produc1996495991257130529ist_fm] :
      ( ( ( finser3446675674286072169ist_fm @ A2 @ ( finser3446675674286072169ist_fm @ B2 @ bot_bo2367426573206113139ist_fm ) )
        = ( finser3446675674286072169ist_fm @ C @ ( finser3446675674286072169ist_fm @ D2 @ bot_bo2367426573206113139ist_fm ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_351_fset__exhaust,axiom,
    ! [S4: fset_P661503646757059847ist_fm] :
      ( ( S4 != bot_bo2367426573206113139ist_fm )
     => ~ ! [X: produc1996495991257130529ist_fm,S6: fset_P661503646757059847ist_fm] :
            ( S4
           != ( finser3446675674286072169ist_fm @ X @ S6 ) ) ) ).

% fset_exhaust
thf(fact_352_FSet_Ofset__induct,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,S4: fset_P661503646757059847ist_fm] :
      ( ( P2 @ bot_bo2367426573206113139ist_fm )
     => ( ! [X: produc1996495991257130529ist_fm,S5: fset_P661503646757059847ist_fm] :
            ( ( P2 @ S5 )
           => ( P2 @ ( finser3446675674286072169ist_fm @ X @ S5 ) ) )
       => ( P2 @ S4 ) ) ) ).

% FSet.fset_induct
thf(fact_353_fsubset__fsingletonD,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) )
     => ( ( A = bot_bo2367426573206113139ist_fm )
        | ( A
          = ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) ) ) ).

% fsubset_fsingletonD
thf(fact_354_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X3: nat,Y3: nat,A2: product_prod_nat_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A2
          = ( product_Pair_nat_nat @ X3 @ Y3 ) )
       => ( P2 @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_355_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: produc859450856879609959at_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A2
          = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc3213797794245857475at_nat @ A2 ) @ ( produc6408287024330202629at_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_356_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_fm > list_fm > $o,X3: list_fm,Y3: list_fm,A2: produc1996495991257130529ist_fm] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A2
          = ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
       => ( P2 @ ( produc1501393135466168645ist_fm @ A2 ) @ ( produc4588648349897876871ist_fm @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_357_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,X3: produc1996495991257130529ist_fm,Y3: rule,A2: produc164195504107695125m_rule] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A2
          = ( produc491467635432902671m_rule @ X3 @ Y3 ) )
       => ( P2 @ ( produc6879501374131015971m_rule @ A2 ) @ ( produc7165828336582415457m_rule @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_358_fthe__felem__eq,axiom,
    ! [X3: produc1996495991257130529ist_fm] :
      ( ( fthe_e1015496435986986014ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) )
      = X3 ) ).

% fthe_felem_eq
thf(fact_359_eq__snd__iff,axiom,
    ! [B2: nat,P: product_prod_nat_nat] :
      ( ( B2
        = ( product_snd_nat_nat @ P ) )
      = ( ? [A3: nat] :
            ( P
            = ( product_Pair_nat_nat @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_360_eq__snd__iff,axiom,
    ! [B2: product_prod_nat_nat,P: produc859450856879609959at_nat] :
      ( ( B2
        = ( produc6408287024330202629at_nat @ P ) )
      = ( ? [A3: product_prod_nat_nat] :
            ( P
            = ( produc6161850002892822231at_nat @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_361_eq__snd__iff,axiom,
    ! [B2: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( B2
        = ( produc4588648349897876871ist_fm @ P ) )
      = ( ? [A3: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_362_eq__snd__iff,axiom,
    ! [B2: rule,P: produc164195504107695125m_rule] :
      ( ( B2
        = ( produc7165828336582415457m_rule @ P ) )
      = ( ? [A3: produc1996495991257130529ist_fm] :
            ( P
            = ( produc491467635432902671m_rule @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_363_sndI,axiom,
    ! [X3: product_prod_nat_nat,Y3: nat,Z3: nat] :
      ( ( X3
        = ( product_Pair_nat_nat @ Y3 @ Z3 ) )
     => ( ( product_snd_nat_nat @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_364_sndI,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ( X3
        = ( produc6161850002892822231at_nat @ Y3 @ Z3 ) )
     => ( ( produc6408287024330202629at_nat @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_365_sndI,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: list_fm,Z3: list_fm] :
      ( ( X3
        = ( produc381145313068854617ist_fm @ Y3 @ Z3 ) )
     => ( ( produc4588648349897876871ist_fm @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_366_sndI,axiom,
    ! [X3: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z3: rule] :
      ( ( X3
        = ( produc491467635432902671m_rule @ Y3 @ Z3 ) )
     => ( ( produc7165828336582415457m_rule @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_367_eq__fst__iff,axiom,
    ! [A2: nat,P: product_prod_nat_nat] :
      ( ( A2
        = ( product_fst_nat_nat @ P ) )
      = ( ? [B3: nat] :
            ( P
            = ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_368_eq__fst__iff,axiom,
    ! [A2: product_prod_nat_nat,P: produc859450856879609959at_nat] :
      ( ( A2
        = ( produc3213797794245857475at_nat @ P ) )
      = ( ? [B3: product_prod_nat_nat] :
            ( P
            = ( produc6161850002892822231at_nat @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_369_eq__fst__iff,axiom,
    ! [A2: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( A2
        = ( produc1501393135466168645ist_fm @ P ) )
      = ( ? [B3: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_370_eq__fst__iff,axiom,
    ! [A2: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( A2
        = ( produc6879501374131015971m_rule @ P ) )
      = ( ? [B3: rule] :
            ( P
            = ( produc491467635432902671m_rule @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_371_fstI,axiom,
    ! [X3: product_prod_nat_nat,Y3: nat,Z3: nat] :
      ( ( X3
        = ( product_Pair_nat_nat @ Y3 @ Z3 ) )
     => ( ( product_fst_nat_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_372_fstI,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ( X3
        = ( produc6161850002892822231at_nat @ Y3 @ Z3 ) )
     => ( ( produc3213797794245857475at_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_373_fstI,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: list_fm,Z3: list_fm] :
      ( ( X3
        = ( produc381145313068854617ist_fm @ Y3 @ Z3 ) )
     => ( ( produc1501393135466168645ist_fm @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_374_fstI,axiom,
    ! [X3: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z3: rule] :
      ( ( X3
        = ( produc491467635432902671m_rule @ Y3 @ Z3 ) )
     => ( ( produc6879501374131015971m_rule @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_375_fset__of__list__subset,axiom,
    ! [Xs: list_fm,Ys2: list_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys2 ) )
     => ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys2 ) ) ) ).

% fset_of_list_subset
thf(fact_376_gen__length__code_I2_J,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( gen_length_fm @ N @ ( cons_fm @ X3 @ Xs ) )
      = ( gen_length_fm @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_377_all__not__in__conv,axiom,
    ! [A: set_fm] :
      ( ( ! [X4: fm] :
            ~ ( member_fm @ X4 @ A ) )
      = ( A = bot_bot_set_fm ) ) ).

% all_not_in_conv
thf(fact_378_all__not__in__conv,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( ! [X4: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ X4 @ A ) )
      = ( A = bot_bo5327735625951526323at_nat ) ) ).

% all_not_in_conv
thf(fact_379_empty__iff,axiom,
    ! [C: fm] :
      ~ ( member_fm @ C @ bot_bot_set_fm ) ).

% empty_iff
thf(fact_380_empty__iff,axiom,
    ! [C: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ C @ bot_bo5327735625951526323at_nat ) ).

% empty_iff
thf(fact_381_fset__of__list__simps_I2_J,axiom,
    ! [X21: produc1996495991257130529ist_fm,X22: list_P5616295576739893671ist_fm] :
      ( ( fset_o3706400737857578983ist_fm @ ( cons_P7841146678257726167ist_fm @ X21 @ X22 ) )
      = ( finser3446675674286072169ist_fm @ X21 @ ( fset_o3706400737857578983ist_fm @ X22 ) ) ) ).

% fset_of_list_simps(2)
thf(fact_382_fset__of__list__simps_I2_J,axiom,
    ! [X21: fm,X22: list_fm] :
      ( ( fset_of_list_fm @ ( cons_fm @ X21 @ X22 ) )
      = ( finsert_fm @ X21 @ ( fset_of_list_fm @ X22 ) ) ) ).

% fset_of_list_simps(2)
thf(fact_383_ex__in__conv,axiom,
    ! [A: set_fm] :
      ( ( ? [X4: fm] : ( member_fm @ X4 @ A ) )
      = ( A != bot_bot_set_fm ) ) ).

% ex_in_conv
thf(fact_384_ex__in__conv,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( ? [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A ) )
      = ( A != bot_bo5327735625951526323at_nat ) ) ).

% ex_in_conv
thf(fact_385_equals0I,axiom,
    ! [A: set_fm] :
      ( ! [Y: fm] :
          ~ ( member_fm @ Y @ A )
     => ( A = bot_bot_set_fm ) ) ).

% equals0I
thf(fact_386_equals0I,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ! [Y: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ Y @ A )
     => ( A = bot_bo5327735625951526323at_nat ) ) ).

% equals0I
thf(fact_387_equals0D,axiom,
    ! [A: set_fm,A2: fm] :
      ( ( A = bot_bot_set_fm )
     => ~ ( member_fm @ A2 @ A ) ) ).

% equals0D
thf(fact_388_equals0D,axiom,
    ! [A: set_Pr8693737435421807431at_nat,A2: produc859450856879609959at_nat] :
      ( ( A = bot_bo5327735625951526323at_nat )
     => ~ ( member8206827879206165904at_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_389_emptyE,axiom,
    ! [A2: fm] :
      ~ ( member_fm @ A2 @ bot_bot_set_fm ) ).

% emptyE
thf(fact_390_emptyE,axiom,
    ! [A2: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ A2 @ bot_bo5327735625951526323at_nat ) ).

% emptyE
thf(fact_391_fset__of__list__elem,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( fmembe6431293532144391214at_nat @ X3 @ ( fset_o2072219007702160685at_nat @ Xs ) )
      = ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_392_fset__of__list__elem,axiom,
    ! [X3: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ ( fset_o3706400737857578983ist_fm @ Xs ) )
      = ( member8102475879199740618ist_fm @ X3 @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_393_fset__of__list__elem,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( fmember_fm @ X3 @ ( fset_of_list_fm @ Xs ) )
      = ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_394_subset__emptyI,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ X @ A )
     => ( ord_le3000389064537975527at_nat @ A @ bot_bo5327735625951526323at_nat ) ) ).

% subset_emptyI
thf(fact_395_subset__emptyI,axiom,
    ! [A: set_fm] :
      ( ! [X: fm] :
          ~ ( member_fm @ X @ A )
     => ( ord_less_eq_set_fm @ A @ bot_bot_set_fm ) ) ).

% subset_emptyI
thf(fact_396_not__in__set__insert,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( insert4532235091570160003at_nat @ X3 @ Xs )
        = ( cons_P8732206157123786781at_nat @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_397_not__in__set__insert,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X3 @ Xs )
        = ( cons_fm @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_398_stake_Osimps_I2_J,axiom,
    ! [N: nat,S: stream_fm] :
      ( ( stake_fm @ ( suc @ N ) @ S )
      = ( cons_fm @ ( shd_fm @ S ) @ ( stake_fm @ N @ ( stl_fm @ S ) ) ) ) ).

% stake.simps(2)
thf(fact_399_stake_Osimps_I2_J,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( stake_1447931197033250628m_rule @ ( suc @ N ) @ S )
      = ( cons_P768676612401224159m_rule @ ( shd_Pr7235097944458474089m_rule @ S ) @ ( stake_1447931197033250628m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S ) ) ) ) ).

% stake.simps(2)
thf(fact_400_remove__code_I1_J,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( remove_fm @ X3 @ ( set_fm2 @ Xs ) )
      = ( set_fm2 @ ( removeAll_fm @ X3 @ Xs ) ) ) ).

% remove_code(1)
thf(fact_401_member__remove,axiom,
    ! [X3: fm,Y3: fm,A: set_fm] :
      ( ( member_fm @ X3 @ ( remove_fm @ Y3 @ A ) )
      = ( ( member_fm @ X3 @ A )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_402_member__remove,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( remove2798020372241314124at_nat @ Y3 @ A ) )
      = ( ( member8206827879206165904at_nat @ X3 @ A )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_403_in__set__insert,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( insert4532235091570160003at_nat @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_404_in__set__insert,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_405_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S: list_fm,R: set_Pr7058068377845519745ist_fm,S7: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S ) @ R )
     => ( ( S7 = S )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S7 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_406_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S7: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
     => ( ( S7 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S7 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_407_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S7: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R )
     => ( ( S7 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S7 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_408_List_Oinsert__def,axiom,
    ( insert4532235091570160003at_nat
    = ( ^ [X4: produc859450856879609959at_nat,Xs3: list_P8469869581646625389at_nat] : ( if_lis7763640049307703347at_nat @ ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P8732206157123786781at_nat @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_409_List_Oinsert__def,axiom,
    ( insert_fm
    = ( ^ [X4: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm @ X4 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_410_bot__empty__eq,axiom,
    ( bot_bot_fm_o
    = ( ^ [X4: fm] : ( member_fm @ X4 @ bot_bot_set_fm ) ) ) ).

% bot_empty_eq
thf(fact_411_bot__empty__eq,axiom,
    ( bot_bo7573314457883560170_nat_o
    = ( ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq
thf(fact_412_is__empty__set,axiom,
    ! [Xs: list_fm] :
      ( ( is_empty_fm @ ( set_fm2 @ Xs ) )
      = ( null_fm @ Xs ) ) ).

% is_empty_set
thf(fact_413_lexord__cons__cons,axiom,
    ! [A2: list_fm,X3: list_list_fm,B2: list_fm,Y3: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ ( cons_list_fm @ A2 @ X3 ) @ ( cons_list_fm @ B2 @ Y3 ) ) @ ( lexord_list_fm @ R2 ) )
      = ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B2 ) @ R2 )
        | ( ( A2 = B2 )
          & ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ X3 @ Y3 ) @ ( lexord_list_fm @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_414_lexord__cons__cons,axiom,
    ! [A2: nat,X3: list_nat,B2: nat,Y3: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A2 @ X3 ) @ ( cons_nat @ B2 @ Y3 ) ) @ ( lexord_nat @ R2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
        | ( ( A2 = B2 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_415_lexord__cons__cons,axiom,
    ! [A2: product_prod_nat_nat,X3: list_P6011104703257516679at_nat,B2: product_prod_nat_nat,Y3: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A2 @ X3 ) @ ( cons_P6512896166579812791at_nat @ B2 @ Y3 ) ) @ ( lexord2841853652668343668at_nat @ R2 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R2 )
        | ( ( A2 = B2 )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y3 ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_416_lexord__cons__cons,axiom,
    ! [A2: fm,X3: list_fm,B2: fm,Y3: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ A2 @ X3 ) @ ( cons_fm @ B2 @ Y3 ) ) @ ( lexord_fm @ R2 ) )
      = ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A2 @ B2 ) @ R2 )
        | ( ( A2 = B2 )
          & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( lexord_fm @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_417_fsubset__finsert__iff,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ B ) )
      = ( ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
         => ( ord_le2064643713053750439ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) @ B ) )
        & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
         => ( ord_le2064643713053750439ist_fm @ A @ B ) ) ) ) ).

% fsubset_finsert_iff
thf(fact_418_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S22: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_419_szip_Ocode,axiom,
    ( szip_list_fm_list_fm
    = ( ^ [S12: stream_list_fm,S22: stream_list_fm] : ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ ( shd_list_fm @ S12 ) @ ( shd_list_fm @ S22 ) ) @ ( szip_list_fm_list_fm @ ( stl_list_fm @ S12 ) @ ( stl_list_fm @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_420_szip_Ocode,axiom,
    ( szip_nat_nat
    = ( ^ [S12: stream_nat,S22: stream_nat] : ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ ( shd_nat @ S12 ) @ ( shd_nat @ S22 ) ) @ ( szip_nat_nat @ ( stl_nat @ S12 ) @ ( stl_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_421_szip_Ocode,axiom,
    ( szip_P4314639285670189082at_nat
    = ( ^ [S12: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat] : ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S12 ) @ ( shd_Pr4260400998323988397at_nat @ S22 ) ) @ ( szip_P4314639285670189082at_nat @ ( stl_Pr5027827701538482609at_nat @ S12 ) @ ( stl_Pr5027827701538482609at_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_422_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_423_fminusI,axiom,
    ! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ A )
     => ( ~ ( fmembe3381613331217039976ist_fm @ C @ B )
       => ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) ) ) ) ).

% fminusI
thf(fact_424_fminus__iff,axiom,
    ! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
      = ( ( fmembe3381613331217039976ist_fm @ C @ A )
        & ~ ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% fminus_iff
thf(fact_425_fempty__fminus,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( minus_8437252545254675822ist_fm @ bot_bo2367426573206113139ist_fm @ A )
      = bot_bo2367426573206113139ist_fm ) ).

% fempty_fminus
thf(fact_426_fminus__cancel,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( minus_8437252545254675822ist_fm @ A @ A )
      = bot_bo2367426573206113139ist_fm ) ).

% fminus_cancel
thf(fact_427_fminus__fempty,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ( ( minus_8437252545254675822ist_fm @ A @ bot_bo2367426573206113139ist_fm )
      = A ) ).

% fminus_fempty
thf(fact_428_finsert__fminus1,axiom,
    ! [X3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ B )
     => ( ( minus_8437252545254675822ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) @ B )
        = ( minus_8437252545254675822ist_fm @ A @ B ) ) ) ).

% finsert_fminus1
thf(fact_429_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_430_finsert__fminus__single,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( finser3446675674286072169ist_fm @ A2 @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ) )
      = ( finser3446675674286072169ist_fm @ A2 @ A ) ) ).

% finsert_fminus_single
thf(fact_431_szip__unfold,axiom,
    ! [A2: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B2: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A2 @ S1 ) @ ( sCons_rule @ B2 @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A2 @ B2 ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_432_szip__unfold,axiom,
    ! [A2: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B2: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A2 @ S1 ) @ ( sCons_5731567480799343806m_rule @ B2 @ S2 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A2 @ B2 ) @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_433_szip__unfold,axiom,
    ! [A2: list_fm,S1: stream_list_fm,B2: list_fm,S2: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A2 @ S1 ) @ ( sCons_list_fm @ B2 @ S2 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B2 ) @ ( szip_list_fm_list_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_434_szip__unfold,axiom,
    ! [A2: nat,S1: stream_nat,B2: nat,S2: stream_nat] :
      ( ( szip_nat_nat @ ( sCons_nat @ A2 @ S1 ) @ ( sCons_nat @ B2 @ S2 ) )
      = ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( szip_nat_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_435_szip__unfold,axiom,
    ! [A2: product_prod_nat_nat,S1: stream6724221391990029191at_nat,B2: product_prod_nat_nat,S2: stream6724221391990029191at_nat] :
      ( ( szip_P4314639285670189082at_nat @ ( sCons_2147322719763279000at_nat @ A2 @ S1 ) @ ( sCons_2147322719763279000at_nat @ B2 @ S2 ) )
      = ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_436_stream_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_437_stream_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_438_fminusE,axiom,
    ! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
     => ~ ( ( fmembe3381613331217039976ist_fm @ C @ A )
         => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% fminusE
thf(fact_439_fminusD1,axiom,
    ! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
     => ( fmembe3381613331217039976ist_fm @ C @ A ) ) ).

% fminusD1
thf(fact_440_fminusD2,axiom,
    ! [C: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A @ B ) )
     => ~ ( fmembe3381613331217039976ist_fm @ C @ B ) ) ).

% fminusD2
thf(fact_441_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_442_finsert__fminus__if,axiom,
    ! [X3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( ( fmembe3381613331217039976ist_fm @ X3 @ B )
       => ( ( minus_8437252545254675822ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) @ B )
          = ( minus_8437252545254675822ist_fm @ A @ B ) ) )
      & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ B )
       => ( ( minus_8437252545254675822ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) @ B )
          = ( finser3446675674286072169ist_fm @ X3 @ ( minus_8437252545254675822ist_fm @ A @ B ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_443_fminus__finsert,axiom,
    ! [A: fset_P661503646757059847ist_fm,A2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ B ) )
      = ( minus_8437252545254675822ist_fm @ ( minus_8437252545254675822ist_fm @ A @ B ) @ ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ) ) ).

% fminus_finsert
thf(fact_444_fminus__finsert2,axiom,
    ! [A: fset_P661503646757059847ist_fm,A2: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ B ) )
      = ( minus_8437252545254675822ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ) @ B ) ) ).

% fminus_finsert2
thf(fact_445_lexord__linear,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,X3: list_fm,Y3: list_fm] :
      ( ! [A4: fm,B4: fm] :
          ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A4 @ B4 ) @ R2 )
          | ( A4 = B4 )
          | ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ B4 @ A4 ) @ R2 ) )
     => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( lexord_fm @ R2 ) )
        | ( X3 = Y3 )
        | ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ X3 ) @ ( lexord_fm @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_446_lexord__linear,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,X3: list_list_fm,Y3: list_list_fm] :
      ( ! [A4: list_fm,B4: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ R2 )
          | ( A4 = B4 )
          | ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ B4 @ A4 ) @ R2 ) )
     => ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ X3 @ Y3 ) @ ( lexord_list_fm @ R2 ) )
        | ( X3 = Y3 )
        | ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Y3 @ X3 ) @ ( lexord_list_fm @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_447_lexord__linear,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X3: list_nat,Y3: list_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R2 )
          | ( A4 = B4 )
          | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A4 ) @ R2 ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R2 ) )
        | ( X3 = Y3 )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y3 @ X3 ) @ ( lexord_nat @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_448_lexord__linear,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X3: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] :
      ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B4 ) @ R2 )
          | ( A4 = B4 )
          | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ A4 ) @ R2 ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y3 ) @ ( lexord2841853652668343668at_nat @ R2 ) )
        | ( X3 = Y3 )
        | ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Y3 @ X3 ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ) ).

% lexord_linear
thf(fact_449_lexord__irreflexive,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,Xs: list_fm] :
      ( ! [X: fm] :
          ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ X ) @ R2 )
     => ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Xs ) @ ( lexord_fm @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_450_lexord__irreflexive,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,Xs: list_list_fm] :
      ( ! [X: list_fm] :
          ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ X ) @ R2 )
     => ~ ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Xs ) @ ( lexord_list_fm @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_451_lexord__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_452_lexord__irreflexive,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ).

% lexord_irreflexive
thf(fact_453_null__rec_I1_J,axiom,
    ! [X3: fm,Xs: list_fm] :
      ~ ( null_fm @ ( cons_fm @ X3 @ Xs ) ) ).

% null_rec(1)
thf(fact_454_subrelI,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,S: set_Pr7058068377845519745ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R2 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ S ) )
     => ( ord_le2055072402069232929ist_fm @ R2 @ S ) ) ).

% subrelI
thf(fact_455_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_456_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_457_finsert__fminus,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ A )
     => ( ( finser3446675674286072169ist_fm @ A2 @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ) )
        = A ) ) ).

% finsert_fminus
thf(fact_458_fminus__finsert__absorb,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( minus_8437252545254675822ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) )
        = A ) ) ).

% fminus_finsert_absorb
thf(fact_459_fminus__single__finsert,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) @ B )
     => ( ord_le2064643713053750439ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ B ) ) ) ).

% fminus_single_finsert
thf(fact_460_subset__code_I2_J,axiom,
    ! [A: set_Pr8693737435421807431at_nat,Ys2: list_P8469869581646625389at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ ( coset_5579199043005364954at_nat @ Ys2 ) )
      = ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Ys2 ) )
           => ~ ( member8206827879206165904at_nat @ X4 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_461_subset__code_I2_J,axiom,
    ! [A: set_fm,Ys2: list_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( coset_fm @ Ys2 ) )
      = ( ! [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ Ys2 ) )
           => ~ ( member_fm @ X4 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_462_lexord__partial__trans,axiom,
    ! [Xs: list_P8469869581646625389at_nat,R2: set_Pr553994874890374343at_nat,Ys2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
      ( ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ Z ) @ R2 )
             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Z ) @ R2 ) ) ) )
     => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys2 ) @ ( lexord5831005462426227802at_nat @ R2 ) )
       => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Ys2 @ Zs ) @ ( lexord5831005462426227802at_nat @ R2 ) )
         => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Zs ) @ ( lexord5831005462426227802at_nat @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_463_lexord__partial__trans,axiom,
    ! [Xs: list_fm,R2: set_Pr4463079037648049377_fm_fm,Ys2: list_fm,Zs: list_fm] :
      ( ! [X: fm,Y: fm,Z: fm] :
          ( ( member_fm @ X @ ( set_fm2 @ Xs ) )
         => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R2 )
           => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y @ Z ) @ R2 )
             => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Z ) @ R2 ) ) ) )
     => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( lexord_fm @ R2 ) )
       => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ys2 @ Zs ) @ ( lexord_fm @ R2 ) )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Zs ) @ ( lexord_fm @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_464_lexord__partial__trans,axiom,
    ! [Xs: list_list_fm,R2: set_Pr7058068377845519745ist_fm,Ys2: list_list_fm,Zs: list_list_fm] :
      ( ! [X: list_fm,Y: list_fm,Z: list_fm] :
          ( ( member_list_fm @ X @ ( set_list_fm2 @ Xs ) )
         => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R2 )
           => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y @ Z ) @ R2 )
             => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Z ) @ R2 ) ) ) )
     => ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Ys2 ) @ ( lexord_list_fm @ R2 ) )
       => ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Ys2 @ Zs ) @ ( lexord_list_fm @ R2 ) )
         => ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Zs ) @ ( lexord_list_fm @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_465_lexord__partial__trans,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Ys2: list_nat,Zs: list_nat] :
      ( ! [X: nat,Y: nat,Z: nat] :
          ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lexord_nat @ R2 ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R2 ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_466_lexord__partial__trans,axiom,
    ! [Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z ) @ R2 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ R2 ) ) ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( lexord2841853652668343668at_nat @ R2 ) )
       => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Zs ) @ ( lexord2841853652668343668at_nat @ R2 ) )
         => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs ) @ ( lexord2841853652668343668at_nat @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_467_diff__shunt__var,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ( minus_minus_set_fm @ X3 @ Y3 )
        = bot_bot_set_fm )
      = ( ord_less_eq_set_fm @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_468_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_469_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_470_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_471_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_472_Diff__eq__empty__iff,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ( minus_minus_set_fm @ A @ B )
        = bot_bot_set_fm )
      = ( ord_less_eq_set_fm @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_473_double__diff,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ( minus_minus_set_fm @ B @ ( minus_minus_set_fm @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_474_Diff__subset,axiom,
    ! [A: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_475_Diff__mono,axiom,
    ! [A: set_fm,C2: set_fm,D: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ C2 )
     => ( ( ord_less_eq_set_fm @ D @ B )
       => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B ) @ ( minus_minus_set_fm @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_476_zero__induct__lemma,axiom,
    ! [P2: nat > $o,K: nat,I: nat] :
      ( ( P2 @ K )
     => ( ! [N2: nat] :
            ( ( P2 @ ( suc @ N2 ) )
           => ( P2 @ N2 ) )
       => ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_477_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_478_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_479_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_480_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_481_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_482_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_483_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_484_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_485_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_486_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( sinter8084399408526207096m_rule @ S2 @ ( stl_Pr950425576149878629m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_487_pfsubset__finsert__iff,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ B ) )
      = ( ( ( fmembe3381613331217039976ist_fm @ X3 @ B )
         => ( ord_le8344328325585982387ist_fm @ A @ B ) )
        & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ B )
         => ( ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
             => ( ord_le8344328325585982387ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) @ B ) )
            & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
             => ( ord_le2064643713053750439ist_fm @ A @ B ) ) ) ) ) ) ).

% pfsubset_finsert_iff
thf(fact_488_fcard__Suc__fminus1,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( suc @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) ) )
        = ( fcard_6234007255513813746ist_fm @ A ) ) ) ).

% fcard_Suc_fminus1
thf(fact_489_fcard__fminus1__le,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] : ( ord_less_eq_nat @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) ) @ ( fcard_6234007255513813746ist_fm @ A ) ) ).

% fcard_fminus1_le
thf(fact_490_fcard__finsert,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fcard_6234007255513813746ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) )
      = ( suc @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) ) ) ) ).

% fcard_finsert
thf(fact_491_Diff__iff,axiom,
    ! [C: fm,A: set_fm,B: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A @ B ) )
      = ( ( member_fm @ C @ A )
        & ~ ( member_fm @ C @ B ) ) ) ).

% Diff_iff
thf(fact_492_Diff__iff,axiom,
    ! [C: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A @ B ) )
      = ( ( member8206827879206165904at_nat @ C @ A )
        & ~ ( member8206827879206165904at_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_493_DiffI,axiom,
    ! [C: fm,A: set_fm,B: set_fm] :
      ( ( member_fm @ C @ A )
     => ( ~ ( member_fm @ C @ B )
       => ( member_fm @ C @ ( minus_minus_set_fm @ A @ B ) ) ) ) ).

% DiffI
thf(fact_494_DiffI,axiom,
    ! [C: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ A )
     => ( ~ ( member8206827879206165904at_nat @ C @ B )
       => ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_495_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_496_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_1292763952458720599ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_1292763952458720599ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_497_sdrop__smap2,axiom,
    ! [N: nat,F: produc1996495991257130529ist_fm > rule > rule,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6133544317484809199e_rule @ F @ S1 @ S2 ) )
      = ( smap2_6133544317484809199e_rule @ F @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_498_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_3943848181979139747m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3943848181979139747m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_499_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_500_sdrop__smap2,axiom,
    ! [N: nat,F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_501_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_502_sdrop__smap2,axiom,
    ! [N: nat,F: produc1996495991257130529ist_fm > rule > produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_4725336451129573111ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_4725336451129573111ist_fm @ F @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_503_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_7773077075198283587ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_7773077075198283587ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_504_sdrop__smap2,axiom,
    ! [N: nat,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > rule,S1: stream8299795917829157543ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_7376420680649992259m_rule @ F @ S1 @ S2 ) )
      = ( smap2_7376420680649992259m_rule @ F @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_505_diff__commute,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_506_DiffD2,axiom,
    ! [C: fm,A: set_fm,B: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A @ B ) )
     => ~ ( member_fm @ C @ B ) ) ).

% DiffD2
thf(fact_507_DiffD2,axiom,
    ! [C: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A @ B ) )
     => ~ ( member8206827879206165904at_nat @ C @ B ) ) ).

% DiffD2
thf(fact_508_DiffD1,axiom,
    ! [C: fm,A: set_fm,B: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A @ B ) )
     => ( member_fm @ C @ A ) ) ).

% DiffD1
thf(fact_509_DiffD1,axiom,
    ! [C: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A @ B ) )
     => ( member8206827879206165904at_nat @ C @ A ) ) ).

% DiffD1
thf(fact_510_DiffE,axiom,
    ! [C: fm,A: set_fm,B: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A @ B ) )
     => ~ ( ( member_fm @ C @ A )
         => ( member_fm @ C @ B ) ) ) ).

% DiffE
thf(fact_511_DiffE,axiom,
    ! [C: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A @ B ) )
     => ~ ( ( member8206827879206165904at_nat @ C @ A )
         => ( member8206827879206165904at_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_512_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_513_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_514_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_515_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_516_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P2: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_517_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_518_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_519_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_520_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_521_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_522_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_523_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_524_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_525_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_526_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_527_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_528_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_529_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_530_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_531_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_532_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_533_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
         => ( P2 @ A4 @ B4 ) )
     => ( ! [A4: nat] : ( P2 @ A4 @ A4 )
       => ( ! [A4: nat,B4: nat] :
              ( ( P2 @ B4 @ A4 )
             => ( P2 @ A4 @ B4 ) )
         => ( P2 @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_534_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X6: nat] : ( P5 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [N4: nat] :
          ( ( P3 @ N4 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ~ ( P3 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_535_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_536_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_537_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_538_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_539_less__induct,axiom,
    ! [P2: nat > $o,A2: nat] :
      ( ! [X: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X )
             => ( P2 @ Y6 ) )
         => ( P2 @ X ) )
     => ( P2 @ A2 ) ) ).

% less_induct
thf(fact_540_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_541_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_542_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_543_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_544_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X3 @ X_12 ) ).

% gt_ex
thf(fact_545_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_546_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_547_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_548_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_549_verit__comp__simplify1_I3_J,axiom,
    ! [B6: extended_enat,A6: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ B6 @ A6 ) )
      = ( ord_le72135733267957522d_enat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_550_leD,axiom,
    ! [Y3: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y3 @ X3 )
     => ~ ( ord_less_set_fm @ X3 @ Y3 ) ) ).

% leD
thf(fact_551_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_552_leD,axiom,
    ! [Y3: extended_enat,X3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
     => ~ ( ord_le72135733267957522d_enat @ X3 @ Y3 ) ) ).

% leD
thf(fact_553_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_554_leI,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y3 )
     => ( ord_le2932123472753598470d_enat @ Y3 @ X3 ) ) ).

% leI
thf(fact_555_nless__le,axiom,
    ! [A2: set_fm,B2: set_fm] :
      ( ( ~ ( ord_less_set_fm @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_fm @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_556_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_557_nless__le,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ A2 @ B2 ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_558_antisym__conv1,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ~ ( ord_less_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_559_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_560_antisym__conv1,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y3 )
     => ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_561_antisym__conv2,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ~ ( ord_less_set_fm @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_562_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_563_antisym__conv2,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_564_less__le__not__le,axiom,
    ( ord_less_set_fm
    = ( ^ [X4: set_fm,Y5: set_fm] :
          ( ( ord_less_eq_set_fm @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_fm @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_565_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_566_less__le__not__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X4 @ Y5 )
          & ~ ( ord_le2932123472753598470d_enat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_567_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_568_not__le__imp__less,axiom,
    ! [Y3: extended_enat,X3: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
     => ( ord_le72135733267957522d_enat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_569_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A3: set_fm,B3: set_fm] :
          ( ( ord_less_set_fm @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_570_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_571_order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_572_order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [A3: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_573_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_574_order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_575_order_Ostrict__trans1,axiom,
    ! [A2: set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_set_fm @ B2 @ C )
       => ( ord_less_set_fm @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_576_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_577_order_Ostrict__trans1,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le72135733267957522d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_578_order_Ostrict__trans2,axiom,
    ! [A2: set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ord_less_set_fm @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_579_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_580_order_Ostrict__trans2,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le72135733267957522d_enat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_581_order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [A3: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A3 @ B3 )
          & ~ ( ord_less_eq_set_fm @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_582_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_583_order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ~ ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_584_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [B3: set_fm,A3: set_fm] :
          ( ( ord_less_set_fm @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_585_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_586_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_587_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [B3: set_fm,A3: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_588_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_589_dual__order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_590_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_fm,A2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ B2 @ A2 )
     => ( ( ord_less_set_fm @ C @ B2 )
       => ( ord_less_set_fm @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_591_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_592_dual__order_Ostrict__trans1,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( ord_le72135733267957522d_enat @ C @ B2 )
       => ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_593_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_fm,A2: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ B2 @ A2 )
     => ( ( ord_less_eq_set_fm @ C @ B2 )
       => ( ord_less_set_fm @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_594_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_595_dual__order_Ostrict__trans2,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ B2 )
       => ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_596_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [B3: set_fm,A3: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A3 )
          & ~ ( ord_less_eq_set_fm @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_597_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_598_dual__order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ~ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_599_order_Ostrict__implies__order,axiom,
    ! [A2: set_fm,B2: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B2 )
     => ( ord_less_eq_set_fm @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_600_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_601_order_Ostrict__implies__order,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ B2 )
     => ( ord_le2932123472753598470d_enat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_602_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_fm,A2: set_fm] :
      ( ( ord_less_set_fm @ B2 @ A2 )
     => ( ord_less_eq_set_fm @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_603_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_604_dual__order_Ostrict__implies__order,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A2 )
     => ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_605_order__le__less,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [X4: set_fm,Y5: set_fm] :
          ( ( ord_less_set_fm @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_606_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_607_order__le__less,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_608_order__less__le,axiom,
    ( ord_less_set_fm
    = ( ^ [X4: set_fm,Y5: set_fm] :
          ( ( ord_less_eq_set_fm @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_609_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_610_order__less__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X4: extended_enat,Y5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_611_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_612_linorder__not__le,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ X3 @ Y3 ) )
      = ( ord_le72135733267957522d_enat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_613_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_614_linorder__not__less,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X3 @ Y3 ) )
      = ( ord_le2932123472753598470d_enat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_615_order__less__imp__le,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y3 )
     => ( ord_less_eq_set_fm @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_616_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_617_order__less__imp__le,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Y3 )
     => ( ord_le2932123472753598470d_enat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_618_order__le__neq__trans,axiom,
    ! [A2: set_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_fm @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_619_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_620_order__le__neq__trans,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le72135733267957522d_enat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_621_order__neq__le__trans,axiom,
    ! [A2: set_fm,B2: set_fm] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_fm @ A2 @ B2 )
       => ( ord_less_set_fm @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_622_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_623_order__neq__le__trans,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2 != B2 )
     => ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
       => ( ord_le72135733267957522d_enat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_624_order__le__less__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_set_fm @ Y3 @ Z3 )
       => ( ord_less_set_fm @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_625_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_626_order__le__less__trans,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_le72135733267957522d_enat @ Y3 @ Z3 )
       => ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_627_order__less__le__trans,axiom,
    ! [X3: set_fm,Y3: set_fm,Z3: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y3 )
     => ( ( ord_less_eq_set_fm @ Y3 @ Z3 )
       => ( ord_less_set_fm @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_628_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_629_order__less__le__trans,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Y3 )
     => ( ( ord_le2932123472753598470d_enat @ Y3 @ Z3 )
       => ( ord_le72135733267957522d_enat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_630_order__le__less__subst1,axiom,
    ! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_631_order__le__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_632_order__le__less__subst1,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_le72135733267957522d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_633_order__le__less__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_634_order__le__less__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_635_order__le__less__subst2,axiom,
    ! [A2: set_fm,B2: set_fm,F: set_fm > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_fm @ A2 @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_636_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_637_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_638_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_639_order__le__less__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > set_fm,C: set_fm] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_less_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_640_order__le__less__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_641_order__le__less__subst2,axiom,
    ! [A2: extended_enat,B2: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_642_order__less__le__subst1,axiom,
    ! [A2: set_fm,F: set_fm > set_fm,B2: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_643_order__less__le__subst1,axiom,
    ! [A2: nat,F: set_fm > nat,B2: set_fm,C: set_fm] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_644_order__less__le__subst1,axiom,
    ! [A2: extended_enat,F: set_fm > extended_enat,B2: set_fm,C: set_fm] :
      ( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_fm @ B2 @ C )
       => ( ! [X: set_fm,Y: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_645_order__less__le__subst1,axiom,
    ! [A2: set_fm,F: nat > set_fm,B2: nat,C: nat] :
      ( ( ord_less_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_646_order__less__le__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_647_order__less__le__subst1,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B2: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_648_order__less__le__subst1,axiom,
    ! [A2: set_fm,F: extended_enat > set_fm,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_set_fm @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_649_order__less__le__subst1,axiom,
    ! [A2: nat,F: extended_enat > nat,B2: extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_650_order__less__le__subst1,axiom,
    ! [A2: extended_enat,F: extended_enat > extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ! [X: extended_enat,Y: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X @ Y )
             => ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_651_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_fm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_652_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_653_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B2 ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_le72135733267957522d_enat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_654_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_655_linorder__le__less__linear,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
      | ( ord_le72135733267957522d_enat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_656_order__le__imp__less__or__eq,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_less_set_fm @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_657_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_658_order__le__imp__less__or__eq,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_le72135733267957522d_enat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_659_bot_Onot__eq__extremum,axiom,
    ! [A2: fset_P661503646757059847ist_fm] :
      ( ( A2 != bot_bo2367426573206113139ist_fm )
      = ( ord_le8344328325585982387ist_fm @ bot_bo2367426573206113139ist_fm @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_660_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_661_bot_Oextremum__strict,axiom,
    ! [A2: fset_P661503646757059847ist_fm] :
      ~ ( ord_le8344328325585982387ist_fm @ A2 @ bot_bo2367426573206113139ist_fm ) ).

% bot.extremum_strict
thf(fact_662_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_663_pfsubsetD,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A )
       => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% pfsubsetD
thf(fact_664_not__pfsubset__fempty,axiom,
    ! [A: fset_P661503646757059847ist_fm] :
      ~ ( ord_le8344328325585982387ist_fm @ A @ bot_bo2367426573206113139ist_fm ) ).

% not_pfsubset_fempty
thf(fact_665_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_666_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3865418324013074943m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_667_fset__card__induct,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,S4: fset_P661503646757059847ist_fm] :
      ( ( P2 @ bot_bo2367426573206113139ist_fm )
     => ( ! [S5: fset_P661503646757059847ist_fm,T3: fset_P661503646757059847ist_fm] :
            ( ( ( suc @ ( fcard_6234007255513813746ist_fm @ S5 ) )
              = ( fcard_6234007255513813746ist_fm @ T3 ) )
           => ( ( P2 @ S5 )
             => ( P2 @ T3 ) ) )
       => ( P2 @ S4 ) ) ) ).

% fset_card_induct
thf(fact_668_fcard__finsert__le,axiom,
    ! [A: fset_P661503646757059847ist_fm,X3: produc1996495991257130529ist_fm] : ( ord_less_eq_nat @ ( fcard_6234007255513813746ist_fm @ A ) @ ( fcard_6234007255513813746ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) ) ) ).

% fcard_finsert_le
thf(fact_669_pfsubset__imp__ex__fmem,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A @ B )
     => ? [B4: produc1996495991257130529ist_fm] : ( fmembe3381613331217039976ist_fm @ B4 @ ( minus_8437252545254675822ist_fm @ B @ A ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_670_fcard__finsert__disjoint,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( fcard_6234007255513813746ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) )
        = ( suc @ ( fcard_6234007255513813746ist_fm @ A ) ) ) ) ).

% fcard_finsert_disjoint
thf(fact_671_fcard__finsert__if,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
       => ( ( fcard_6234007255513813746ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) )
          = ( fcard_6234007255513813746ist_fm @ A ) ) )
      & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
       => ( ( fcard_6234007255513813746ist_fm @ ( finser3446675674286072169ist_fm @ X3 @ A ) )
          = ( suc @ ( fcard_6234007255513813746ist_fm @ A ) ) ) ) ) ).

% fcard_finsert_if
thf(fact_672_fset__linorder__max__induct,axiom,
    ! [P2: fset_nat > $o,S4: fset_nat] :
      ( ( P2 @ bot_bot_fset_nat )
     => ( ! [X: nat,S5: fset_nat] :
            ( ! [Y6: nat] :
                ( ( fmember_nat @ Y6 @ S5 )
               => ( ord_less_nat @ Y6 @ X ) )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finsert_nat @ X @ S5 ) ) ) )
       => ( P2 @ S4 ) ) ) ).

% fset_linorder_max_induct
thf(fact_673_fset__linorder__min__induct,axiom,
    ! [P2: fset_nat > $o,S4: fset_nat] :
      ( ( P2 @ bot_bot_fset_nat )
     => ( ! [X: nat,S5: fset_nat] :
            ( ! [Y6: nat] :
                ( ( fmember_nat @ Y6 @ S5 )
               => ( ord_less_nat @ X @ Y6 ) )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finsert_nat @ X @ S5 ) ) ) )
       => ( P2 @ S4 ) ) ) ).

% fset_linorder_min_induct
thf(fact_674_fcard__fminus__finsert,axiom,
    ! [A2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ A )
     => ( ~ ( fmembe3381613331217039976ist_fm @ A2 @ B )
       => ( ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ A2 @ B ) ) )
          = ( minus_minus_nat @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ B ) ) @ one_one_nat ) ) ) ) ).

% fcard_fminus_finsert
thf(fact_675_fcard__fminus__fsingleton,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) )
        = ( minus_minus_nat @ ( fcard_6234007255513813746ist_fm @ A ) @ one_one_nat ) ) ) ).

% fcard_fminus_fsingleton
thf(fact_676_fcard__fminus__fsingleton__if,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
       => ( ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) )
          = ( minus_minus_nat @ ( fcard_6234007255513813746ist_fm @ A ) @ one_one_nat ) ) )
      & ( ~ ( fmembe3381613331217039976ist_fm @ X3 @ A )
       => ( ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) )
          = ( fcard_6234007255513813746ist_fm @ A ) ) ) ) ).

% fcard_fminus_fsingleton_if
thf(fact_677_minf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ~ ( ord_less_eq_nat @ T2 @ X7 ) ) ).

% minf(8)
thf(fact_678_minf_I8_J,axiom,
    ! [T2: extended_enat] :
    ? [Z: extended_enat] :
    ! [X7: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X7 @ Z )
     => ~ ( ord_le2932123472753598470d_enat @ T2 @ X7 ) ) ).

% minf(8)
thf(fact_679_minf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ X7 @ Z )
     => ( ord_less_eq_nat @ X7 @ T2 ) ) ).

% minf(6)
thf(fact_680_minf_I6_J,axiom,
    ! [T2: extended_enat] :
    ? [Z: extended_enat] :
    ! [X7: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X7 @ Z )
     => ( ord_le2932123472753598470d_enat @ X7 @ T2 ) ) ).

% minf(6)
thf(fact_681_pinf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ( ord_less_eq_nat @ T2 @ X7 ) ) ).

% pinf(8)
thf(fact_682_pinf_I8_J,axiom,
    ! [T2: extended_enat] :
    ? [Z: extended_enat] :
    ! [X7: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X7 )
     => ( ord_le2932123472753598470d_enat @ T2 @ X7 ) ) ).

% pinf(8)
thf(fact_683_psubsetI,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% psubsetI
thf(fact_684_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_685_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_686_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_687_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_688_linorder__neqE__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_689_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P2 @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P2 @ M2 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_690_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P2 @ M2 ) )
         => ( P2 @ N2 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_691_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_692_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( ( ord_less_nat @ S @ T2 )
     => ( S != T2 ) ) ).

% less_not_refl3
thf(fact_693_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_694_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_695_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_696_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
          ( ( ord_less_set_fm @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_697_subset__psubset__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ B @ C2 )
       => ( ord_less_set_fm @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_698_subset__not__subset__eq,axiom,
    ( ord_less_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B5 )
          & ~ ( ord_less_eq_set_fm @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_699_psubset__subset__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_set_fm @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_700_psubset__imp__subset,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_701_psubset__eq,axiom,
    ( ord_less_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_702_psubsetE,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ~ ( ( ord_less_eq_set_fm @ A @ B )
         => ( ord_less_eq_set_fm @ B @ A ) ) ) ).

% psubsetE
thf(fact_703_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_704_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I2: nat] :
            ( ( J2
              = ( suc @ I2 ) )
           => ( P2 @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( P2 @ ( suc @ I2 ) )
               => ( P2 @ I2 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_705_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( ord_less_nat @ J @ K2 )
               => ( ( P2 @ I2 @ J )
                 => ( ( P2 @ J @ K2 )
                   => ( P2 @ I2 @ K2 ) ) ) ) )
         => ( P2 @ I @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_706_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ J2 @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_707_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_708_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_709_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_710_All__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_711_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_712_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_713_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_714_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_715_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_716_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_717_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( K
             != ( suc @ J ) ) ) ) ).

% Suc_lessE
thf(fact_718_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_719_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( K
               != ( suc @ J ) ) ) ) ) ).

% Nat.lessE
thf(fact_720_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J2: nat] :
      ( ! [I2: nat,J: nat] :
          ( ( ord_less_nat @ I2 @ J )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
     => ( ( ord_less_eq_nat @ I @ J2 )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_721_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_722_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_723_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
          | ( M5 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_724_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_725_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
          & ( M5 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_726_psubset__imp__ex__mem,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ? [B4: fm] : ( member_fm @ B4 @ ( minus_minus_set_fm @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_727_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A @ B )
     => ? [B4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B4 @ ( minus_8321449233255521966at_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_728_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_729_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_730_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_731_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_732_dec__induct,axiom,
    ! [I: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P2 @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P2 @ N2 )
                 => ( P2 @ ( suc @ N2 ) ) ) ) )
         => ( P2 @ J2 ) ) ) ) ).

% dec_induct
thf(fact_733_inc__induct,axiom,
    ! [I: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P2 @ J2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P2 @ ( suc @ N2 ) )
                 => ( P2 @ N2 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_734_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_735_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_736_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_737_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_738_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_739_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_740_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_741_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_742_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_743_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_744_fcard__fminus2__less,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ( fmembe3381613331217039976ist_fm @ Y3 @ A )
       => ( ord_less_nat @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) @ ( finser3446675674286072169ist_fm @ Y3 @ bot_bo2367426573206113139ist_fm ) ) ) @ ( fcard_6234007255513813746ist_fm @ A ) ) ) ) ).

% fcard_fminus2_less
thf(fact_745_fcard__fminus1__less,axiom,
    ! [X3: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X3 @ A )
     => ( ord_less_nat @ ( fcard_6234007255513813746ist_fm @ ( minus_8437252545254675822ist_fm @ A @ ( finser3446675674286072169ist_fm @ X3 @ bot_bo2367426573206113139ist_fm ) ) ) @ ( fcard_6234007255513813746ist_fm @ A ) ) ) ).

% fcard_fminus1_less
thf(fact_746_pinf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z: nat] :
    ! [X7: nat] :
      ( ( ord_less_nat @ Z @ X7 )
     => ~ ( ord_less_eq_nat @ X7 @ T2 ) ) ).

% pinf(6)
thf(fact_747_pinf_I6_J,axiom,
    ! [T2: extended_enat] :
    ? [Z: extended_enat] :
    ! [X7: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ X7 )
     => ~ ( ord_le2932123472753598470d_enat @ X7 @ T2 ) ) ).

% pinf(6)
thf(fact_748_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_749_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_750_le__numeral__extra_I4_J,axiom,
    ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).

% le_numeral_extra(4)
thf(fact_751_complete__interval,axiom,
    ! [A2: nat,B2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( P2 @ A2 )
       => ( ~ ( P2 @ B2 )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A2 @ C3 )
              & ( ord_less_eq_nat @ C3 @ B2 )
              & ! [X7: nat] :
                  ( ( ( ord_less_eq_nat @ A2 @ X7 )
                    & ( ord_less_nat @ X7 @ C3 ) )
                 => ( P2 @ X7 ) )
              & ! [D3: nat] :
                  ( ! [X: nat] :
                      ( ( ( ord_less_eq_nat @ A2 @ X )
                        & ( ord_less_nat @ X @ D3 ) )
                     => ( P2 @ X ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_752_complete__interval,axiom,
    ! [A2: extended_enat,B2: extended_enat,P2: extended_enat > $o] :
      ( ( ord_le72135733267957522d_enat @ A2 @ B2 )
     => ( ( P2 @ A2 )
       => ( ~ ( P2 @ B2 )
         => ? [C3: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ A2 @ C3 )
              & ( ord_le2932123472753598470d_enat @ C3 @ B2 )
              & ! [X7: extended_enat] :
                  ( ( ( ord_le2932123472753598470d_enat @ A2 @ X7 )
                    & ( ord_le72135733267957522d_enat @ X7 @ C3 ) )
                 => ( P2 @ X7 ) )
              & ! [D3: extended_enat] :
                  ( ! [X: extended_enat] :
                      ( ( ( ord_le2932123472753598470d_enat @ A2 @ X )
                        & ( ord_le72135733267957522d_enat @ X @ D3 ) )
                     => ( P2 @ X ) )
                 => ( ord_le2932123472753598470d_enat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_753_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_754_psubsetD,axiom,
    ! [A: set_fm,B: set_fm,C: fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ( member_fm @ C @ A )
       => ( member_fm @ C @ B ) ) ) ).

% psubsetD
thf(fact_755_psubsetD,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A @ B )
     => ( ( member8206827879206165904at_nat @ C @ A )
       => ( member8206827879206165904at_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_756_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa @ X3 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X3 @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X3 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_757_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( enumerate_fm @ N @ ( cons_fm @ X3 @ Xs ) )
      = ( cons_P9056855792931809079nat_fm @ ( product_Pair_nat_fm @ N @ X3 ) @ ( enumerate_fm @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_758_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X3 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_759_in__measures_I2_J,axiom,
    ! [X3: list_fm,Y3: list_fm,F: list_fm > nat,Fs: list_list_fm_nat] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ ( cons_list_fm_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X3 )
            = ( F @ Y3 ) )
          & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_760_in__measures_I2_J,axiom,
    ! [X3: nat,Y3: nat,F: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X3 )
            = ( F @ Y3 ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_761_in__measures_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X3 )
            = ( F @ Y3 ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_762_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S ) @ ( product_Pair_nat_nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_763_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_764_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_765_le__zero__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% le_zero_eq
thf(fact_766_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_767_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_768_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_769_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_770_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_771_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_772_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_773_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_774_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_775_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_776_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_777_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_778_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_779_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_780_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_781_le__numeral__extra_I3_J,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).

% le_numeral_extra(3)
thf(fact_782_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X3
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_783_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_784_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_785_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_786_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_787_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_788_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P2 @ N2 )
           => ( P2 @ ( suc @ N2 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_789_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N: nat] :
      ( ! [X: nat] : ( P2 @ X @ zero_zero_nat )
     => ( ! [Y: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y ) )
       => ( ! [X: nat,Y: nat] :
              ( ( P2 @ X @ Y )
             => ( P2 @ ( suc @ X ) @ ( suc @ Y ) ) )
         => ( P2 @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_790_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N2: nat] :
            ( ( P2 @ ( suc @ N2 ) )
           => ( P2 @ N2 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_791_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_792_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_793_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_794_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_795_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_796_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_797_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_798_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_799_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_800_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_801_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P2 @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P2 @ M2 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_802_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_803_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_804_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_805_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_806_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_807_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_808_sdrop_Osimps_I1_J,axiom,
    ! [S: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ zero_zero_nat @ S )
      = S ) ).

% sdrop.simps(1)
thf(fact_809_sdrop_Osimps_I1_J,axiom,
    ! [S: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ zero_zero_nat @ S )
      = S ) ).

% sdrop.simps(1)
thf(fact_810_sdrop_Osimps_I1_J,axiom,
    ! [S: stream_rule] :
      ( ( sdrop_rule @ zero_zero_nat @ S )
      = S ) ).

% sdrop.simps(1)
thf(fact_811_sdrop_Osimps_I1_J,axiom,
    ! [S: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ zero_zero_nat @ S )
      = S ) ).

% sdrop.simps(1)
thf(fact_812_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_813_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_814_zero__le,axiom,
    ! [X3: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X3 ) ).

% zero_le
thf(fact_815_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_816_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_817_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_818_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_819_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_820_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_821_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_822_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_823_fcard__fempty,axiom,
    ( ( fcard_6234007255513813746ist_fm @ bot_bo2367426573206113139ist_fm )
    = zero_zero_nat ) ).

% fcard_fempty
thf(fact_824_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_825_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_826_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P2 @ N2 )
               => ( P2 @ ( suc @ N2 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_827_measures__less,axiom,
    ! [F: list_fm > nat,X3: list_fm,Y3: list_fm,Fs: list_list_fm_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ ( cons_list_fm_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_828_measures__less,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_829_measures__less,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_830_measures__lesseq,axiom,
    ! [F: list_fm > nat,X3: list_fm,Y3: list_fm,Fs: list_list_fm_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ Fs ) )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ ( cons_list_fm_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_831_measures__lesseq,axiom,
    ! [F: nat > nat,X3: nat,Y3: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_832_measures__lesseq,axiom,
    ! [F: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ Fs ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_833_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_834_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_835_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_836_not__one__le__zero,axiom,
    ~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_le_zero
thf(fact_837_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_838_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_839_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_840_zero__less__one__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one_class.zero_le_one
thf(fact_841_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S ) @ ( product_Pair_nat_nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_842_exists__least__lemma,axiom,
    ! [P2: nat > $o] :
      ( ~ ( P2 @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P2 @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P2 @ N2 )
            & ( P2 @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_843_nth__Cons__pos,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
        = ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_844_set__removeAll,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( set_fm2 @ ( removeAll_fm @ X3 @ Xs ) )
      = ( minus_minus_set_fm @ ( set_fm2 @ Xs ) @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ) ).

% set_removeAll
thf(fact_845_Cons__replicate__eq,axiom,
    ! [X3: fm,Xs: list_fm,N: nat,Y3: fm] :
      ( ( ( cons_fm @ X3 @ Xs )
        = ( replicate_fm @ N @ Y3 ) )
      = ( ( X3 = Y3 )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs
          = ( replicate_fm @ ( minus_minus_nat @ N @ one_one_nat ) @ X3 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_846_insert__iff,axiom,
    ! [A2: fm,B2: fm,A: set_fm] :
      ( ( member_fm @ A2 @ ( insert_fm2 @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_fm @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_847_insert__iff,axiom,
    ! [A2: produc859450856879609959at_nat,B2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member8206827879206165904at_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_848_insertCI,axiom,
    ! [A2: fm,B: set_fm,B2: fm] :
      ( ( ~ ( member_fm @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_fm @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).

% insertCI
thf(fact_849_insertCI,axiom,
    ! [A2: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat,B2: produc859450856879609959at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_850_singletonI,axiom,
    ! [A2: fm] : ( member_fm @ A2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ).

% singletonI
thf(fact_851_singletonI,axiom,
    ! [A2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ A2 @ bot_bo5327735625951526323at_nat ) ) ).

% singletonI
thf(fact_852_insert__subset,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( insert5050368324300391991at_nat @ X3 @ A ) @ B )
      = ( ( member8206827879206165904at_nat @ X3 @ B )
        & ( ord_le3000389064537975527at_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_853_insert__subset,axiom,
    ! [X3: fm,A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ ( insert_fm2 @ X3 @ A ) @ B )
      = ( ( member_fm @ X3 @ B )
        & ( ord_less_eq_set_fm @ A @ B ) ) ) ).

% insert_subset
thf(fact_854_Diff__insert0,axiom,
    ! [X3: fm,A: set_fm,B: set_fm] :
      ( ~ ( member_fm @ X3 @ A )
     => ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ X3 @ B ) )
        = ( minus_minus_set_fm @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_855_Diff__insert0,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ A )
     => ( ( minus_8321449233255521966at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ B ) )
        = ( minus_8321449233255521966at_nat @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_856_insert__Diff1,axiom,
    ! [X3: fm,B: set_fm,A: set_fm] :
      ( ( member_fm @ X3 @ B )
     => ( ( minus_minus_set_fm @ ( insert_fm2 @ X3 @ A ) @ B )
        = ( minus_minus_set_fm @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_857_insert__Diff1,axiom,
    ! [X3: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ B )
     => ( ( minus_8321449233255521966at_nat @ ( insert5050368324300391991at_nat @ X3 @ A ) @ B )
        = ( minus_8321449233255521966at_nat @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_858_singleton__insert__inj__eq_H,axiom,
    ! [A2: fm,A: set_fm,B2: fm] :
      ( ( ( insert_fm2 @ A2 @ A )
        = ( insert_fm2 @ B2 @ bot_bot_set_fm ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_859_singleton__insert__inj__eq,axiom,
    ! [B2: fm,A2: fm,A: set_fm] :
      ( ( ( insert_fm2 @ B2 @ bot_bot_set_fm )
        = ( insert_fm2 @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_860_list_Osimps_I15_J,axiom,
    ! [X21: fm,X22: list_fm] :
      ( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
      = ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_861_nth__Cons__Suc,axiom,
    ! [X3: fm,Xs: list_fm,N: nat] :
      ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_fm @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_862_nth__Cons__0,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_863_in__set__replicate,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,Y3: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( replic6713244433751818279at_nat @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_864_in__set__replicate,axiom,
    ! [X3: fm,N: nat,Y3: fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ ( replicate_fm @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_865_Bex__set__replicate,axiom,
    ! [N: nat,A2: fm,P2: fm > $o] :
      ( ( ? [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
            & ( P2 @ X4 ) ) )
      = ( ( P2 @ A2 )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_866_Ball__set__replicate,axiom,
    ! [N: nat,A2: fm,P2: fm > $o] :
      ( ( ! [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ ( replicate_fm @ N @ A2 ) ) )
           => ( P2 @ X4 ) ) )
      = ( ( P2 @ A2 )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_867_List_Oset__insert,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( set_fm2 @ ( insert_fm @ X3 @ Xs ) )
      = ( insert_fm2 @ X3 @ ( set_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_868_set__replicate,axiom,
    ! [N: nat,X3: fm] :
      ( ( N != zero_zero_nat )
     => ( ( set_fm2 @ ( replicate_fm @ N @ X3 ) )
        = ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ) ).

% set_replicate
thf(fact_869_insert__subsetI,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,X8: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ A )
     => ( ( ord_le3000389064537975527at_nat @ X8 @ A )
       => ( ord_le3000389064537975527at_nat @ ( insert5050368324300391991at_nat @ X3 @ X8 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_870_insert__subsetI,axiom,
    ! [X3: fm,A: set_fm,X8: set_fm] :
      ( ( member_fm @ X3 @ A )
     => ( ( ord_less_eq_set_fm @ X8 @ A )
       => ( ord_less_eq_set_fm @ ( insert_fm2 @ X3 @ X8 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_871_subset__insertI2,axiom,
    ! [A: set_fm,B: set_fm,B2: fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_872_subset__insertI,axiom,
    ! [B: set_fm,A2: fm] : ( ord_less_eq_set_fm @ B @ ( insert_fm2 @ A2 @ B ) ) ).

% subset_insertI
thf(fact_873_subset__insert,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ A )
     => ( ( ord_le3000389064537975527at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ B ) )
        = ( ord_le3000389064537975527at_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_874_subset__insert,axiom,
    ! [X3: fm,A: set_fm,B: set_fm] :
      ( ~ ( member_fm @ X3 @ A )
     => ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X3 @ B ) )
        = ( ord_less_eq_set_fm @ A @ B ) ) ) ).

% subset_insert
thf(fact_875_insert__mono,axiom,
    ! [C2: set_fm,D: set_fm,A2: fm] :
      ( ( ord_less_eq_set_fm @ C2 @ D )
     => ( ord_less_eq_set_fm @ ( insert_fm2 @ A2 @ C2 ) @ ( insert_fm2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_876_insert__Diff__if,axiom,
    ! [X3: fm,B: set_fm,A: set_fm] :
      ( ( ( member_fm @ X3 @ B )
       => ( ( minus_minus_set_fm @ ( insert_fm2 @ X3 @ A ) @ B )
          = ( minus_minus_set_fm @ A @ B ) ) )
      & ( ~ ( member_fm @ X3 @ B )
       => ( ( minus_minus_set_fm @ ( insert_fm2 @ X3 @ A ) @ B )
          = ( insert_fm2 @ X3 @ ( minus_minus_set_fm @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_877_insert__Diff__if,axiom,
    ! [X3: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( ( member8206827879206165904at_nat @ X3 @ B )
       => ( ( minus_8321449233255521966at_nat @ ( insert5050368324300391991at_nat @ X3 @ A ) @ B )
          = ( minus_8321449233255521966at_nat @ A @ B ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X3 @ B )
       => ( ( minus_8321449233255521966at_nat @ ( insert5050368324300391991at_nat @ X3 @ A ) @ B )
          = ( insert5050368324300391991at_nat @ X3 @ ( minus_8321449233255521966at_nat @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_878_mk__disjoint__insert,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm @ A2 @ A )
     => ? [B7: set_fm] :
          ( ( A
            = ( insert_fm2 @ A2 @ B7 ) )
          & ~ ( member_fm @ A2 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_879_mk__disjoint__insert,axiom,
    ! [A2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ A )
     => ? [B7: set_Pr8693737435421807431at_nat] :
          ( ( A
            = ( insert5050368324300391991at_nat @ A2 @ B7 ) )
          & ~ ( member8206827879206165904at_nat @ A2 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_880_insert__eq__iff,axiom,
    ! [A2: fm,A: set_fm,B2: fm,B: set_fm] :
      ( ~ ( member_fm @ A2 @ A )
     => ( ~ ( member_fm @ B2 @ B )
       => ( ( ( insert_fm2 @ A2 @ A )
            = ( insert_fm2 @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: set_fm] :
                  ( ( A
                    = ( insert_fm2 @ B2 @ C4 ) )
                  & ~ ( member_fm @ B2 @ C4 )
                  & ( B
                    = ( insert_fm2 @ A2 @ C4 ) )
                  & ~ ( member_fm @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_881_insert__eq__iff,axiom,
    ! [A2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B2: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ A2 @ A )
     => ( ~ ( member8206827879206165904at_nat @ B2 @ B )
       => ( ( ( insert5050368324300391991at_nat @ A2 @ A )
            = ( insert5050368324300391991at_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: set_Pr8693737435421807431at_nat] :
                  ( ( A
                    = ( insert5050368324300391991at_nat @ B2 @ C4 ) )
                  & ~ ( member8206827879206165904at_nat @ B2 @ C4 )
                  & ( B
                    = ( insert5050368324300391991at_nat @ A2 @ C4 ) )
                  & ~ ( member8206827879206165904at_nat @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_882_insert__absorb,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm @ A2 @ A )
     => ( ( insert_fm2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_883_insert__absorb,axiom,
    ! [A2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ A )
     => ( ( insert5050368324300391991at_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_884_insert__ident,axiom,
    ! [X3: fm,A: set_fm,B: set_fm] :
      ( ~ ( member_fm @ X3 @ A )
     => ( ~ ( member_fm @ X3 @ B )
       => ( ( ( insert_fm2 @ X3 @ A )
            = ( insert_fm2 @ X3 @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_885_insert__ident,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ A )
     => ( ~ ( member8206827879206165904at_nat @ X3 @ B )
       => ( ( ( insert5050368324300391991at_nat @ X3 @ A )
            = ( insert5050368324300391991at_nat @ X3 @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_886_Set_Oset__insert,axiom,
    ! [X3: fm,A: set_fm] :
      ( ( member_fm @ X3 @ A )
     => ~ ! [B7: set_fm] :
            ( ( A
              = ( insert_fm2 @ X3 @ B7 ) )
           => ( member_fm @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_887_Set_Oset__insert,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ A )
     => ~ ! [B7: set_Pr8693737435421807431at_nat] :
            ( ( A
              = ( insert5050368324300391991at_nat @ X3 @ B7 ) )
           => ( member8206827879206165904at_nat @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_888_insertI2,axiom,
    ! [A2: fm,B: set_fm,B2: fm] :
      ( ( member_fm @ A2 @ B )
     => ( member_fm @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).

% insertI2
thf(fact_889_insertI2,axiom,
    ! [A2: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat,B2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ B )
     => ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_890_insertI1,axiom,
    ! [A2: fm,B: set_fm] : ( member_fm @ A2 @ ( insert_fm2 @ A2 @ B ) ) ).

% insertI1
thf(fact_891_insertI1,axiom,
    ! [A2: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat] : ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_892_insertE,axiom,
    ! [A2: fm,B2: fm,A: set_fm] :
      ( ( member_fm @ A2 @ ( insert_fm2 @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_fm @ A2 @ A ) ) ) ).

% insertE
thf(fact_893_insertE,axiom,
    ! [A2: produc859450856879609959at_nat,B2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ ( insert5050368324300391991at_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member8206827879206165904at_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_894_singletonD,axiom,
    ! [B2: fm,A2: fm] :
      ( ( member_fm @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_895_singletonD,axiom,
    ! [B2: produc859450856879609959at_nat,A2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B2 @ ( insert5050368324300391991at_nat @ A2 @ bot_bo5327735625951526323at_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_896_singleton__iff,axiom,
    ! [B2: fm,A2: fm] :
      ( ( member_fm @ B2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_897_singleton__iff,axiom,
    ! [B2: produc859450856879609959at_nat,A2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B2 @ ( insert5050368324300391991at_nat @ A2 @ bot_bo5327735625951526323at_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_898_set__replicate__conv__if,axiom,
    ! [N: nat,X3: fm] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_fm2 @ ( replicate_fm @ N @ X3 ) )
          = bot_bot_set_fm ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_fm2 @ ( replicate_fm @ N @ X3 ) )
          = ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ) ) ).

% set_replicate_conv_if
thf(fact_899_set__replicate__Suc,axiom,
    ! [N: nat,X3: fm] :
      ( ( set_fm2 @ ( replicate_fm @ ( suc @ N ) @ X3 ) )
      = ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ).

% set_replicate_Suc
thf(fact_900_replicate__Suc,axiom,
    ! [N: nat,X3: fm] :
      ( ( replicate_fm @ ( suc @ N ) @ X3 )
      = ( cons_fm @ X3 @ ( replicate_fm @ N @ X3 ) ) ) ).

% replicate_Suc
thf(fact_901_subset__singletonD,axiom,
    ! [A: set_fm,X3: fm] :
      ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) )
     => ( ( A = bot_bot_set_fm )
        | ( A
          = ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) ) ) ).

% subset_singletonD
thf(fact_902_subset__singleton__iff,axiom,
    ! [X8: set_fm,A2: fm] :
      ( ( ord_less_eq_set_fm @ X8 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
      = ( ( X8 = bot_bot_set_fm )
        | ( X8
          = ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ) ).

% subset_singleton_iff
thf(fact_903_insert__Diff,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm @ A2 @ A )
     => ( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_904_insert__Diff,axiom,
    ! [A2: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A2 @ A )
     => ( ( insert5050368324300391991at_nat @ A2 @ ( minus_8321449233255521966at_nat @ A @ ( insert5050368324300391991at_nat @ A2 @ bot_bo5327735625951526323at_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_905_Diff__insert__absorb,axiom,
    ! [X3: fm,A: set_fm] :
      ( ~ ( member_fm @ X3 @ A )
     => ( ( minus_minus_set_fm @ ( insert_fm2 @ X3 @ A ) @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_906_Diff__insert__absorb,axiom,
    ! [X3: produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ A )
     => ( ( minus_8321449233255521966at_nat @ ( insert5050368324300391991at_nat @ X3 @ A ) @ ( insert5050368324300391991at_nat @ X3 @ bot_bo5327735625951526323at_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_907_subset__Diff__insert,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ ( minus_8321449233255521966at_nat @ B @ ( insert5050368324300391991at_nat @ X3 @ C2 ) ) )
      = ( ( ord_le3000389064537975527at_nat @ A @ ( minus_8321449233255521966at_nat @ B @ C2 ) )
        & ~ ( member8206827879206165904at_nat @ X3 @ A ) ) ) ).

% subset_Diff_insert
thf(fact_908_subset__Diff__insert,axiom,
    ! [A: set_fm,B: set_fm,X3: fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ ( insert_fm2 @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_fm @ A @ ( minus_minus_set_fm @ B @ C2 ) )
        & ~ ( member_fm @ X3 @ A ) ) ) ).

% subset_Diff_insert
thf(fact_909_insert__code_I2_J,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( insert_fm2 @ X3 @ ( coset_fm @ Xs ) )
      = ( coset_fm @ ( removeAll_fm @ X3 @ Xs ) ) ) ).

% insert_code(2)
thf(fact_910_subset__insert__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ B ) )
      = ( ( ( member8206827879206165904at_nat @ X3 @ A )
         => ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ bot_bo5327735625951526323at_nat ) ) @ B ) )
        & ( ~ ( member8206827879206165904at_nat @ X3 @ A )
         => ( ord_le3000389064537975527at_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_911_subset__insert__iff,axiom,
    ! [A: set_fm,X3: fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X3 @ B ) )
      = ( ( ( member_fm @ X3 @ A )
         => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) @ B ) )
        & ( ~ ( member_fm @ X3 @ A )
         => ( ord_less_eq_set_fm @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_912_Diff__single__insert,axiom,
    ! [A: set_fm,X3: fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) @ B )
     => ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X3 @ B ) ) ) ).

% Diff_single_insert
thf(fact_913_nth__Cons_H,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
          = X3 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
          = ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_914_psubset__insert__iff,axiom,
    ! [A: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat,B: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ B ) )
      = ( ( ( member8206827879206165904at_nat @ X3 @ B )
         => ( ord_le6428140832669894131at_nat @ A @ B ) )
        & ( ~ ( member8206827879206165904at_nat @ X3 @ B )
         => ( ( ( member8206827879206165904at_nat @ X3 @ A )
             => ( ord_le6428140832669894131at_nat @ ( minus_8321449233255521966at_nat @ A @ ( insert5050368324300391991at_nat @ X3 @ bot_bo5327735625951526323at_nat ) ) @ B ) )
            & ( ~ ( member8206827879206165904at_nat @ X3 @ A )
             => ( ord_le3000389064537975527at_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_915_psubset__insert__iff,axiom,
    ! [A: set_fm,X3: fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ ( insert_fm2 @ X3 @ B ) )
      = ( ( ( member_fm @ X3 @ B )
         => ( ord_less_set_fm @ A @ B ) )
        & ( ~ ( member_fm @ X3 @ B )
         => ( ( ( member_fm @ X3 @ A )
             => ( ord_less_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X3 @ bot_bot_set_fm ) ) @ B ) )
            & ( ~ ( member_fm @ X3 @ A )
             => ( ord_less_eq_set_fm @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_916_nth__non__equal__first__eq,axiom,
    ! [X3: fm,Y3: fm,Xs: list_fm,N: nat] :
      ( ( X3 != Y3 )
     => ( ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_fm @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_917_nth__Cons__numeral,axiom,
    ! [X3: fm,Xs: list_fm,V: num] :
      ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_fm @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_918_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_919_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_920_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_921_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_922_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_923_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_le_numeral
thf(fact_924_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_925_one__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% one_le_numeral
thf(fact_926_is__singletonI_H,axiom,
    ! [A: set_fm] :
      ( ( A != bot_bot_set_fm )
     => ( ! [X: fm,Y: fm] :
            ( ( member_fm @ X @ A )
           => ( ( member_fm @ Y @ A )
             => ( X = Y ) ) )
       => ( is_singleton_fm @ A ) ) ) ).

% is_singletonI'
thf(fact_927_is__singletonI_H,axiom,
    ! [A: set_Pr8693737435421807431at_nat] :
      ( ( A != bot_bo5327735625951526323at_nat )
     => ( ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A )
           => ( ( member8206827879206165904at_nat @ Y @ A )
             => ( X = Y ) ) )
       => ( is_sin3461952106595595099at_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_928_nth__equal__first__eq,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,N: nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( ( nth_Pr6744343527793145070at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_929_nth__equal__first__eq,axiom,
    ! [X3: fm,Xs: list_fm,N: nat] :
      ( ~ ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Xs ) )
       => ( ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_930_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X3 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X3 @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X3 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa @ ( suc @ X3 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_931_size__neq__size__imp__neq,axiom,
    ! [X3: rule,Y3: rule] :
      ( ( ( size_size_rule @ X3 )
       != ( size_size_rule @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_932_size__neq__size__imp__neq,axiom,
    ! [X3: fm,Y3: fm] :
      ( ( ( size_size_fm @ X3 )
       != ( size_size_fm @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_933_length__Cons,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( size_size_list_fm @ ( cons_fm @ X3 @ Xs ) )
      = ( suc @ ( size_size_list_fm @ Xs ) ) ) ).

% length_Cons
thf(fact_934_length__Suc__conv,axiom,
    ! [Xs: list_fm,N: nat] :
      ( ( ( size_size_list_fm @ Xs )
        = ( suc @ N ) )
      = ( ? [Y5: fm,Ys3: list_fm] :
            ( ( Xs
              = ( cons_fm @ Y5 @ Ys3 ) )
            & ( ( size_size_list_fm @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_935_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_fm] :
      ( ( ( suc @ N )
        = ( size_size_list_fm @ Xs ) )
      = ( ? [Y5: fm,Ys3: list_fm] :
            ( ( Xs
              = ( cons_fm @ Y5 @ Ys3 ) )
            & ( ( size_size_list_fm @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_936_impossible__Cons,axiom,
    ! [Xs: list_fm,Ys2: list_fm,X3: fm] :
      ( ( ord_less_eq_nat @ ( size_size_list_fm @ Xs ) @ ( size_size_list_fm @ Ys2 ) )
     => ( Xs
       != ( cons_fm @ X3 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_937_replicate__eqI,axiom,
    ! [Xs: list_P8469869581646625389at_nat,N: nat,X3: produc859450856879609959at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
        = N )
     => ( ! [Y: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( Y = X3 ) )
       => ( Xs
          = ( replic6713244433751818279at_nat @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_938_replicate__eqI,axiom,
    ! [Xs: list_fm,N: nat,X3: fm] :
      ( ( ( size_size_list_fm @ Xs )
        = N )
     => ( ! [Y: fm] :
            ( ( member_fm @ Y @ ( set_fm2 @ Xs ) )
           => ( Y = X3 ) )
       => ( Xs
          = ( replicate_fm @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_939_replicate__length__same,axiom,
    ! [Xs: list_fm,X3: fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ ( set_fm2 @ Xs ) )
         => ( X = X3 ) )
     => ( ( replicate_fm @ ( size_size_list_fm @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_940_length__removeAll__less__eq,axiom,
    ! [X3: fm,Xs: list_fm] : ( ord_less_eq_nat @ ( size_size_list_fm @ ( removeAll_fm @ X3 @ Xs ) ) @ ( size_size_list_fm @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_941_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_fm] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_fm @ Xs ) )
      = ( ? [X4: fm,Ys3: list_fm] :
            ( ( Xs
              = ( cons_fm @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_942_length__pos__if__in__set,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_943_length__pos__if__in__set,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_944_all__set__conv__all__nth,axiom,
    ! [Xs: list_fm,P2: fm > $o] :
      ( ( ! [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
           => ( P2 @ ( nth_fm @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_945_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P2: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs ) )
         => ( P2 @ ( nth_Pr6744343527793145070at_nat @ Xs @ I2 ) ) )
     => ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_946_all__nth__imp__all__set,axiom,
    ! [Xs: list_fm,P2: fm > $o,X3: fm] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
         => ( P2 @ ( nth_fm @ Xs @ I2 ) ) )
     => ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_947_in__set__conv__nth,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( ( nth_Pr6744343527793145070at_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_948_in__set__conv__nth,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
            & ( ( nth_fm @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_949_list__ball__nth,axiom,
    ! [N: nat,Xs: list_fm,P2: fm > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ ( set_fm2 @ Xs ) )
           => ( P2 @ X ) )
       => ( P2 @ ( nth_fm @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_950_nth__mem,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_951_nth__mem,axiom,
    ! [N: nat,Xs: list_fm] :
      ( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
     => ( member_fm @ ( nth_fm @ Xs @ N ) @ ( set_fm2 @ Xs ) ) ) ).

% nth_mem
thf(fact_952_length__removeAll__less,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ord_less_nat @ ( size_s3679842834875189465at_nat @ ( remove2181804207701385843at_nat @ X3 @ Xs ) ) @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_953_length__removeAll__less,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_fm @ ( removeAll_fm @ X3 @ Xs ) ) @ ( size_size_list_fm @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_954_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_955_Cons__lenlex__iff,axiom,
    ! [M: list_fm,Ms: list_list_fm,N: list_fm,Ns: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ ( cons_list_fm @ M @ Ms ) @ ( cons_list_fm @ N @ Ns ) ) @ ( lenlex_list_fm @ R2 ) )
      = ( ( ord_less_nat @ ( size_s4563186235979089028ist_fm @ Ms ) @ ( size_s4563186235979089028ist_fm @ Ns ) )
        | ( ( ( size_s4563186235979089028ist_fm @ Ms )
            = ( size_s4563186235979089028ist_fm @ Ns ) )
          & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Ms @ Ns ) @ ( lenlex_list_fm @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_956_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_957_Cons__lenlex__iff,axiom,
    ! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
      = ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
        | ( ( ( size_s5460976970255530739at_nat @ Ms )
            = ( size_s5460976970255530739at_nat @ Ns ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_958_Cons__lenlex__iff,axiom,
    ! [M: fm,Ms: list_fm,N: fm,Ns: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ M @ Ms ) @ ( cons_fm @ N @ Ns ) ) @ ( lenlex_fm @ R2 ) )
      = ( ( ord_less_nat @ ( size_size_list_fm @ Ms ) @ ( size_size_list_fm @ Ns ) )
        | ( ( ( size_size_list_fm @ Ms )
            = ( size_size_list_fm @ Ns ) )
          & ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ M @ N ) @ R2 ) )
        | ( ( M = N )
          & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ms @ Ns ) @ ( lenlex_fm @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_959_Cons__in__lex,axiom,
    ! [X3: list_fm,Xs: list_list_fm,Y3: list_fm,Ys2: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ ( cons_list_fm @ X3 @ Xs ) @ ( cons_list_fm @ Y3 @ Ys2 ) ) @ ( lex_list_fm @ R2 ) )
      = ( ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ R2 )
          & ( ( size_s4563186235979089028ist_fm @ Xs )
            = ( size_s4563186235979089028ist_fm @ Ys2 ) ) )
        | ( ( X3 = Y3 )
          & ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Ys2 ) @ ( lex_list_fm @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_960_Cons__in__lex,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys2 ) ) )
        | ( ( X3 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_961_Cons__in__lex,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
          & ( ( size_s5460976970255530739at_nat @ Xs )
            = ( size_s5460976970255530739at_nat @ Ys2 ) ) )
        | ( ( X3 = Y3 )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_962_Cons__in__lex,axiom,
    ! [X3: fm,Xs: list_fm,Y3: fm,Ys2: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ X3 @ Xs ) @ ( cons_fm @ Y3 @ Ys2 ) ) @ ( lex_fm @ R2 ) )
      = ( ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R2 )
          & ( ( size_size_list_fm @ Xs )
            = ( size_size_list_fm @ Ys2 ) ) )
        | ( ( X3 = Y3 )
          & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( lex_fm @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_963_lenlex__irreflexive,axiom,
    ! [R2: set_Pr4463079037648049377_fm_fm,Xs: list_fm] :
      ( ! [X: fm] :
          ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ X ) @ R2 )
     => ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Xs ) @ ( lenlex_fm @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_964_lenlex__irreflexive,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,Xs: list_list_fm] :
      ( ! [X: list_fm] :
          ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ X ) @ R2 )
     => ~ ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Xs ) @ ( lenlex_list_fm @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_965_lenlex__irreflexive,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_966_lenlex__irreflexive,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ).

% lenlex_irreflexive
thf(fact_967_rule_Osize_I14_J,axiom,
    ! [X61: fm,X62: fm] :
      ( ( size_size_rule @ ( impR @ X61 @ X62 ) )
      = zero_zero_nat ) ).

% rule.size(14)
thf(fact_968_lexord__lex,axiom,
    ! [X3: list_fm,Y3: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( lex_fm @ R2 ) )
      = ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( lexord_fm @ R2 ) )
        & ( ( size_size_list_fm @ X3 )
          = ( size_size_list_fm @ Y3 ) ) ) ) ).

% lexord_lex
thf(fact_969_lenlex__length,axiom,
    ! [Ms: list_fm,Ns: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ms @ Ns ) @ ( lenlex_fm @ R2 ) )
     => ( ord_less_eq_nat @ ( size_size_list_fm @ Ms ) @ ( size_size_list_fm @ Ns ) ) ) ).

% lenlex_length
thf(fact_970_listrel__iff__nth,axiom,
    ! [Xs: list_list_fm,Ys2: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Ys2 ) @ ( listre6567353182834750131ist_fm @ R2 ) )
      = ( ( ( size_s4563186235979089028ist_fm @ Xs )
          = ( size_s4563186235979089028ist_fm @ Ys2 ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s4563186235979089028ist_fm @ Xs ) )
           => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( nth_list_fm @ Xs @ N4 ) @ ( nth_list_fm @ Ys2 @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_971_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys2 ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ ( nth_nat @ Ys2 @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_972_listrel__iff__nth,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre818007680106770737at_nat @ R2 ) )
      = ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys2 ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N4 ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_973_listrel__iff__nth,axiom,
    ! [Xs: list_fm,Ys2: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( listrel_fm_fm @ R2 ) )
      = ( ( ( size_size_list_fm @ Xs )
          = ( size_size_list_fm @ Ys2 ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_fm @ Xs ) )
           => ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( nth_fm @ Xs @ N4 ) @ ( nth_fm @ Ys2 @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_974_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_975_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_976_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_977_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_978_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_979_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_980_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_981_diff__diff__left,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_982_sdrop__add,axiom,
    ! [N: nat,M: nat,S: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( sdrop_7224736112439592940m_rule @ M @ S ) )
      = ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ M @ N ) @ S ) ) ).

% sdrop_add
thf(fact_983_sdrop__add,axiom,
    ! [N: nat,M: nat,S: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( sdrop_4442373711808556042ist_fm @ M @ S ) )
      = ( sdrop_4442373711808556042ist_fm @ ( plus_plus_nat @ M @ N ) @ S ) ) ).

% sdrop_add
thf(fact_984_sdrop__add,axiom,
    ! [N: nat,M: nat,S: stream_rule] :
      ( ( sdrop_rule @ N @ ( sdrop_rule @ M @ S ) )
      = ( sdrop_rule @ ( plus_plus_nat @ M @ N ) @ S ) ) ).

% sdrop_add
thf(fact_985_sdrop__add,axiom,
    ! [N: nat,M: nat,S: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( sdrop_7373388980841566196m_rule @ M @ S ) )
      = ( sdrop_7373388980841566196m_rule @ ( plus_plus_nat @ M @ N ) @ S ) ) ).

% sdrop_add
thf(fact_986_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_987_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_988_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_989_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_990_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_991_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_992_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_993_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_994_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_995_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_996_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_997_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_998_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_999_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: extended_enat,J2: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J2 )
        & ( K = L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1000_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1001_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: extended_enat,J2: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J2 )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1002_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1003_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: extended_enat,J2: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J2 )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1004_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_1005_add__mono,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat,D2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ D2 )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ C ) @ ( plus_p3455044024723400733d_enat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_1006_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_1007_add__left__mono,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A2 ) @ ( plus_p3455044024723400733d_enat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_1008_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ~ ! [C3: nat] :
            ( B2
           != ( plus_plus_nat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_1009_less__eqE,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ~ ! [C3: extended_enat] :
            ( B2
           != ( plus_p3455044024723400733d_enat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_1010_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_1011_add__right__mono,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ C ) @ ( plus_p3455044024723400733d_enat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_1012_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C5: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_1013_le__iff__add,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
        ? [C5: extended_enat] :
          ( B3
          = ( plus_p3455044024723400733d_enat @ A3 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_1014_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_1015_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_1016_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1017_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_le_add2
thf(fact_1018_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_le_add1
thf(fact_1019_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_le_mono1
thf(fact_1020_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_1021_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_1022_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_1023_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_1024_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_1025_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_1026_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_1027_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1028_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1029_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,A2: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( suc @ A )
        = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).

% nat_arith.suc1
thf(fact_1030_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1031_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1032_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1033_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1034_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_1035_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_1036_not__add__less1,axiom,
    ! [I: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).

% not_add_less1
thf(fact_1037_not__add__less2,axiom,
    ! [J2: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).

% not_add_less2
thf(fact_1038_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_less_mono1
thf(fact_1039_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_less_add1
thf(fact_1040_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_less_add2
thf(fact_1041_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_1042_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1043_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1044_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_1045_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1046_add__nonpos__eq__0__iff,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ Y3 @ zero_z5237406670263579293d_enat )
       => ( ( ( plus_p3455044024723400733d_enat @ X3 @ Y3 )
            = zero_z5237406670263579293d_enat )
          = ( ( X3 = zero_z5237406670263579293d_enat )
            & ( Y3 = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1047_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1048_add__nonneg__eq__0__iff,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X3 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y3 )
       => ( ( ( plus_p3455044024723400733d_enat @ X3 @ Y3 )
            = zero_z5237406670263579293d_enat )
          = ( ( X3 = zero_z5237406670263579293d_enat )
            & ( Y3 = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1049_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1050_add__nonpos__nonpos,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1051_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1052_add__nonneg__nonneg,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1053_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1054_add__increasing2,axiom,
    ! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
       => ( ord_le2932123472753598470d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1055_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1056_add__decreasing2,axiom,
    ! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_1057_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_1058_add__increasing,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ C )
       => ( ord_le2932123472753598470d_enat @ B2 @ ( plus_p3455044024723400733d_enat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_1059_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1060_add__decreasing,axiom,
    ! [A2: extended_enat,C: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ C @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_1061_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1062_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1063_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1064_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1065_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1066_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1067_diff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% diff_add
thf(fact_1068_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_1069_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1070_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1071_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1072_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1073_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1074_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1075_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1076_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A2 )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1077_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1078_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1079_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1080_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1081_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1082_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1083_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1084_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_1085_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1086_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1087_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).

% less_diff_conv
thf(fact_1088_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1089_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ( minus_minus_nat @ J2 @ I )
          = K )
        = ( J2
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1090_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1091_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1092_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1093_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
      = ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1094_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1095_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1096_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1097_listrel__eq__len,axiom,
    ! [Xs: list_fm,Ys2: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( listrel_fm_fm @ R2 ) )
     => ( ( size_size_list_fm @ Xs )
        = ( size_size_list_fm @ Ys2 ) ) ) ).

% listrel_eq_len
thf(fact_1098_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1099_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1100_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1101_add__pos__nonneg,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_1102_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1103_add__nonpos__neg,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_neg
thf(fact_1104_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1105_add__nonneg__pos,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A2 )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B2 )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_1106_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1107_add__neg__nonpos,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B2 ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_nonpos
thf(fact_1108_nat__diff__split,axiom,
    ! [P2: nat > $o,A2: nat,B2: nat] :
      ( ( P2 @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ( ( ord_less_nat @ A2 @ B2 )
         => ( P2 @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A2
              = ( plus_plus_nat @ B2 @ D4 ) )
           => ( P2 @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_1109_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A2: nat,B2: nat] :
      ( ( P2 @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B2 @ D4 ) )
                & ~ ( P2 @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1110_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1111_fm_Osize_I7_J,axiom,
    ! [X31: fm,X32: fm] :
      ( ( size_size_fm @ ( imp @ X31 @ X32 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X31 ) @ ( size_size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% fm.size(7)
thf(fact_1112_listrel_OCons,axiom,
    ! [X3: fm,Y3: fm,R2: set_Pr4463079037648049377_fm_fm,Xs: list_fm,Ys2: list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R2 )
     => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( listrel_fm_fm @ R2 ) )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ X3 @ Xs ) @ ( cons_fm @ Y3 @ Ys2 ) ) @ ( listrel_fm_fm @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1113_listrel_OCons,axiom,
    ! [X3: list_fm,Y3: list_fm,R2: set_Pr7058068377845519745ist_fm,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ R2 )
     => ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Ys2 ) @ ( listre6567353182834750131ist_fm @ R2 ) )
       => ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ ( cons_list_fm @ X3 @ Xs ) @ ( cons_list_fm @ Y3 @ Ys2 ) ) @ ( listre6567353182834750131ist_fm @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1114_listrel_OCons,axiom,
    ! [X3: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1115_listrel_OCons,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre818007680106770737at_nat @ R2 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1116_listrel__Cons1,axiom,
    ! [Y3: list_fm,Ys2: list_list_fm,Xs: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ ( cons_list_fm @ Y3 @ Ys2 ) @ Xs ) @ ( listre6567353182834750131ist_fm @ R2 ) )
     => ~ ! [Y: list_fm,Ys: list_list_fm] :
            ( ( Xs
              = ( cons_list_fm @ Y @ Ys ) )
           => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Y3 @ Y ) @ R2 )
             => ~ ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Ys2 @ Ys ) @ ( listre6567353182834750131ist_fm @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1117_listrel__Cons1,axiom,
    ! [Y3: nat,Ys2: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [Y: nat,Ys: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y @ Ys ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1118_listrel__Cons1,axiom,
    ! [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) @ Xs ) @ ( listre818007680106770737at_nat @ R2 ) )
     => ~ ! [Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ Y @ Ys ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Y ) @ R2 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1119_listrel__Cons1,axiom,
    ! [Y3: fm,Ys2: list_fm,Xs: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( cons_fm @ Y3 @ Ys2 ) @ Xs ) @ ( listrel_fm_fm @ R2 ) )
     => ~ ! [Y: fm,Ys: list_fm] :
            ( ( Xs
              = ( cons_fm @ Y @ Ys ) )
           => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ Y3 @ Y ) @ R2 )
             => ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Ys2 @ Ys ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1120_listrel__Cons2,axiom,
    ! [Xs: list_list_fm,Y3: list_fm,Ys2: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ ( cons_list_fm @ Y3 @ Ys2 ) ) @ ( listre6567353182834750131ist_fm @ R2 ) )
     => ~ ! [X: list_fm,Xs2: list_list_fm] :
            ( ( Xs
              = ( cons_list_fm @ X @ Xs2 ) )
           => ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ R2 )
             => ~ ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs2 @ Ys2 ) @ ( listre6567353182834750131ist_fm @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1121_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel_nat_nat @ R2 ) )
     => ~ ! [X: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1122_listrel__Cons2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) @ ( listre818007680106770737at_nat @ R2 ) )
     => ~ ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R2 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1123_listrel__Cons2,axiom,
    ! [Xs: list_fm,Y3: fm,Ys2: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ ( cons_fm @ Y3 @ Ys2 ) ) @ ( listrel_fm_fm @ R2 ) )
     => ~ ! [X: fm,Xs2: list_fm] :
            ( ( Xs
              = ( cons_fm @ X @ Xs2 ) )
           => ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R2 )
             => ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs2 @ Ys2 ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1124_list_Osize_I4_J,axiom,
    ! [X21: fm,X22: list_fm] :
      ( ( size_size_list_fm @ ( cons_fm @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_fm @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_1125_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_1126_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1127_size__prod__simp,axiom,
    ( basic_7103594793270586102ist_fm
    = ( ^ [F2: list_fm > nat,G: list_fm > nat,P4: produc1996495991257130529ist_fm] : ( plus_plus_nat @ ( plus_plus_nat @ ( F2 @ ( produc1501393135466168645ist_fm @ P4 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ P4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_1128_size__prod__simp,axiom,
    ( basic_7266071961737426738m_rule
    = ( ^ [F2: produc1996495991257130529ist_fm > nat,G: rule > nat,P4: produc164195504107695125m_rule] : ( plus_plus_nat @ ( plus_plus_nat @ ( F2 @ ( produc6879501374131015971m_rule @ P4 ) ) @ ( G @ ( produc7165828336582415457m_rule @ P4 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_1129_fm_Osize__gen_I3_J,axiom,
    ! [X31: fm,X32: fm] :
      ( ( size_fm @ ( imp @ X31 @ X32 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X31 ) @ ( size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% fm.size_gen(3)
thf(fact_1130_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_1131_nth__zip,axiom,
    ! [I: nat,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( ord_less_nat @ I @ ( size_s4563186235979089028ist_fm @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s4563186235979089028ist_fm @ Ys2 ) )
       => ( ( nth_Pr4548706283066508072ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) @ I )
          = ( produc381145313068854617ist_fm @ ( nth_list_fm @ Xs @ I ) @ ( nth_list_fm @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1132_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1133_nth__zip,axiom,
    ! [I: nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) @ I )
          = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1134_stake__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stake_1447931197033250628m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( zip_Pr2138331976196620527m_rule @ ( stake_2930890243849202354ist_fm @ N @ S1 ) @ ( stake_rule @ N @ S2 ) ) ) ).

% stake_szip
thf(fact_1135_stake__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stake_8908532221384226460m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( zip_Pr1780431989191363299m_rule @ ( stake_1447931197033250628m_rule @ N @ S1 ) @ ( stake_1447931197033250628m_rule @ N @ S2 ) ) ) ).

% stake_szip
thf(fact_1136_zip__Cons__Cons,axiom,
    ! [X3: fm,Xs: list_fm,Y3: fm,Ys2: list_fm] :
      ( ( zip_fm_fm @ ( cons_fm @ X3 @ Xs ) @ ( cons_fm @ Y3 @ Ys2 ) )
      = ( cons_P2486850454927788855_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ ( zip_fm_fm @ Xs @ Ys2 ) ) ) ).

% zip_Cons_Cons
thf(fact_1137_zip__Cons__Cons,axiom,
    ! [X3: list_fm,Xs: list_list_fm,Y3: list_fm,Ys2: list_list_fm] :
      ( ( zip_list_fm_list_fm @ ( cons_list_fm @ X3 @ Xs ) @ ( cons_list_fm @ Y3 @ Ys2 ) )
      = ( cons_P7841146678257726167ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) ) ).

% zip_Cons_Cons
thf(fact_1138_zip__Cons__Cons,axiom,
    ! [X3: nat,Xs: list_nat,Y3: nat,Ys2: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y3 @ Ys2 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ).

% zip_Cons_Cons
thf(fact_1139_zip__Cons__Cons,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) ) ).

% zip_Cons_Cons
thf(fact_1140_zip__same,axiom,
    ! [A2: produc859450856879609959at_nat,B2: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A2 @ B2 ) @ ( set_Pr1322963821424748924at_nat @ ( zip_Pr935030979083031159at_nat @ Xs @ Xs ) ) )
      = ( ( member8206827879206165904at_nat @ A2 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1141_zip__same,axiom,
    ! [A2: fm,B2: fm,Xs: list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ A2 @ B2 ) @ ( set_Pr6624747810323803798_fm_fm @ ( zip_fm_fm @ Xs @ Xs ) ) )
      = ( ( member_fm @ A2 @ ( set_fm2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1142_zip__same,axiom,
    ! [A2: list_fm,B2: list_fm,Xs: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B2 ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Xs ) ) )
      = ( ( member_list_fm @ A2 @ ( set_list_fm2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1143_zip__same,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1144_zip__same,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1145_in__set__zipE,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ ( set_Pr1322963821424748924at_nat @ ( zip_Pr935030979083031159at_nat @ Xs @ Ys2 ) ) )
     => ~ ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ~ ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1146_in__set__zipE,axiom,
    ! [X3: produc859450856879609959at_nat,Y3: fm,Xs: list_P8469869581646625389at_nat,Ys2: list_fm] :
      ( ( member9198537383846477821nat_fm @ ( produc6439020385818810464nat_fm @ X3 @ Y3 ) @ ( set_Pr3075457875423956113nat_fm @ ( zip_Pr9151751282906081600nat_fm @ Xs @ Ys2 ) ) )
     => ~ ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ~ ( member_fm @ Y3 @ ( set_fm2 @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1147_in__set__zipE,axiom,
    ! [X3: fm,Y3: produc859450856879609959at_nat,Xs: list_fm,Ys2: list_P8469869581646625389at_nat] :
      ( ( member6781037255463946659at_nat @ ( produc4700841676774948030at_nat @ X3 @ Y3 ) @ ( set_Pr657957747041424951at_nat @ ( zip_fm7413572573862219166at_nat @ Xs @ Ys2 ) ) )
     => ~ ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
         => ~ ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1148_in__set__zipE,axiom,
    ! [X3: fm,Y3: fm,Xs: list_fm,Ys2: list_fm] :
      ( ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ ( set_Pr6624747810323803798_fm_fm @ ( zip_fm_fm @ Xs @ Ys2 ) ) )
     => ~ ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
         => ~ ( member_fm @ Y3 @ ( set_fm2 @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1149_in__set__zipE,axiom,
    ! [X3: list_fm,Y3: list_fm,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) )
     => ~ ( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
         => ~ ( member_list_fm @ Y3 @ ( set_list_fm2 @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1150_in__set__zipE,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1151_in__set__zipE,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ).

% in_set_zipE
thf(fact_1152_set__zip__leftD,axiom,
    ! [X3: list_fm,Y3: list_fm,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) )
     => ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1153_set__zip__leftD,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1154_set__zip__leftD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1155_set__zip__rightD,axiom,
    ! [X3: list_fm,Y3: list_fm,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) )
     => ( member_list_fm @ Y3 @ ( set_list_fm2 @ Ys2 ) ) ) ).

% set_zip_rightD
thf(fact_1156_set__zip__rightD,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) ) ) ).

% set_zip_rightD
thf(fact_1157_set__zip__rightD,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) )
     => ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ).

% set_zip_rightD
thf(fact_1158_zip__eq__ConsE,axiom,
    ! [Xs: list_fm,Ys2: list_fm,Xy: product_prod_fm_fm,Xys: list_P6934817935912052487_fm_fm] :
      ( ( ( zip_fm_fm @ Xs @ Ys2 )
        = ( cons_P2486850454927788855_fm_fm @ Xy @ Xys ) )
     => ~ ! [X: fm,Xs4: list_fm] :
            ( ( Xs
              = ( cons_fm @ X @ Xs4 ) )
           => ! [Y: fm,Ys4: list_fm] :
                ( ( Ys2
                  = ( cons_fm @ Y @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_fm_fm @ X @ Y ) )
                 => ( Xys
                   != ( zip_fm_fm @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1159_zip__eq__ConsE,axiom,
    ! [Xs: list_list_fm,Ys2: list_list_fm,Xy: produc1996495991257130529ist_fm,Xys: list_P5616295576739893671ist_fm] :
      ( ( ( zip_list_fm_list_fm @ Xs @ Ys2 )
        = ( cons_P7841146678257726167ist_fm @ Xy @ Xys ) )
     => ~ ! [X: list_fm,Xs4: list_list_fm] :
            ( ( Xs
              = ( cons_list_fm @ X @ Xs4 ) )
           => ! [Y: list_fm,Ys4: list_list_fm] :
                ( ( Ys2
                  = ( cons_list_fm @ Y @ Ys4 ) )
               => ( ( Xy
                    = ( produc381145313068854617ist_fm @ X @ Y ) )
                 => ( Xys
                   != ( zip_list_fm_list_fm @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1160_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys2 )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Xs4 ) )
           => ! [Y: nat,Ys4: list_nat] :
                ( ( Ys2
                  = ( cons_nat @ Y @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X @ Y ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1161_zip__eq__ConsE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
      ( ( ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 )
        = ( cons_P8732206157123786781at_nat @ Xy @ Xys ) )
     => ~ ! [X: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X @ Xs4 ) )
           => ! [Y: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
                ( ( Ys2
                  = ( cons_P6512896166579812791at_nat @ Y @ Ys4 ) )
               => ( ( Xy
                    = ( produc6161850002892822231at_nat @ X @ Y ) )
                 => ( Xys
                   != ( zip_Pr4664179122662387191at_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1162_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_list_fm,Ys2: list_list_fm,X3: list_fm] :
      ( ( ( size_s4563186235979089028ist_fm @ Xs )
        = ( size_s4563186235979089028ist_fm @ Ys2 ) )
     => ( ( member_list_fm @ X3 @ ( set_list_fm2 @ Xs ) )
       => ~ ! [Y: list_fm] :
              ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1163_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys2: list_nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1164_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys2 ) )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ~ ! [Y: product_prod_nat_nat] :
              ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1165_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_list_fm,Ys2: list_list_fm,Y3: list_fm] :
      ( ( ( size_s4563186235979089028ist_fm @ Xs )
        = ( size_s4563186235979089028ist_fm @ Ys2 ) )
     => ( ( member_list_fm @ Y3 @ ( set_list_fm2 @ Ys2 ) )
       => ~ ! [X: list_fm] :
              ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1166_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Y3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) )
       => ~ ! [X: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1167_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys2 ) )
     => ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
       => ~ ! [X: product_prod_nat_nat] :
              ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1168_in__set__zip,axiom,
    ! [P: produc859450856879609959at_nat,Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ P @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys2 ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_Pr7617993195940197384at_nat @ Xs @ N4 )
              = ( produc3213797794245857475at_nat @ P ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Ys2 @ N4 )
              = ( produc6408287024330202629at_nat @ P ) )
            & ( ord_less_nat @ N4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_s5460976970255530739at_nat @ Ys2 ) ) ) ) ) ).

% in_set_zip
thf(fact_1169_in__set__zip,axiom,
    ! [P: produc1996495991257130529ist_fm,Xs: list_list_fm,Ys2: list_list_fm] :
      ( ( member8102475879199740618ist_fm @ P @ ( set_Pr8767716839810916150ist_fm @ ( zip_list_fm_list_fm @ Xs @ Ys2 ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_list_fm @ Xs @ N4 )
              = ( produc1501393135466168645ist_fm @ P ) )
            & ( ( nth_list_fm @ Ys2 @ N4 )
              = ( produc4588648349897876871ist_fm @ P ) )
            & ( ord_less_nat @ N4 @ ( size_s4563186235979089028ist_fm @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_s4563186235979089028ist_fm @ Ys2 ) ) ) ) ) ).

% in_set_zip
thf(fact_1170_in__set__zip,axiom,
    ! [P: produc164195504107695125m_rule,Xs: list_P5616295576739893671ist_fm,Ys2: list_rule] :
      ( ( member4220325220686508332m_rule @ P @ ( set_Pr8505323785428441536m_rule @ ( zip_Pr2138331976196620527m_rule @ Xs @ Ys2 ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_Pr4548706283066508072ist_fm @ Xs @ N4 )
              = ( produc6879501374131015971m_rule @ P ) )
            & ( ( nth_rule @ Ys2 @ N4 )
              = ( produc7165828336582415457m_rule @ P ) )
            & ( ord_less_nat @ N4 @ ( size_s2722402132374190611ist_fm @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_rule @ Ys2 ) ) ) ) ) ).

% in_set_zip
thf(fact_1171_lex__take__index,axiom,
    ! [Xs: list_list_fm,Ys2: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ Xs @ Ys2 ) @ ( lex_list_fm @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s4563186235979089028ist_fm @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s4563186235979089028ist_fm @ Ys2 ) )
             => ( ( ( take_list_fm @ I2 @ Xs )
                  = ( take_list_fm @ I2 @ Ys2 ) )
               => ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( nth_list_fm @ Xs @ I2 ) @ ( nth_list_fm @ Ys2 @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1172_lex__take__index,axiom,
    ! [Xs: list_nat,Ys2: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
             => ( ( ( take_nat @ I2 @ Xs )
                  = ( take_nat @ I2 @ Ys2 ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1173_lex__take__index,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Ys2 ) )
             => ( ( ( take_P2173866234530122223at_nat @ I2 @ Xs )
                  = ( take_P2173866234530122223at_nat @ I2 @ Ys2 ) )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1174_lex__take__index,axiom,
    ! [Xs: list_fm,Ys2: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ Xs @ Ys2 ) @ ( lex_fm @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Ys2 ) )
             => ( ( ( take_fm @ I2 @ Xs )
                  = ( take_fm @ I2 @ Ys2 ) )
               => ~ ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( nth_fm @ Xs @ I2 ) @ ( nth_fm @ Ys2 @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_1175_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1176_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1177_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_1178_take__Suc__Cons,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( take_fm @ ( suc @ N ) @ ( cons_fm @ X3 @ Xs ) )
      = ( cons_fm @ X3 @ ( take_fm @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1179_take__Cons__numeral,axiom,
    ! [V: num,X3: fm,Xs: list_fm] :
      ( ( take_fm @ ( numeral_numeral_nat @ V ) @ ( cons_fm @ X3 @ Xs ) )
      = ( cons_fm @ X3 @ ( take_fm @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).

% take_Cons_numeral
thf(fact_1180_in__set__takeD,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1181_in__set__takeD,axiom,
    ! [X3: fm,N: nat,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) )
     => ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1182_set__take__subset,axiom,
    ! [N: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).

% set_take_subset
thf(fact_1183_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_fm] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ M @ Xs ) ) @ ( set_fm2 @ ( take_fm @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1184_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1185_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_1186_lexord__take__index__conv,axiom,
    ! [X3: list_list_fm,Y3: list_list_fm,R2: set_Pr7058068377845519745ist_fm] :
      ( ( member197921522497012330ist_fm @ ( produc8668676954508811513ist_fm @ X3 @ Y3 ) @ ( lexord_list_fm @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_s4563186235979089028ist_fm @ X3 ) @ ( size_s4563186235979089028ist_fm @ Y3 ) )
          & ( ( take_list_fm @ ( size_s4563186235979089028ist_fm @ X3 ) @ Y3 )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_s4563186235979089028ist_fm @ X3 ) @ ( size_s4563186235979089028ist_fm @ Y3 ) ) )
            & ( ( take_list_fm @ I3 @ X3 )
              = ( take_list_fm @ I3 @ Y3 ) )
            & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ ( nth_list_fm @ X3 @ I3 ) @ ( nth_list_fm @ Y3 @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1187_lexord__take__index__conv,axiom,
    ! [X3: list_nat,Y3: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ ( lexord_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y3 ) )
          & ( ( take_nat @ ( size_size_list_nat @ X3 ) @ Y3 )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y3 ) ) )
            & ( ( take_nat @ I3 @ X3 )
              = ( take_nat @ I3 @ Y3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X3 @ I3 ) @ ( nth_nat @ Y3 @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1188_lexord__take__index__conv,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y3 ) @ ( lexord2841853652668343668at_nat @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y3 ) )
          & ( ( take_P2173866234530122223at_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ Y3 )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y3 ) ) )
            & ( ( take_P2173866234530122223at_nat @ I3 @ X3 )
              = ( take_P2173866234530122223at_nat @ I3 @ Y3 ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ X3 @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Y3 @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1189_lexord__take__index__conv,axiom,
    ! [X3: list_fm,Y3: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( lexord_fm @ R2 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_fm @ X3 ) @ ( size_size_list_fm @ Y3 ) )
          & ( ( take_fm @ ( size_size_list_fm @ X3 ) @ Y3 )
            = X3 ) )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_fm @ X3 ) @ ( size_size_list_fm @ Y3 ) ) )
            & ( ( take_fm @ I3 @ X3 )
              = ( take_fm @ I3 @ Y3 ) )
            & ( member8474499337054950954_fm_fm @ ( product_Pair_fm_fm @ ( nth_fm @ X3 @ I3 ) @ ( nth_fm @ Y3 @ I3 ) ) @ R2 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1190_min__bot,axiom,
    ! [X3: fset_P661503646757059847ist_fm] :
      ( ( ord_mi3151844185881495662ist_fm @ bot_bo2367426573206113139ist_fm @ X3 )
      = bot_bo2367426573206113139ist_fm ) ).

% min_bot
thf(fact_1191_min__bot,axiom,
    ! [X3: nat] :
      ( ( ord_min_nat @ bot_bot_nat @ X3 )
      = bot_bot_nat ) ).

% min_bot
thf(fact_1192_min__bot2,axiom,
    ! [X3: fset_P661503646757059847ist_fm] :
      ( ( ord_mi3151844185881495662ist_fm @ X3 @ bot_bo2367426573206113139ist_fm )
      = bot_bo2367426573206113139ist_fm ) ).

% min_bot2
thf(fact_1193_min__bot2,axiom,
    ! [X3: nat] :
      ( ( ord_min_nat @ X3 @ bot_bot_nat )
      = bot_bot_nat ) ).

% min_bot2
thf(fact_1194_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_1195_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_1196_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_1197_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_1198_min__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_mi8085742599997312461d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_mi8085742599997312461d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% min_number_of(1)
thf(fact_1199_zip__replicate,axiom,
    ! [I: nat,X3: list_fm,J2: nat,Y3: list_fm] :
      ( ( zip_list_fm_list_fm @ ( replicate_list_fm @ I @ X3 ) @ ( replicate_list_fm @ J2 @ Y3 ) )
      = ( replic4812330464126889441ist_fm @ ( ord_min_nat @ I @ J2 ) @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) ) ).

% zip_replicate
thf(fact_1200_zip__replicate,axiom,
    ! [I: nat,X3: nat,J2: nat,Y3: nat] :
      ( ( zip_nat_nat @ ( replicate_nat @ I @ X3 ) @ ( replicate_nat @ J2 @ Y3 ) )
      = ( replic4235873036481779905at_nat @ ( ord_min_nat @ I @ J2 ) @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).

% zip_replicate
thf(fact_1201_zip__replicate,axiom,
    ! [I: nat,X3: product_prod_nat_nat,J2: nat,Y3: product_prod_nat_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( replic4235873036481779905at_nat @ I @ X3 ) @ ( replic4235873036481779905at_nat @ J2 @ Y3 ) )
      = ( replic6713244433751818279at_nat @ ( ord_min_nat @ I @ J2 ) @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ).

% zip_replicate
thf(fact_1202_in__set__butlastD,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( butlas8493624449043409337at_nat @ Xs ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1203_in__set__butlastD,axiom,
    ! [X3: fm,Xs: list_fm] :
      ( ( member_fm @ X3 @ ( set_fm2 @ ( butlast_fm @ Xs ) ) )
     => ( member_fm @ X3 @ ( set_fm2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1204_min__def,axiom,
    ( ord_min_set_fm
    = ( ^ [A3: set_fm,B3: set_fm] : ( if_set_fm @ ( ord_less_eq_set_fm @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_1205_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_1206_min__def,axiom,
    ( ord_mi8085742599997312461d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_1207_min__absorb1,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y3 )
     => ( ( ord_min_set_fm @ X3 @ Y3 )
        = X3 ) ) ).

% min_absorb1
thf(fact_1208_min__absorb1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_min_nat @ X3 @ Y3 )
        = X3 ) ) ).

% min_absorb1
thf(fact_1209_min__absorb1,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_mi8085742599997312461d_enat @ X3 @ Y3 )
        = X3 ) ) ).

% min_absorb1
thf(fact_1210_min__absorb2,axiom,
    ! [Y3: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y3 @ X3 )
     => ( ( ord_min_set_fm @ X3 @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_1211_min__absorb2,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_min_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_1212_min__absorb2,axiom,
    ! [Y3: extended_enat,X3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
     => ( ( ord_mi8085742599997312461d_enat @ X3 @ Y3 )
        = Y3 ) ) ).

% min_absorb2
thf(fact_1213_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_1214_min_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% min.bounded_iff
thf(fact_1215_min_Obounded__iff,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( ord_mi8085742599997312461d_enat @ B2 @ C ) )
      = ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
        & ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% min.bounded_iff
thf(fact_1216_min_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_1217_min_Oabsorb2,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( ord_mi8085742599997312461d_enat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_1218_min_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_1219_min_Oabsorb1,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_mi8085742599997312461d_enat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_1220_min__le__iff__disj,axiom,
    ! [X3: nat,Y3: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X3 @ Y3 ) @ Z3 )
      = ( ( ord_less_eq_nat @ X3 @ Z3 )
        | ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ).

% min_le_iff_disj
thf(fact_1221_min__le__iff__disj,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ X3 @ Y3 ) @ Z3 )
      = ( ( ord_le2932123472753598470d_enat @ X3 @ Z3 )
        | ( ord_le2932123472753598470d_enat @ Y3 @ Z3 ) ) ) ).

% min_le_iff_disj
thf(fact_1222_min_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI2
thf(fact_1223_min_OcoboundedI2,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ C )
     => ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI2
thf(fact_1224_min_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI1
thf(fact_1225_min_OcoboundedI1,axiom,
    ! [A2: extended_enat,C: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ C )
     => ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) @ C ) ) ).

% min.coboundedI1
thf(fact_1226_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_1227_min_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_mi8085742599997312461d_enat @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_1228_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_1229_min_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_mi8085742599997312461d_enat @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_1230_min_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_1231_min_Ocobounded2,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_1232_min_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_1233_min_Ocobounded1,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_1234_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( ord_min_nat @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_1235_min_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( A3
          = ( ord_mi8085742599997312461d_enat @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_1236_min_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ) ).

% min.boundedI
thf(fact_1237_min_OboundedI,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( ord_le2932123472753598470d_enat @ A2 @ C )
       => ( ord_le2932123472753598470d_enat @ A2 @ ( ord_mi8085742599997312461d_enat @ B2 @ C ) ) ) ) ).

% min.boundedI
thf(fact_1238_min_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% min.boundedE
thf(fact_1239_min_OboundedE,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( ord_mi8085742599997312461d_enat @ B2 @ C ) )
     => ~ ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
         => ~ ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% min.boundedE
thf(fact_1240_min_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( ord_min_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_1241_min_OorderI,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2
        = ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) )
     => ( ord_le2932123472753598470d_enat @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_1242_min_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( ord_min_nat @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_1243_min_OorderE,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( A2
        = ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_1244_min_Omono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).

% min.mono
thf(fact_1245_min_Omono,axiom,
    ! [A2: extended_enat,C: extended_enat,B2: extended_enat,D2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ C )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ D2 )
       => ( ord_le2932123472753598470d_enat @ ( ord_mi8085742599997312461d_enat @ A2 @ B2 ) @ ( ord_mi8085742599997312461d_enat @ C @ D2 ) ) ) ) ).

% min.mono
thf(fact_1246_fMin_Osemilattice__order__fset__axioms,axiom,
    semila6541227223090250811et_nat @ ord_min_nat @ ord_less_eq_nat @ ord_less_nat ).

% fMin.semilattice_order_fset_axioms
thf(fact_1247_fMin_Osemilattice__order__fset__axioms,axiom,
    semila7129917228889371579d_enat @ ord_mi8085742599997312461d_enat @ ord_le2932123472753598470d_enat @ ord_le72135733267957522d_enat ).

% fMin.semilattice_order_fset_axioms
thf(fact_1248_fMin_Oinsert,axiom,
    ! [A: fset_nat,X3: nat] :
      ( ( A != bot_bot_fset_nat )
     => ( ( linorder_fMin_nat @ ( finsert_nat @ X3 @ A ) )
        = ( ord_min_nat @ X3 @ ( linorder_fMin_nat @ A ) ) ) ) ).

% fMin.insert
thf(fact_1249_fMin__finsert,axiom,
    ! [A: fset_nat,X3: nat] :
      ( ( ( A = bot_bot_fset_nat )
       => ( ( linorder_fMin_nat @ ( finsert_nat @ X3 @ A ) )
          = X3 ) )
      & ( ( A != bot_bot_fset_nat )
       => ( ( linorder_fMin_nat @ ( finsert_nat @ X3 @ A ) )
          = ( ord_min_nat @ X3 @ ( linorder_fMin_nat @ A ) ) ) ) ) ).

% fMin_finsert
thf(fact_1250_fMin__le,axiom,
    ! [X3: nat,A: fset_nat] :
      ( ( fmember_nat @ X3 @ A )
     => ( ord_less_eq_nat @ ( linorder_fMin_nat @ A ) @ X3 ) ) ).

% fMin_le
thf(fact_1251_fMin__le,axiom,
    ! [X3: extended_enat,A: fset_Extended_enat] :
      ( ( fmembe9107328845036146477d_enat @ X3 @ A )
     => ( ord_le2932123472753598470d_enat @ ( linord1867481200929749078d_enat @ A ) @ X3 ) ) ).

% fMin_le
thf(fact_1252_fMin__eqI,axiom,
    ! [A: fset_nat,X3: nat] :
      ( ! [Y: nat] :
          ( ( fmember_nat @ Y @ A )
         => ( ord_less_eq_nat @ X3 @ Y ) )
     => ( ( fmember_nat @ X3 @ A )
       => ( ( linorder_fMin_nat @ A )
          = X3 ) ) ) ).

% fMin_eqI
thf(fact_1253_fMin__eqI,axiom,
    ! [A: fset_Extended_enat,X3: extended_enat] :
      ( ! [Y: extended_enat] :
          ( ( fmembe9107328845036146477d_enat @ Y @ A )
         => ( ord_le2932123472753598470d_enat @ X3 @ Y ) )
     => ( ( fmembe9107328845036146477d_enat @ X3 @ A )
       => ( ( linord1867481200929749078d_enat @ A )
          = X3 ) ) ) ).

% fMin_eqI
thf(fact_1254_fMin_Oin__idem,axiom,
    ! [X3: nat,A: fset_nat] :
      ( ( fmember_nat @ X3 @ A )
     => ( ( ord_min_nat @ X3 @ ( linorder_fMin_nat @ A ) )
        = ( linorder_fMin_nat @ A ) ) ) ).

% fMin.in_idem
thf(fact_1255_fMin_Oinsert__not__elem,axiom,
    ! [X3: nat,A: fset_nat] :
      ( ~ ( fmember_nat @ X3 @ A )
     => ( ( A != bot_bot_fset_nat )
       => ( ( linorder_fMin_nat @ ( finsert_nat @ X3 @ A ) )
          = ( ord_min_nat @ X3 @ ( linorder_fMin_nat @ A ) ) ) ) ) ).

% fMin.insert_not_elem
thf(fact_1256_take__Cons_H,axiom,
    ! [N: nat,X3: fm,Xs: list_fm] :
      ( ( ( N = zero_zero_nat )
       => ( ( take_fm @ N @ ( cons_fm @ X3 @ Xs ) )
          = nil_fm ) )
      & ( ( N != zero_zero_nat )
       => ( ( take_fm @ N @ ( cons_fm @ X3 @ Xs ) )
          = ( cons_fm @ X3 @ ( take_fm @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).

% take_Cons'
thf(fact_1257_set__rotate1,axiom,
    ! [Xs: list_fm] :
      ( ( set_fm2 @ ( rotate1_fm @ Xs ) )
      = ( set_fm2 @ Xs ) ) ).

% set_rotate1
thf(fact_1258_in__measures_I1_J,axiom,
    ! [X3: list_fm,Y3: list_fm] :
      ~ ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ ( measures_list_fm @ nil_list_fm_nat ) ) ).

% in_measures(1)
thf(fact_1259_in__measures_I1_J,axiom,
    ! [X3: nat,Y3: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_1260_in__measures_I1_J,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( measur2679027848233739777at_nat @ nil_Pr2865493887535707976at_nat ) ) ).

% in_measures(1)
thf(fact_1261_set__empty2,axiom,
    ! [Xs: list_fm] :
      ( ( bot_bot_set_fm
        = ( set_fm2 @ Xs ) )
      = ( Xs = nil_fm ) ) ).

% set_empty2
thf(fact_1262_set__empty,axiom,
    ! [Xs: list_fm] :
      ( ( ( set_fm2 @ Xs )
        = bot_bot_set_fm )
      = ( Xs = nil_fm ) ) ).

% set_empty
thf(fact_1263_Nil__lenlex__iff1,axiom,
    ! [Ns: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ nil_fm @ Ns ) @ ( lenlex_fm @ R2 ) )
      = ( Ns != nil_fm ) ) ).

% Nil_lenlex_iff1
thf(fact_1264_fset__of__list__simps_I1_J,axiom,
    ( ( fset_o3706400737857578983ist_fm @ nil_Pr6600767949821390631ist_fm )
    = bot_bo2367426573206113139ist_fm ) ).

% fset_of_list_simps(1)
thf(fact_1265_insert__Nil,axiom,
    ! [X3: fm] :
      ( ( insert_fm @ X3 @ nil_fm )
      = ( cons_fm @ X3 @ nil_fm ) ) ).

% insert_Nil
thf(fact_1266_lexord__Nil__left,axiom,
    ! [Y3: list_fm,R2: set_Pr4463079037648049377_fm_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ nil_fm @ Y3 ) @ ( lexord_fm @ R2 ) )
      = ( ? [A3: fm,X4: list_fm] :
            ( Y3
            = ( cons_fm @ A3 @ X4 ) ) ) ) ).

% lexord_Nil_left

% Helper facts (19)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Syntax__Ofm_J_T,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( if_set_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Syntax__Ofm_J_T,axiom,
    ! [X3: set_fm,Y3: set_fm] :
      ( ( if_set_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
    ! [X3: list_fm,Y3: list_fm] :
      ( ( if_list_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Syntax__Ofm_J_T,axiom,
    ! [X3: list_fm,Y3: list_fm] :
      ( ( if_list_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X3: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: list_P8469869581646625389at_nat,Y3: list_P8469869581646625389at_nat] :
      ( ( if_lis7763640049307703347at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: list_P8469869581646625389at_nat,Y3: list_P8469869581646625389at_nat] :
      ( ( if_lis7763640049307703347at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X3: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( member_fm @ p @ ( treeA @ ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) )
    & ( member_fm @ q @ ( treeB @ ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ j @ steps ) ) ) ) ) ).

%------------------------------------------------------------------------------